Contents

1 Introduction 5

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Formulation of boundary value problems for wave-ice interaction 12

2.1 Laplace’s equation of velocity potential and its

boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Boundary conditions for Laplace’s equation . . . . . . . . . . . . . . 15

2.1.3 Natural transition conditions . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.4 Boundary conditions on the surface . . . . . . . . . . . . . . . . . . . 17

2.2 Thin plate equation and its boundary conditions . . . . . . . . . . . . . . . . 19

2.2.1 Thin plate equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Boundary conditions for the plate equation . . . . . . . . . . . . . . . 23

2.2.3 Transition conditions for an elastic plate . . . . . . . . . . . . . . . . 25

2.3 Time-harmonic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Time-harmonic forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.2 Radiation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Harmonic forcing of an infinite floating plate 30

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Mittag-Leffler expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Mittag-Leffler expansion . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.2 Expansion of w (γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Fundamental solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.2 Non-dimensional formulation . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.3 Spatial Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.4 The inverse Fourier transform . . . . . . . . . . . . . . . . . . . . . . 42

1

3.3.5 Modal expansion of the solutions . . . . . . . . . . . . . . . . . . . . 45

3.3.6 Summary of the analytic structure of w (γ) . . . . . . . . . . . . . . . 47

3.4 Deep water solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Computation of the solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.1 Static load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.2 Deflection at the location of forcing . . . . . . . . . . . . . . . . . . . 53

3.5.3 Derivatives of the deflection . . . . . . . . . . . . . . . . . . . . . . . 57

3.6 Scaling of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6.1 Scaled solutions and physical solutions . . . . . . . . . . . . . . . . . 61

3.6.2 General scaling law of a floating ice sheet . . . . . . . . . . . . . . . . 64

3.7 Determining characteristic length from field measurements . . . . . . . . . . 66

3.7.1 Characteristic length . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Wave propagation in semi-infinite floating plates 74

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Methods of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.2 Mode matching by Fox and Squire . . . . . . . . . . . . . . . . . . . 79

4.2.3 Approximation by Wadhams . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 The Wiener-Hopf technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.2 Weierstrass’s factor theorem . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.3 Derivation of the Wiener-Hopf equation . . . . . . . . . . . . . . . . . 83

4.4 Determination of J (α) from the transition conditions . . . . . . . . . . . . . 89

4.4.1 Dock problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.2 Ocean wave and ice sheet . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4.3 Open crack problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4.4 Two semi-infinite ice sheets . . . . . . . . . . . . . . . . . . . . . . . 94

4.5 Reflection and transmission coefficients . . . . . . . . . . . . . . . . . . . . . 94

4.6 Deep water solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.7 Scaled solution for wave-ice interaction . . . . . . . . . . . . . . . . . . . . . 100

4.7.1 Derivation of the Wiener-Hopf Equation . . . . . . . . . . . . . . . . 101

4.7.2 Determination of the solutions . . . . . . . . . . . . . . . . . . . . . . 106

4.7.3 Computation of the reflection and transmission coefficients . . . . . . 109

4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2

5 The Wiener-Hopf technique and Boundary integral equations 114

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Formulation of BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.3 Semi-infinite plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.4 Finite plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Conclusions and review 128

A Integrals and Special functions 132

A.1 Calculations of bending and shear . . . . . . . . . . . . . . . . . . . . . . . . 132

A.2 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B Deep water solution 135

C Relationship between R and T 137

3

Preface

This monograph is a result my Ph.D. research which began with a study of the analytic

properties of the dispersion function of a thin elastic floating plate. That function has a

countably infinite number of isolated zeros and singularities on the complex plane. The

purpose of that study was to formerly verify the Mittag-Leffler expansion in section 3.2.2.

However, at the time I did not know anything about plate equations and surface waves on

water. Even Laplace’s equation seemed unfamiliar in the context of surface waves coupled

with the plate equation. It was surprising that from all these complicated partial differential

equations, I was able to derive the solutions using a classical mathematical tool, the Fourier

transform. Furthermore, the inverse Fourier transforms could be calculated explicitly. The

only numerical computation required to calculate solutions is to find the roots of the disper-

sion equation. It became clear that the dispersion equation which is derived for an infinite

ice sheet is also relevant to any shape of ice sheet, or floe, along with the edge conditions

for the floe. An example of that is the Wiener-Hopf technique. The solutions derived using

the Wiener-Hopf technique have been known for more than thirty years and thought to be

too complicated for practical computational use. By re-examining the solutions with the

knowledge of the roots of the dispersion equations and their analytic properties, it became

apparent that the computationally practical formulas were only a step away. Again the

only mathematical requirement was classical complex analysis. I was able to find explicit

solutions whose numerical computation requires only algebraic calculations involving the

roots of the dispersion equations for the cases of infinite and semi-infinite ice sheets.

I would like to thank my supervisor Dr. C. Fox and advisor Dr. M. Meylan for their

patience and support during my Ph.D. study.

Auckland, New Zealand, November, 2002

Hyuck Chung

4

Chapter 1

Introduction

1.1 Background

There are mathematical, physical, and engineering aspects in researches on ice and each area

has a long history. A comprehensive list of references on ice mechanics is given by Dempsey

[16]. This monograph presents a study of a mathematical model describing the flexural

motion of floating sea-ice such as the large sea ice sheet that forms the shore-fast ice around

the coast of Antarctica during each winter. The process of formation and break-up of shore-

fast ice sheets has interested scientists of many disciplines, including mathematicians. That

interest is in part due to it being the largest natural seasonal phenomenon on the surface

of the Earth. Despite the simple nature of the mathematical models used to describe the

floating ice sheet, there have been few analytical formulae for the flexural deflection of an

ice sheet.

There are several forces that potentially cause the motions of an ice sheet. The natural

ones may be wind, ocean current or swell, while the man-made forces can be moving vehicles

such as trucks, airplanes or vibrations generated by a machine. We focus on ocean waves

and swell and man-made flexural motion generated by a mechanical device. The flexural

waves caused by ocean swell typically have a range of wave period around 2 to 20 seconds

and wavelengths around 10 to 200 metres. For a thin elastic plate floating on the water,

many mathematical methods have been applied to obtain either numerical or approximate

analytical solutions (Kheisin [32], Nevel [37]). For a few special cases analytical solutions for

the dynamics of a thin homogeneous plate are well known (Shames and Dym [41], Wyman

[51]). However, when hydrodynamic effects are significant as a force acting on the plate,

the dynamics of the plate becomes much more complicated.

In terms of mathematical models for sea ice dynamics, at least within the thin plate

approximation, Bernoulli’s equation and Laplace’s equation are required to properly model

a floating ice sheet, although it is idealized in many ways. Then, the mathematical tools

which we use to deal with those equations are the Fourier transform and theories of analytic

5

functions in the complex plane that can be found in various text books (Carrier, Krook

and Pearson [7], Churchill, Brown and Verhey [12], Noble [38], Roos [40]). We need the

classical theory of linear elasticity to derive the partial differential equations of an elastic

plate found in the books by Shames and Dym [41] and Timoshenko [48], which give the

justification for the thin plate approximation and an equation of vertical deflection of a

plate. In addition, we need to understand waves in fluids and floating object in the context

of Bernoulli’s equation at the surface shape of the water, the linearization of it and the

interaction conditions between water and the floating object. These ideas can be found in

the articles by Greenhill [25], John [28, 29] and Stoker [46]. Each of these is a very classical

mathematical techniques and there are many text books on the subjects.

Modeling an ice sheet using a thin elastic plate is a well established practice as shown

by Evans and Davies [17], Fox and Squire [21] and Kerr and Palmer [31], particularly for

the relatively featureless ice sheets that forms the fast ice around the coast of Antarctica.

Although an ice sheet is in reality neither homogeneous nor isotropic, it has been observed

that the thin plate model can capture basic properties of the ice sheet, such as the existence

of evanescent waves near the edge of the ice sheet. Our aim here is to explore analytical

aspects of the mathematical study of a floating elastic plate in the context of ice sheet dy-

namics, which determines the ranges of the physical parameters relevant to sea ice. Studies

of a fluid-loaded elastic plate, i.e., the plate equation coupled with either Laplace’s equation

or Helmholtz’s equation, can be found in acoustical research papers (Abrahams [3], Crighton

[13], Lawrie and [33]). As a mathematical problem, the floating plate problem presents in-

teresting difficulties despite its apparent simplicity. As we deal with boundary problems

formulated in the water on which an ice sheet is placed, we must solve Laplace’s equation

for the velocity potential of the water, which is perhaps the most studied differential equa-

tion because of its representation of the law of conservation of mass or electrical charge. A

difficulty of dealing with the fully hydrodynamic problem is that the water pressure acting

on the plate is dependent not only on the buoyancy due to the vertical displacement of the

plate, but also on the changing surface pressure due to the motion at the surface.

The primary achievement in this monograph is finding analytical solutions for the dy-

namics of floating elastic plate in simple geometries such as infinite or semi-infinite plates.

The analytical descriptions of a thin elastic floating plate with hydrodynamic effects taken

into account has not been known until recently, even for the simple geometries mentioned

above. In the case of semi-infinite plate, there have been various forms of solutions which

consist of either numerical integrations or computationally impractical representations of

the solutions. The latter case, even though the analytical method of solution was shown by

Evans and Davies [17] more than thirty years ago, due to lack of computing power then and

knowledge of mode expansion of the flexural waves in an elastic plate. The similar boundary

value problem is studied by Balmforth and Craster [5] and Gol’dshtein and Marchenko [24],

although neither paper fully exploits the mathematical techniques that are available to us.

6

The solutions are an improvement on these articles in terms of numerical computation and

mathematical clarity in the method of solution.

Another important feature of this monograph is the characteristic length and character-

istic time which are used to non-dimensionalize (or scale) the system of partial differential

equations, and then the resulting solutions. Equations are often non-dimensionalized merely

to simplify the appearance of the equations. However, an effective scaling regime enables us

to derive a solution that is unaffected by changes of physical parameters. Therefore, we are

able to represent the solutions of various scales of ice sheets with one solution and a given

pair of characteristic length and characteristic time. Although, any mathematical model is

only an approximation of the reality, and advantage of having an analytical solution is that

it enables us to study qualitative effects of particular physical parameters such as thick-

ness of the ice, wave period and ocean depth, on the motion of an ice sheet. Because the

mathematical model used here is a simple form of an idealization of a complex geophysical

phenomena, our analytical solution cannot be used to predict, for example, when or where

in the ice sheet a crack might appear. However, besides the pure mathematical interests,

the solution may tell us relationships between the flexural response in scaled experiments

in the laboratory and real size field experiments.

1.2 Preview

We preview here the differential equations and mathematical methods of solution that will

be discussed in chapters 2 to 5.

In chapter 2, we derive differential equations that describe the motion of an ice sheet,

the water under the ice and the associated boundary conditions. The ice sheet is modeled

as a thin elastic plate as the deflection of the ice is assumed to be small. We will further

assume that the plate is isotropic and homogeneous so that the linear elasticity theory can

be used to formulate a linear differential equation for the vertical displacement of the plate.

The thin plate equation derived using the classical theory of thin plates is a fourth order

partial differential equation of vertical displacement, which has the following form

D∇4w (x, y, t) +m∂2

∂t2w (x, y, t) = p (x, y, t)

where D, m and p are the flexural rigidity of the plate, the mass density per unit area and

the pressure acting on the plate, respectively. We will find that the boundary conditions

at the edge of the plate are expressed by the displacement, the slope, the bending moment

and the effective shear force intensity, that are present at the edge of the plate. The water

is assumed to be incompressible and inviscid. This is reasonable because we are dealing

with surface waves of long wave length and small frequency (Phillips [39]). Under these

assumptions, the motion of the water is represented by Laplace’s equation in the velocity

7

potential,

∇2φ (x, y, z, t) = 0.

Necessary boundary conditions at the bottom and surface of the ocean for Laplace’s equation

are introduced in order for Laplace’s equation to have a unique solution. We will show the

process of linearization of Bernoulli’s equation and kinematic condition at the surface, thus

the final formulae of the system are linear. The boundary value problems of the simple time

harmonic oscillation are formulated for solutions with single radial frequency dependence

φ (x, y, z, t) = Re[φ (x, y, z, ω) ei ωt

],

w (x, y, t) = Re[w (x, y, ω) eiωt

].

In chapter 3, we study the case when the ice sheet extends to infinity, i.e., the whole

surface satisfies the plate equation. Analytical solutions for simple harmonic waves are

derived when external force either at a point or an infinite line, is applied to an ice sheet.

The Fourier transforms in polar coordinates and one dimensional space

w (γ) =

∫ ∞

0

w (r) rJ0 (γr) dr,

w (γ) =

∫ ∞

−∞

w (x) ei γx dx,

respectively, are used to derive a dispersion relation between wavenumbers and forcing

frequency. We find that the Fourier transform is written as

w (γ) =1

d (γ)

where d (γ) is an analytic function of a complex variable γ with an infinite number of zeros.

The equation d (γ) = 0 is commonly called the dispersion equation. It will be shown that

the solutions are expressed by infinite series of wave modes that exist in an ice sheet and

the analytical formulae for the coefficients of the modes are easy to compute using simple

computer codes. We will see that for a point forcing each mode is a Hankel function of the

first kind, giving

w (r) =∑

q∈Kˆ

a (q)H(1)0 (qr)

and for line loading each mode is an exponential function giving

w (x) =∑

q∈Kˆ

b (q) exp (i q |x|)

where Kˆ is an infinite set of wavenumbers and the coefficients a and b are determined by

8

the dispersion equations. Besides deriving the solutions, we will introduce a scaling of the

solution using characteristic length lc and time tc which eliminate the physical dimensions

from the spatial and time variables, i.e.,

1

lc(x, y, z) ,

t

tc.

We will show that by choosing appropriate values for lc and tc, the system of equations

can be made simple and the resulting solutions become insensitive to a wide range of ice

thickness. Using the analytical solution, we are able to use field measurements to good

estimates of an effective Young’s modulus of an ice sheet from mechanically generated

flexural motion of the ice sheet. A few examples of actual implementation of the method

of finding the effective Young’s modulus from an experimental data set are shown and

the resulting Young’s modulus is compared with the value that is widely in use in other

literatures.

In chapter 4, we consider wave propagation in two semi-infinite plates that are joined by

a straight discontinuity, i.e., the regions x < 0 and x > 0 are occupied by two plates with

different flexural rigidity D1 and D2 respectively. An incoming plane wave that is incident

at an angle from infinity forces the waves in the ice sheets.

I exp i (λx+ ky)

where λ and k are real wavenumbers. Although, the boundary value problem is in three

dimensional space, the plane incident wave reduces the system to two dimensional, and the

solutions are

φ (x, y, z, t) = Re [φ (x, z) exp i (ky + ωt)] .

We derive analytical solutions that can be implemented directly to computer codes for

numerical computation. The solutions are derived using the Wiener-Hopf technique, which

is an extension of the Fourier transform method for half-space problems. We will use

Fourier-type integrals in half spaces

Φ+ (α, z) =

∫ ∞

0

φ (x, z) eiαxdx,

Φ− (α, z) =

∫ 0

−∞

φ (x, z) ei αxdx,

where α is a complex variable. We find that the solutions are again expressed by an infinite

series of natural modes of the ice sheets by solving an algebraic equation in the complex

plane of the type

f1 (α)Φ− (α, z) + f2 (α) Φ+ (α, z) + C (α) = 0

9

called a Wiener-Hopf equation, which is derived from the Fourier transforms of the system

of differential equations in the two regions, x < 0 and x > 0. This Wiener-Hopf equation

can be solved by factorizing the dispersion functions f1 and f2 into two regular functions in

the lower half and the upper half planes. We will find that the coefficients of the solutions

consist of four unknown constants, which must be determined from the boundary conditions

at x = 0. The solutions are expressed in the following form,

φ (x, 0) =

∑λ c (λ) ei λx, for x > 0,∑µ d (µ) ei µx, for x < 0,

where µ and λ are the wavenumbers, which are determined by the dispersion equations of

the ice sheets for x > 0 and x < 0.

Incorporation of conditions at the discontinuity, which we call the transition, are shown

to be represented by a simple system of linear algebraic equations

Md = b

where M and b are respectively a 4 × 4 matrix and a vector representing the conditions

at x = 0, and d is the vector whose elements are the four unknown constants in d (µ) and

c (λ). The solutions are derived using the Wiener-Hopf technique, first applied by Evans

and Davies [17]. We modify and simplify the original method in the aspects of incorporation

of incoming waves from infinity. With these modifications, it becomes apparent that we are

able to deal with the transition conditions using a single universal formula, only changing

the matrix M and vector b.

In chapter 5, the further extension of the wave-ice interaction problem is discussed using

a combination of the Wiener-Hopf technique and the boundary integral method. Chapter 5

gives a different perspective on calculating the dynamics of a floating ice sheet, other than

using the Fourier transform. We will show that the boundary element method expressing

the solution can be reduced using the Wiener-Hopf technique. The solution given in chapter

4 using the Wiener-Hopf technique can be represented by a linear summation of a special

solution G (x, 0) and its derivative Gx (x, 0),

φ (x, 0) = a1G (x, 0) + a2Gx (x, 0) + I (x)

where a1 and a2 are unknown constants to be determined from the edge conditions of the

ice sheet and I (x) is a term due to the incident wave. The above expression inspires us

to find a connection between the boundary integral equation method and the Wiener-Hopf

technique. In that process we find a wider use of the fundamental solutions found in chapter

3 for other than an infinite ice sheet.

We find that the response of a finite ice sheet can be represented by a linear summation

10

of the boundary integral of a fundamental solution,

φ (r, 0) =

∫

∂Ω

[b1 (ρ)G (r − ρ, 0) + b2 (ρ)

∂G (r − ρ, 0)

∂nρ

]dσρ + I (r)

where r = (x, y), and ρ = (ξ, η) in R2 and b1 and b2 are unknown functions defined on the

boundary of the ice sheet ∂Ω to be determined from the edge conditions of the ice sheet.

We will show that the fundamental solution that makes the above representation of the

solution possible may be found using the convolution of the split fundamental solutions of

the infinite free surface and the ice sheet in the two dimensional plane. This splitting and

convolution of the fundamental solutions is equivalent to the factorization and the inverse

Fourier transform of the Wiener-Hopf technique.

The monograph is concluded by chapter 6 and appendices in which we show the steps of

calculations of variational form and Green’s theorem. Analytic properties of the dispersion

function are introduced and proved.

11

Chapter 2

Formulation of boundary value

problems for wave-ice interaction

In this chapter we derive differential equations that describe the dynamics of ice sheets

floating on an ocean wave field. Fig. (2.1) is a general schematic of the physical situation

that we will study. As shown in Fig. (2.1) the origin of the coordinate system is placed

on the surface and the z-axis is pointing upward. Thus the (x, y)-plane is the surface of

the ocean. In all the boundary value problems (BVPs) studied here, it is always assumed

that the depth of the ocean is constant H metres everywhere. Hence the ocean floor is at

z = −H .

xy z

xy z

z=-H

water

ice

W

Figure 2.1: Schematic of the coordinate system and a floating piece of ice sheet on water ofconstant depth H .

In the following sections, we will derive differential equations that govern the motion in

the water, ocean surface, ocean floor and the ice sheet. We will show how to approximate

the physical conditions of the sea ice sheet and simplify the differential equations describing

the interaction between the ice sheet and ocean so that the resulting system of differential

equations can be solved analytically.

In section 2.1, we derive Laplace’s equation for the velocity potential of water and the

boundary conditions associated with Laplace’s equation. In section 2.2, a thin plate equation

12

is derived to model a relatively featureless ice sheet. We will also introduce the boundary

conditions required to solve the plate equation. In section 2.3, we summarize the differential

equations introduced in sections 2.1 and 2.2, and formulate BVPs that will be solved in the

following chapters.

2.1 Laplace’s equation of velocity potential and its

boundary conditions

2.1.1 Laplace’s equation

A standard way of deriving the equations describing the motion of water is to study the

equilibrium relations of forces acting on an infinitesimally small cube of water and then

consider its velocity (Stoker [46]). First we make an assumption of incompressibility of

water (Phillips [39] chapter 2), that is, the mass density of water is constant everywhere for

all time. This assumption can be made because we are studying ocean waves which have

small amplitude and long wave length. From the assumption of incompressibility we can

derive a differential equation expressing the law of mass conservation.

We consider the velocity field of water at time t and location (x, y, z) in some domain Vand we denote it by

v (x, y, z, t) =

vx (x, y, z, t)

vy (x, y, z, t)

vz (x, y, z, t)

where vx, vy and vz are the x, y and z components of the vector v respectively. Since there

is no source or sink in V, we have following equation

0 =

∫

∂V

ρvndσ (2.1)

where ρ is constant mass density of water and dσ is the area element. By vn we denote

outward normal velocity component of the velocity vector v (x, y, z, t) of water on the bound-

ary of V denoted by ∂V. Eqn. (2.1) states that the net mass flux crossing ∂V in unit time

length is zero, which is equivalent to saying that water mass in V is conserved. By Gauss’s

divergence theorem, we have

0 =

∫

∂V

ρvndσ =

∫

V

div (ρv) dV (2.2)

where dV is the volume element. Notice that we used the fact that ρis constant. Since

13

Eqn. (2.2) holds for the arbitrary region V, we must have

div v = 0. (2.3)

This equation is called the equation of continuity.

For our work, it is also assumed that the water has zero viscosity (Phillips [39] chapter 2),

which means that the water has no internal friction. This assumption can be made because

we deal with the gradual movement of water. For rapid movement of fluid, viscosity becomes

a primary physical factor in fluid dynamics. The assumption of zero viscosity leads to the

law of conservation of circulation. The circulation is defined as the net mass flow along a

closed curve C that moves with water, thus the water in C does not change. The circulation

at some time t can be expressed by following equation

Γ (t) =

∮

C

v · sds (2.4)

where s = s (x, y, z, t) is the unit counter clockwise tangential vector and ds is the line

element on C. Taking the time derivative denoted by Γ of the both sides of Eqn. (2.4), we

have

Γ (t) =

∮

C

(v · s + v · s) ds

=

∮

C

(−1

ρ(∇p) · s + F · s + v · vs

)ds

=

∮

C

(−1

ρps+Fs + (v · v)s

)ds

where ps is tangential derivative of p along C, and Fs is tangential projection of F on C.

From the first to the second line, we have used the force equilibrium relation for an inviscid

fluid, i.e.,

−1

ρ∇p+F = a (2.5)

where a = v is acceleration of the fluid, p is pressure and F is the acceleration due to external

forces. We notice that the first term of the left hand side of Eqn. (2.5) is acceleration due

to the pressure gradient. Eqn. (2.5) is called the equation of motion. Since, C is closed,

Γ ≡ 0, which leads to the conclusion that the circulation of a non-viscous fluid is conserved.

This result is called Kelvin’s theorem. Therefore, if at some time Γ (t) = 0, for example

v = Const. everywhere at some time, then Γ ≡ 0 always. Hence, using Stoke’s theorem, we

have

0 =

∮

C

v · sds =

∫

∂V

(rotv) · ndσ. (2.6)

where ∂V is any surface spanning the curve C and n is outward unit normal vector on ∂V.

14

Since, C is arbitrary, we conclude that

rotv = 0. (2.7)

The vector field satisfying the above equation is said to irrotational, and hence inviscid fluid

flow is irrotational.

From Eqn. (2.7) and the vector identity for a scalar function φ, namely,

rot (gradφ) = 0

the vector field v can be expressed by

v = gradφ. (2.8)

for some function φ, traditionally called the velocity potential. From Eqn. (2.3), we find

that φ satisfies Laplace’s equation

∇2φ (x, y, z, t) = 0. (2.9)

We have shown that velocity vector field of an incompressible non-viscous fluid can be

expressed by the gradient of a velocity potential, which satisfies Laplace’s equation (2.9).

Hence, in all our BVPs Eqn. (2.9) holds in the water in the region −∞ < x, y <∞, −H <

z < 0.

2.1.2 Boundary conditions for Laplace’s equation

We discuss here boundary conditions for Laplace’s equation that give a unique solution.

The most direct and simple way to find the boundary conditions for Laplace’s equation may

be to use the variational form of Laplace’s equation.

We multiply both sides of Eqn. (2.9) by a variation δφ and integrate the both sides over

a space domain V and a time interval [t1, t2] which are for now unspecified. Then we have

0 =

∫ t2

t1

∫

V

(∇2φ

)δφdV dt.

Using integration by parts and variational principles, we find

0 = δ

∫ t2

t1

∫

V

1

2|∇φ|2 dV dt−

∫ t2

t1

∫

∂V

φnδφdσdt (2.10)

where φn is the normal derivative on ∂V. From the previous subsection, the first integral

represents total kinetic energy of water in V. Specifying the boundary value of φ lets the

15

variation δφ on ∂V become zero, hence the second term of Eqn. (2.10) become zero. Then,

the solution of Laplace’s equation can be found by solving

0 = δ

∫ t2

t1

∫

V

1

2|∇φ|2 dV dt. (2.11)

The boundary condition φ|∂V = 0 is called a Dirichlet condition.

If we have φn|∂V = 0, then the solution is found by the same variational equation (2.11).

When we have a specific function φn|∂V = g then the solution of Laplace’s equation can be

found by solving

0 = δ

∫ t2

t1

[∫

V

1

2|∇φ|2 dV −

∫

∂V

gφdσ

]dt

.

The boundary condition φn|∂V = g or φn|∂V = 0 is called a Neumann condition. Strictly

speaking, the solution using a Neumann condition is unique to the extent of an added

constant, i.e., no movement of water, but we are not interested in such case.

We have shown that in order to uniquely solve Laplace’s equation, we need either Dirich-

let or Neumann boundary conditions. However, the boundary conditions may be given in

the form of differential equations in φn and φ on ∂V. In such a case, it will be shown in

the next section 2.2 that the necessary conditions for uniqueness can again be obtained by

a similar variational calculations as to that above.

2.1.3 Natural transition conditions

In order to solve a BVP involving mixed surface conditions, such as part ice sheet and

part free surface, we may need to divide the region of water into two or more subdomains,

then consider the BVP in each subdomain with interaction conditions between the subdo-

mains. We here derive necessary interaction conditions between subdomains in water, which

correspond to natural transition conditions.

Ice OceanWW

VV VV

s Vs

c

c c

Figure 2.2: Schematic of the domains of the BVP and notations for the subdomains in thewater and the surface.

Fig. (2.2) shows a side view of Fig. (2.1) divided into two subdomains, an ice covered

region and free surface regions denoted by V and Vc respectively. The vertical boundary

16

between V and Vc is denoted by Vs+ (boundary in side the ice covered region) and Vs

−

(boundary out side the ice covered region). Then the second term in Eqn. (2.10) for V ∪Vc

becomes

∫ t2

t1

∫

∂V

φnδφdσ +

∫

∂Vc

φnδφdσ

dt =

∫ t2

t1

∫

∂Vs+

φnδφdσ +

∫

∂Vs−

φnδφdσ

+

∫

∂V\∂Vs+

φnδφdσ +

∫

∂Vc\∂Vs−

φnδφdσ

dt.

The third and fourth integrals become zero by the given boundary conditions. Since, we

specify neither φn nor φ on ∂Vs±, φn and φ must be continuous

φn|∂Vs+

= φn|∂Vs−

, φ|∂Vs+

= φ|∂Vs−

,

in order for the first and second integrals to become zero. We call these two continuity

conditions natural transition conditions of water. In terms of the physical properties of

water, the natural transition conditions are equivalent to continuity of normal velocity and

pressure.

2.1.4 Boundary conditions on the surface

Boundary conditions on the surface of ocean are derived by considering the shape of the

surface which changes in the response to the equilibrium of force at the surface.

Consider an implicit equation representing the shape of the surface,

0 = ζ (x, y, z, t) (2.12)

where ζ is assumed to be smooth and continuous everywhere in space and time. Taking the

time derivative of Eqn. (2.12) gives

0 =dζ

dt= ζxvx + ζyvy + ζzvz + ζt (2.13)

where ζx, ζy and ζz denote the x, y and z derivatives of ζ , respectively. Notice that the

vector (ζx, ζy, ζz) is normal to the surface, hence the unit normal vector on the surface is

n =(ζx, ζy, ζz)√ζ2x + ζ2

y + ζ2z

.

From Eqn. (2.8) and Eqn. (2.13), we find that

φn = − ζt√ζ2x + ζ2

y + ζ2z

.

17

The above equation shows that if the surface shape is fixed, for example the ocean floor,

i.e., time independent, then we have a boundary condition

φz = 0 (2.14)

which is the ocean floor condition in all our BVPs at z = −H .

When the shape of the surface is allowed to change, for example, at a free surface or when

bounding an elastic plate (ice sheet), we can rewrite the equation of the surface (2.12) using

the vertical displacement of the surface denoted by w (x, y, t). Then, we have an equation

of surface

z = w (x, y, t) . (2.15)

Since, ζ = z − w from Eqn. (2.13), we have a boundary condition at z = 0

wxφx + wyφy − φz + wt = 0. (2.16)

For this situation the equation of motion (2.5) can be rewritten using Eqn. (2.3) as

−1

ρ∇pw − g∇z =

d

dt(∇φ) =

1

2∇(|∇φ|2

)+ ∇φt

where pw is pressure in the water. We note that g is the gravitational acceleration and the

only external force is gravity, which is how the second term is obtained. We now integrate

the both sides of above equation in space to obtain a form of Bernoulli’s equation

1

ρpw + gz +

1

2|∇φ|2 + φt = A (t) (2.17)

where A is a function independent of space variables, which we normally set to zero. Since we

assumed that the water moves gradually, we may omit the higher order terms in Eqn. (2.16)

and Eqn. (2.17), then we have

wt = φz (2.18)

which we call the no-cavitation condition, and

pw

ρ+ gw + φt = 0 (2.19)

which is a linearized Bernoulli’s equation. Notice that we replace z with the vertical dis-

placement function. All the Eqns. (2.9, 2.18, 2.19, 2.14) are linear.

If there is no force other than constant atmospheric pressure at the free surface, we

can obtain a free surface equation from Eqn. (2.18), and differentiating the both sides of

Eqn. (2.19) with respect to time t

φtt + gφz = 0. (2.20)

18

If there is a plate (ice sheet) floating on the surface, then pw on the surface is unknown,

then it must be found using the differential equation of plate which is introduced in the

following section.

2.2 Thin plate equation and its boundary conditions

In this section, we derive the differential equation that we use to describe an ice sheet and

the boundary conditions associated with this plate equation. Since the ice sheet which we

study here is assumed to be relatively featureless and almost homogeneous in its thickness

over a large area, we model the ice sheet as an thin elastic plate that has a flexural response

to wave like external forcing.

The justification for modeling an ice sheet in this way is given by Kerr and Palmer [31]

who derive an effective Young’s modulus for the plate with properties that vary through its

thickness. In the following subsections, we will give a brief theoretical background of the

thin plate equation and the derivation of the necessary boundary conditions in connection

with the physical properties of the ice sheet. The partial differential equation of flexural

motion of a thin elastic plate can be derived from equilibrium relations of the forces acting

in the plate.

We assume that the plate is isotropic and homogeneous, that is, strains and stresses are

independent of direction and location in the plate. These assumptions enable us to use linear

elasticity theory to formulate a thin plate equation. We only deal with small deflections of

the plate, thus the thin plate equation can be formulated for the vertical displacement of

the plate using all the assumptions.

2.2.1 Thin plate equation

A differential equation for a thin elastic plate may be derived from equilibrium of forces

acting on the plate. The details are given by Shames and Dym [41]. The accuracy of the

equation depends on how many forces are included in the equilibrium relations. We use here

horizontal stress components only, that is, vertical or transverse components of the stress

are omitted and deformation of the ice sheet occurs only in the (x, y) plane, thus we only

deal with bending due to vertical displacement of the plate.

We can formulate a well defined BVP for the dynamics of a plate using Hamilton’s

principle and variational calculation similar to that in the previous section. Let Ω denote

the domain occupied by the plate on R2 and h be the thickness of the plate, hence the plate

is in Ω × [−h/2, h/2] ⊂ R3. Hamilton’s principle for an elastic plate Ω × [−h/2, h/2] is

δ

∫ t2

t1

(U −W −K) dt = 0

19

where U , W and K are strain energy, work potential of the external forces and kinetic

energy in the plate respectively.

First, we find U using Hooke’s law for a three dimensional object, here it is a thin plate.

Hooke’s law relates strains and stresses in the object. Strain represents the amount of

deformation of the material under stress. Strain is usually denoted by ǫ with two subscripts

that are combinations of x, y and z. The strains ǫxx, ǫyy and ǫzz denote change of length

of the material in x, y and z directions divided by the original length, which lead to simple

linear relations between the strains and stresses in the same directions

ǫxx = 1Eτxx,

ǫyy = 1Eτyy,

ǫzz = 1Eτzz,

(2.21)

where τxx, τyy and τzz are stresses in x, y and z directions, and E is Young’s modulus. It is

easy to imagine that if an elastic material is pulled in one direction then it must shrink in

other directions and the rate of the shrinkage should be dependent of the physical properties

of the material. We intuitively find the contraction relations

ǫyy = − νEτxx, ǫzz = − ν

Eτxx,

ǫxx = − νEτyy, ǫzz = − ν

Eτyy,

ǫxx = − νEτzz, ǫyy = − ν

Eτzz,

(2.22)

where ν is a constant Poisson’s ratio. Hence, if we superpose the strain components produced

by each of the three stresses, we obtain from Eqn. (2.21) and Eqn. (2.22)

ǫxx = 1Eτxx − ν (τyy + τzz) ,

ǫyy = 1Eτyy − ν (τxx + τzz) ,

ǫzz = 1Eτzz − ν (τxx + τyy) .

(2.23)

We note that by adding both sides of three equations above, we have

ǫxx + ǫyy + ǫzz =1 − 2ν

E(τxx + τyy + τzz) .

Since, a material must contract when uniform isotropic forces are applied to a material,

(1 − 2ν) /E must be positive. Hence, Poisson’s ratio must be −1 < ν < 1/2 and for sea ice,

the accepted value is ν = 0.3.

In order to express the deformation of a material, we need, in addition to the length

changes, quantities for the skewing or shearing of the material in (x, y), (y, z) and (x, z)

planes, which are denoted by ǫxy, ǫyz and ǫxz respectively.

ǫxy =1 + ν

Eτxy, ǫyz =

1 + ν

Eτyz , ǫxz =

1 + ν

Eτxz. (2.24)

20

We have now expressed Hooke’s law, relating strains and stresses, by Eqn. (2.23) and

Eqn. (2.24).

For the case of small deflections of a thin plate, horizontal (parallel to the plate) de-

formation is much greater than vertical deformation, thus we may assume that ǫzz ≪ ǫxx,

ǫzz ≪ ǫyy, ǫxz ≪ ǫxy and ǫyz ≪ ǫxy, and τzz ≪ τxx, τzz ≪ τyy, τxz ≪ τxy and τyz ≪ τxy.

Therefore, the z-components of the strains and stresses in Eqn. (2.23) and Eqn. (2.24) can be

omitted. Furthermore, the strains can be expressed by the curvature of the plate, which is

approximately the second derivative of the displacement w (x, y, t), as the shear deformation

is small compared to bending deformation. Hence, we have the formulas for strains

ǫxx = −zwxx (x, y, t) ,

ǫyy = −zwyy (x, y, t) ,

ǫxy = −zwxy (x, y, t) .

(2.25)

for −h/2 < z < h/2 where h is thickness of the plate. We note that the neutral plane

of bending is placed at the middle of the plate since Young’s modulus is assumed to be

constant. In reality Young’s modulus of ice sheet changes through the ice sheet due to the

temperature gradient and sensitivity of ice properties near the freezing temperature. Kerr

and Palmer [31] showed that the flexural rigidity of the ice sheet could be approximated

using a constant Young’s modulus.

The strain energy of a plate is expressed by the following formula

U =1

2

∫ h/2

−h/2

∫

Ω

(τxxǫxx + τyyǫyy + τxyǫxy) dxdydz.

We use Eqn. (2.23) and Eqn. (2.24) to write U in terms of strain,

U =1

2

∫ h/2

−h/2

∫

Ω

E

2 (1 − ν2)

ǫ2xx + 2νǫxxǫyy + ǫ2yy + 2 (1 − ν) ǫ2xy

dxdydz.

We then substitute Eqn. (2.25) to write U in terms of the vertical displacement

U =D

2

∫

Ω

(∇2w

)2+ 2 (1 − ν)

(w2

xy − wxxwyy

)dxdy (2.26)

where D is a constant called flexural rigidity, defined D = Eh3/12 (1 − ν2).

We can express the kinetic energy K due to the motion of the plate using the vertical

displacement

K =

∫

Ω

1

2mw2

t dxdy (2.27)

where m is mass density of the plate per unit area calculated by m = ρih, ρi being the mass

density of sea ice.

21

Finally, the work potential W due to external force p (x, y, t) is

W =

∫

Ω

pwdxdy. (2.28)

Therefore, from Eqns. (2.26, 2.27, 2.28) and Hamilton’s principle, we have

0 = δ

∫ t2

t1

∫

Ω

[D

2

(∇2w

)2+ 2 (1 − ν)

(w2

xy − wxxwyy

)− mw2

t

2− pw

]dxdydt. (2.29)

Using the variational calculations (Hildebrand [27], Shames and Dym [41]), we find that

0 =

∫ t2

t1

[∫

Ω

(D∇4

x,yw +mwtt − p)δwdxdy −

∫

∂Ω

(B1w) δw − (B2w) δwnds

]dt. (2.30)

where

B1w = D (∇2w)n +D (1 − ν) ∂∂s

(n2

x − n2y

)wxy − nxny (wxx − wyy)

,

B2w = Dν∇2w +D (1 − ν)n2

xwxx + n2ywyy + 2nxnywxy

.

(2.31)

The unit out-going normal vector is denoted by n = (nx, ny), and ∂/∂s = −ny∂/∂x+nx∂/∂y

is the counter clockwise tangential derivative along the boundary. The differential operator

∇4 is called the bi-harmonic operator

∇4 =∂4

∂x4+ 2

∂4

∂x2∂y2+

∂4

∂y4.

Eqn. (2.30) shows that if we have boundary conditions

either B1w = 0 or w = prescribed,

and

either B2w = 0 or wn = prescribed

on the boundary ∂Ω, then w (x, y, t) satisfies the differential equation

D∇4w (x, y, t) +mwtt (x, y, t) = p (x, y, t) (2.32)

for (x, y) ∈ Ω. The external force p consists of forces from water, pw in Eqn. (2.19) and

from elsewhere, for example, wind, static load or moving vehicles, etc. We denote external

force other than water pressure by pa, then we have

p = pw + pa. (2.33)

22

2.2.2 Boundary conditions for the plate equation

We study here the physical interpretations of the boundary conditions shown in the previous

subsection. The values B1w and B2w represent effective shear force intensity (shear force

and twisting moment) and bending moment that are present on the edge of the plate. For

simplicity, we consider a straight edge on the y axis, then we have

B1w = D (wxxx + (2 − ν)wxyy) ,

B2w = D (wxx + νwyy) .(2.34)

The bending moment on the edge is commonly denoted by Mx and defined as

Mx =

∫ h/2

−h/2

zτxxdz.

We notice by using Hooke’s law and the displacement representations of the strains, we can

confirm the formula of B2w. The rest of the components of bending moment are defined as

My =

∫ h/2

−h/2

zτyydz, Mxy =

∫ h/2

−h/2

zτxydz.

The shear force intensity is commonly denoted by Qx and defined as

Qx =

∫ h/2

−h/2

τxzdz.

Although, we have omitted the shearing stresses and strains in order to formulate the thin

plate equation, we cannot ignore the effect of shear force intensity on the edge. Qx can be

obtained using the equilibrium relations of forces,

∂τxx

∂x+∂τxy

∂y+∂τxz

∂z= 0.

Multiplying the equation above by z and integrating over [−h/2, h/2], we find that from

the definitions of the bending moments that

∂Mx

∂x+∂Mxy

∂y+

∫ h/2

−h/2

z∂τxz

∂zdz = 0.

We use the definition of Qx and the fact that the shear stress on the top and bottom surface

of the plate is zero, to get∂Mx

∂x+∂Mxy

∂y−Qx = 0.

Hence, the effective shear intensity can be written by summing Qx and the force due to the

23

twisting motion,

B1w = Qx +∂Mxy

∂y

which confirms the formula given by Eqn. (2.34). Fig (2.3) is a diagram of the forces that

are present on the edge of the plate.

yx

MM

Q

xxy

x

Figure 2.3: Diagram of bending moments and shear force acting the edge of a plate. Theedge is on the y axis and the x axis is pointing outward from the plate.

The homogeneous boundary conditions

B1w = 0, B2w = 0

are often called free edge conditions since there are no forces applied at the edge of the

plate. We note that the free edge conditions apply to the edges of a floating ice sheet that

abuts open water. There are many more combinations of boundary conditions associated

with physical constraints at the edge of the plate. For example, if the edge of the plate is

clamped (immobile) then clearly we have

w = 0, wn = 0.

If the edge is on a type of hinge (simply supported) then

w = 0, B2w = 0.

If the edge is attached to a type of roller that allows the edge to move vertically, then

wn = 0, B1w = 0.

There are more edge conditions such as linear combinations of these four values, displace-

ment, slope of the plate, bending moment and effective shear force intensity, but we have

just presented the most obvious ones here.

24

2.2.3 Transition conditions for an elastic plate

We consider a plate with a discontinuity that may be caused by cracking or a change of

thickness, etc. We then must find interaction conditions at the discontinuity in order to

solve the plate equation, similar to those of the previous section for the natural transition

conditions. As an example, Fig. (2.4) shows two ice sheets separated by a discontinuity

denoted by Λ1 from Ω1 side and Λ2 from Ω2 side, i.e., the values of w, wn, B1w and B2w

may be different on Λ1 and Λ2 but both represent the same boundary.

z=-H

ice W2

W1L2

L1

Figure 2.4: Schematic drawing of an example of a BVP of two semi-infinite ice sheets withdifferent thickness. Water is covered with two semi-infinite ice sheets which share theiredges at x = 0, −∞ < y <∞.

We can derive the conditions for w, wn, B1w and B2w at Λ1 and Λ2 using the same

calculations shown in Eqn. (2.29) and Eqn. (2.30), in two subdomains Ω1 and Ω2. The

second spatial integration of Eqn. (2.30) becomes

−∫

∂Ω1\Λ1

[(B1w) δw − (B2w) δwn] ds−∫

∂Ω2\Λ2

[(B1w) δw − (B2w) δwn] ds

−∫

Λ1

[(B1w) δw − (B2w) δwn] ds−∫

Λ2

[(B1w) δw − (B2w) δwn] ds.

This formula tells us the permissible combinations of conditions for w, wn, B1w and B2w at

Λ1 and Λ2. The first and second terms show the same boundary conditions at the edge of

the plates except Λ1 and Λ2 as discussed in the previous subsection. The third and fourth

terms show that if w and wn are continuous, that is, (remembering the sense of the normal)

w|Λ1= w|Λ2

and wn|Λ1= − wn|Λ2

then we must have the continuity relations

B1w|Λ1= − B1w|Λ2

and B2w|Λ1= B2w|Λ2

.

Again we note the signs of the normal and tangential derivatives in B1w. We are able to

find more combinations of conditions using the same argument. We notice that there must

be four conditions, either specifying or requiring continuity of the four elements given in

25

Eqn. (2.38) and Eqn. (2.39). As examples of transition conditions, we have

B1w|Λ1= − B1w|Λ2

w|Λ1= w|Λ2

B2w|Λ1= specified

B2w|Λ2= specified

,

B2w|Λ1= B2w|Λ2

wn|Λ1= − wn|Λ2

B1w|Λ1= specified

B1w|Λ2= specified

.

The transition and edge conditions can be used to express various physical situations

involving ocean waves and a floating objects. I provide several examples when the discontinu-

ity Λ1 (Λ2) is an infinite straight line on the y axis, i.e., Λ1 = x = 0− and Λ2 = x = 0+.

Example 1 When a rigid semi-infinite plate occupies x > 0, the edge condition is

w|x=0+ = 0.

Example 2 When an elastic semi-infinite plate is occupying x > 0, we have the well-known

free edge conditions

B1w|x=0+ = 0, B2w|x=0+ = 0.

Example 3 When two semi-infinite plates are occupying x < 0 and x > 0, respectively with

a gap between them,

B1w|x=0± = 0, B2w|x=0± = 0.

Example 4 When two continuously joined semi-infinite plates are in x < 0 and x > 0, the

transition conditions are

B1w|x=0− = B1w|x=0+, B2w|x=0− = B2w|x=0+ ,

w|x=0− = w|x=0+, wx|x=0− = wx|x=0+ .

2.3 Time-harmonic waves

We conclude this chapter by summarizing the differential equations and boundary condi-

tions introduced in the previous sections. Since we will consider only the time-harmonic

response of the water and the surface, we re-formulate the equations for time-harmonic

external forcing. Thus, using the linearity of the equations we are able to eliminate the

time dependent terms from the system of equation. In order to complete the BVPs, we

introduce boundary conditions at r =√x2 + y2 = ∞, which are commonly called radiation

conditions.

26

2.3.1 Time-harmonic forcing

We rewrite Eqns. (2.9, 2.18, 2.19, 2.32) for time harmonic forcing, i.e., pa has sinusoidal

time dependence and is expressed using a complex valued function,

pa (x, y, t) = Re[pa (x, y, ω) ei ωt

](2.35)

where ω is radial frequency. Then, the solutions φ and w also have sinusoidal time de-

pendence because of the linearity of the equations. Hence, under the forcing given by

Eqn. (2.35), the solutions can be expressed using complex-valued functions of frequency ω,

φ (x, y, z, t) = Re[φ (x, y, z, ω) ei ωt

], (2.36)

w (x, y, t) = Re[w (x, y, ω) eiωt

]. (2.37)

We note that the same notations are used to denote functions of time and frequency because

there is no risk of confusion as from this point on the BVPs will be solved for φ (x, y, z, ω)

and w (x, y, ω).

We may consider the functions φ (x, y, z, ω) and w (x, y, ω) as the Fourier transforms of

φ (x, y, z, t) and w (x, y, t) in time, i.e.,

φ (x, y, z, ω) =

∫ ∞

−∞

φ (x, y, z, t) e− i ωtdt,

w (x, y, ω) =

∫ ∞

−∞

w (x, y, t) e− i ωtdt.

Then the original time dependent functions can be calculated using the usual inverse Fourier

transform

φ (x, y, z, t) =1

2π

∫ ∞

−∞

φ (x, y, z, ω) ei ωtdω,

w (x, y, t) =1

2π

∫ ∞

−∞

w (x, y, ω) ei ωtdω.

Notice that we choose to use the time factor exp (iωt), which is consistent with the defi-

nitions of the Fourier transform, but we could have used exp (− iωt) which would change

the definition of the Fourier transform accordingly. If the forcing is transient in time, then

we can construct time dependent solutions from the single-frequency dependent solutions

using the inverse Fourier transform since the system of equations is linear. The factor 1/2π

is introduced in the various definitions of the Fourier transform also.

In summary, the new system of equations of φ (x, y, z, ω) and w (x, y, ω) which we will

27

study are, on the surface z = 0

D∇4w − ω2w = pw + pa (thin plate equation) (2.38)

iωw = φz (kinematic condition) (2.39)

i ρωφ+ ρgw + pw = 0 (linearized Bernoulli equation) (2.40)

−ω2φ+ gφz = 0 (free surface condition) (2.41)

and at the ocean floor z = −H

φz = 0 (solid ocean floor) (2.42)

and in the water −∞ < x, y <∞, −H < z < 0:

∇2φ = 0, (2.43)

which is Laplace’s equation. It is often convenient to combine Eqn. (2.38) and Eqn. (2.40),

[D∇4 −mω2 + ρg

]w + iωρφ = pa (2.44)

and, if pa = 0, we have an equation of φ from Eqn. (2.39) so multiplying the both sides of

Eqn. (2.44) by iω, [D∇4 −mω2 + ρg

]φz = ρω2φ. (2.45)

2.3.2 Radiation conditions

So far we have introduced boundary conditions of Laplace’s equation on the top and bottom

surface of the ocean, but not yet specified any boundary conditions at r = ∞, for 3-

dimensional problem, and |x| = ∞ for 2-dimensional problems in the (x, z)-plane. The

condition regarding the behavior of waves at infinity is called the Sommerfeld radiation

condition originally formulated for electromagnetic wave problems. A well known form of

the Sommerfeld radiation condition is, in three dimensional space

limr→∞

√r

(∂φ

∂r+ iβφ

)= 0

and in two dimensional space, assuming that waves are travelling along the x axis,

lim|x|→∞

(∂φ

∂x+ i βφ

)= 0

where β = ω/c, c being depending on the medium, the speed of light or sound which is

set to be infinite in our case. Hence, we will not use the radiation condition in this form.

28

Instead we impose an equivalent condition for our solutions at infinity stated as ‘there is no

source of waves at infinity and only out-going waves exist at infinity.’ Without the radiation

condition there can be more that one solution satisfying all the other boundary conditions

since we have not included the effects of dissipation of waves in the water or ice, i.e., wave

can reach infinity.

2.4 Summary

In this chapter, we have formulated the differential equations and associated boundary

conditions for the velocity potential φ (x, y, z, t) of water and the vertical displacement of

the surface w (x, y, t). In section 1, we derived Laplace’s equation (2.9) of the velocity

potential of water from the assumptions of incompressibility and zero-viscosity, which lead

to the laws of conservation of mass and circulation. We then derived two conditions for

the water, Bernoulli’s equation (2.19) and the kinematic condition (2.18). We were able to

linearize the two surface conditions from the assumption that waves are smooth and have

small amplitude.

In section 2, we modeled an ice sheet as an elastic plate that is isotropic and homoge-

neous. We derived a differential equation for a thin elastic plate using Hamilton’s principle.

The assumption of small displacement gradient, thus small shear deformations of the plate

in the z-direction, enabled us to simplify Hooke’s law and express the strain energy of the

plate in terms of only the vertical displacement w. Thus the dynamics of a thin plate

were represented by the differential equation (2.32) of w with an unknown pressure from

the water underneath, pw. The boundary conditions (edge conditions Eqn. (2.31)) of the

plate were found simultaneously with the plate equation using Hamilton’s principle and a

variational calculation (Eqn. (2.29), Eqn. (2.30)). We found that the edge conditions are

combinations of slope wn, displacement w, effective shear force intensity B1w and bending

moment B2w. We showed a few examples of possible physical situations and the associated

edge conditions. We also derived transition conditions at a discontinuity of a plate and

showed examples of permissible continuity conditions at the discontinuity.

In section 3, we summarized the differential equations derived in sections 1 and 2, then

rewrote them for time-harmonic forcing as shown in Eqn. (2.38) to Eqn. (2.41). The system

of equations are written for time harmonic solution φ (x, y, z, ω) and w (x, y, ω), so the

equations are dependent only on the spatial variables. The Sommerfeld radiation condition

is required at r = ∞ or |x| = ∞ in order to ensure the uniqueness of the solution.

29

Chapter 3

Harmonic forcing of an infinite

floating plate

3.1 Background

In this chapter we consider the simple harmonic oscillation of an infinite floating elastic plate

using the method of solution reported in Fox [18], Fox and Chung [19], and Fox, Haskell

and Chung [20]. We consider wave propagation in an infinite elastic plate that is floating

on water, i.e., the whole surface of the water is occupied by an elastic plate, in our case an

ice sheet satisfies the thin plate equation (2.32) given in chapter 2.

There are two main objectives of this chapter. First, we derive analytical formulae for

the deflection of an infinite ice sheet under a localized force. Second, we scale the system

of equations and the solutions to dimensionless values so they are largely independent of

physical parameter values within the ranges of our interest. In order to derive the analytical

solutions, we use the Fourier transform in one and two dimensional space. Radially symmet-

ric waves in a thin plate without the hydrodynamic support, have studied by Sneddon [42]

using the two dimensional radial Fourier transform. This integral transform is commonly

called the Hankel transform

w (γ) =1

2π

∫ ∞

0

w (r) rJ0 (γr) dr.

A complete analytical solution for deflection of an infinite plate due to a localized static

load was derived by Wyman [51] by choosing the bounded solutions of

wrr +1

rwr + iw = 0 (3.1)

30

whose solutions satisfy the radial form of the (non-dimensional) thin plate equation

[∂2

∂r2+

1

r

∂

∂r

]2

w + w = 0.

The hydrodynamic effects that appear as φt in the thin plate equation force us to solve

Laplace’s equation, and thus the application of the Fourier transform is not as straightfor-

ward as in the two examples cited (Sneddon [42] and Wyman [51]). Response of an infinite

thin elastic plate under fluid loading is studied in a series of papers by Crighton [13, 14, 15].

In the following sections, the deflection of a plate is expressed by the infinite number of

modes that exist in the vibrating plate floating on the water, instead of a finite number of

modes that represent the free oscillation of an elastic plate without restraints. The fact that

waves in an incompressible fluid can be expressed by an infinite series of natural modes is

shown by John [28, 29], each mode being a Hankel function of the first kind. It is known

that the oscillation of an infinite plate can also be expressed in terms of Hankel functions.

Kheisin, in 1967 ([32] chapter IV), studied the same problem solved in this chapter. Kheisin

derived the same dispersion equation for the physical variables, and then studied the prop-

erties of the inverse transform for the simple shallow water and static load cases, which are

approximated versions of the solutions reported here.

The most important consequence of being able to derive analytical solutions may be

confirmation of the suitability of the scaling scheme for the range of wave frequencies in which

our geophysical interest lies (Fox [18]). Following Fox [18], we show that the characteristic

length that appears in Wyman [51] for the static-loading problem is also a natural length

scale for a floating ice sheet which is oscillating. In section 3.6, we show that solutions scaled

by the characteristic length and the corresponding characteristic time are independent of

physical parameters, which enables us to find scaling laws relating various scales of ice

sheets.

In section 3.7, we propose several methods of finding the characteristic length, and thus

the effective Young’s modulus of a fast ice sheet using the non-dimensional solutions. In

Fox, et al. [20], characteristic length and effective Young’s modulus are estimated from

actual experimental data sets collected by Fox at McMurdo Sound, Antarctica.

3.2 Mittag-Leffler expansion

Before we formulate and solve the boundary value problem and solve it, we introduce a tool

for expressing functions of a complex variable called the Mittag-Leffler expansion (Carrier,

Krook and Pearson [7]). This tool for expressing functions of a complex variable will be

31

required in the following sections. In particular, we need to show that the function

w (γ) =1

d (γ, ω)

where

d (γ, ω) = γ4 + 1 −mω2 − ω2

γ tanh γH,

can be expressed by a linear sum of terms like 1/ (γ − a), a being a zero of d (γ, ω). We first

remind ourselves of the Mittag-Leffler expansion that can be found in most text books on

complex analysis, and then show that it can indeed be applied to w (γ).

3.2.1 Mittag-Leffler expansion

Consider a function that is regular in the whole plane except at isolated points. A set of

points is called isolated if there exists an open disk around each point that contains none

other of the isolated points. Such a function is known as fractional function. We show that

a fractional function that has an infinite number of poles can be expressed by infinite series

of polynomials (Carrier, Krook and Pearson [7]).

Let f (γ) be a fractional function that has an infinite number of poles. We note that

a number of poles that are situated within a bounded region is always finite since the set

of poles does not have limit-points. Indeed, if there is a limit-point γ = c then any small

circle with centre at γ = c would contain an infinite number of poles. Once we have a finite

number of poles in a confined part of the plane we can number them in the order of their

non-decreasing moduli, so that denoting the poles by ai we have

|a1| ≤ |a2| ≤ |a3| ≤ ...,

where |ai| → ∞ as i → ∞. At every pole γ = ai the function f (γ) will have a definite

infinite part, which will be a polynomial with respect to the argument 1/ (γ − ai) without

the constant term. We denote this polynomial term by

Gi

(1

γ − ai

), i = 1, 2, 3, ... . (3.2)

We show that the fractional function f (γ) can be represented by a simple infinite series of

Gi by making certain additional assumptions. Suppose that a sequence of closed contours

Cn which surround the origin exists and satisfies following conditions.

Condition 1. None of poles of f (γ) are on the contours Cn, n = 1, 2, 3, ...

Condition 2. Every contour Cn lies inside the contour Cn+1.

Condition 3. Let ln be length of the contour Cn and δn be its shortest distance from

the origin then δn → ∞ as n→ ∞, i.e., the contours Cn widen indefinitely in all directions

32

as n increases.

Condition 4. A positive number m exists such that

lnδn

≤ m for any n = 1, 2, 3, ....

We now suppose that given such a sequence of contours, there exists a positive number

M, such that on any contour Cn our fractional function f (γ) satisfies |f (γ)| ≤M . Consider

the integral1

2π i

∫

Cn

f (γ′)

γ′ − γdγ′ (3.3)

where the point γ lies inside Cn and is other than ai (the poles inside Cn.) We also consider

the sum of the polynomials (3.2) for the poles ai, inside Cn,

ωn (γ) =∑

(Cn)

Gi

(1

γ − ai

). (3.4)

The integrand of (3.3) has a pole γ′ = γ and poles γ′ = ai. We can calculate the residue at

the pole γ′ = γ byf (γ′)

(γ′ − γ)′

∣∣∣∣γ′=γ

= f (γ′)|γ′=γ = f (γ) .

The residues at the poles γ′ = ai are, by the definition (3.4), the same as the residues of the

functionωn (γ′)

γ′ − γ. (3.5)

We note that all poles of this function are situated inside Cn. We now show that the sum

of residues of function (3.5) at the poles ai is

−ωn (γ) = −∑

(Cn)

Gi

(1

γ − ai

). (3.6)

Since the definition of ωn and Gi is a polynomial of 1/ (γ − ai) , the order of the denominator

of function (3.5) is at least two units higher than that of the numerator of function (3.5).

Hence, for a circle with a sufficiently large radius R, we have

2π i∑

(Cn)

Resγ′=ai

ωn (γ′)

γ′ − γ=

∮

CR

ωn (γ′)

γ′ − γdγ′.

The LHS of this does not change as the radius R increases, and the RHS→ 0 as R → ∞.

Indeed, ∣∣∣∣∮

CR

ωn (γ′)

γ′ − γdγ′∣∣∣∣ ≤

∮

CR

∣∣∣∣γ′ωn (γ′)

γ′ − γ

1

γ′dγ′∣∣∣∣ ≤ max

|γ′|=R

∣∣∣∣γ′ωn (γ′)

γ′ − γ

∣∣∣∣2πR

R

33

and the term |·| tends to zero as R→ ∞. Thus, the sum of residues at poles within a finite

distance is zero. Since we know that the residue of (3.5) at γ′ = γ is ωn (γ), the sum of the

rest is formula (3.6). Thus, we have an expression for the integral (3.3),

1

2π i

∫

Cn

f (γ′)

γ′ − γdγ′ = f (γ) −

∑

(Cn)

Gi

(1

γ − ai

). (3.7)

Also, when γ = 0 we have

1

2π i

∫

Cn

f (γ′)

γ′dγ′ = f (0) −

∑

(Cn)

Gi

(− 1

ai

). (3.8)

Subtracting Eqn. (3.7) from Eqn. (3.8) gives

γ

2π i

∫

Cn

f (γ′)

γ′ (γ′ − γ)dγ′ = f (γ) − f (0) −

∑

(Cn)

[Gi

(1

γ − ai

)−Gi

(− 1

ai

)].

We now prove that the integrand on the LHS of this expression tends to zero as n→ ∞.

Since, |γ′| ≥ δn, |γ′ − γ| ≥ |γ′| − |γ| ≥ δn − |γ| , we have

∣∣∣∣∫

Cn

f (γ′)

γ′ (γ′ − γ)dγ′∣∣∣∣ ≤

Mlnδn (δn − |γ|)

<Mm

δn − |γ| . (3.9)

Since δn → ∞ as n→ ∞ and condition 4, the integral in inequality (3.9) tends to zero as

n increases.

Finally, we have formula for f (γ),

f (γ) = f (0) + limn→∞

∑

(Cn)

[Gi

(1

γ − ai

)−Gi

(− 1

ai

)].

Since, the contour Cn will widen indefinitely as n increases, the second term is a sum over

all poles, so we have f (γ) in the form of an infinite series

f (γ) = f (0) +∞∑

i=1

[Gi

(1

γ − ai

)−Gi

(− 1

ai

)]. (3.10)

For the expansion formula of w (γ), the polynomial term (3.2) is

Gi

(1

γ − qi

)=R (qi)

γ − qi.

34

3.2.2 Expansion of w (γ)

Now we show that the function w (γ) satisfies the conditions for the Mittag-Leffler expansion.

Define a sequence of square contours Cn, square with its four corners at ǫn− i ǫn, ǫn +i ǫn,

−ǫn + i ǫn and −ǫn − i ǫn, where ǫn =(n + 1

2

)π/H, n = N,N + 1, .... We start by showing

that |w (γ)| is bounded on any Cn in order to follow the proof of Mittag-Leffler expansion

given in the previous subsection.

For the sake of simplicity, write u = 1 − mω2. When Im γ is large the poles of w are

almost ± inπ/H. In fact, the poles i qnn=1,2,..., qn ∈ R of w satisfy

1

(qn + u) qn= tan (qnH) ,

so γn → ±nπ/H as n increases. Thus, by choosing a large N , the contour Cn is always a

certain distance away from the poles for any n ≥ N . We prove the boundedness of |w| by

showing that |w (x+ i y)| is bounded for y = ±ǫn, n = N,N + 1, ..., and x, y ∈ R, and then

for x = ±ǫn, n = N,N + 1, ..., y ∈ [−ǫn, ǫn].

The detailed observation on the poles of w (γ) will be given in the next section, but

for now we only need to know that w has two real, four complex and an infinite countable

number of imaginary poles. Let K be the set of all poles of w and Kˆ be the set of a positive

real pole and poles with positive imaginary parts.

For any n > N we have

∣∣γ4 + u∣∣ > |γ|4 + C = |x+ i y|4 + C

≥ ǫ4n + C for any x ∈ R, y = ǫn, (3.11)

where C is a constant determined by u. When y = ǫn we have

∣∣∣∣1

γ tanh (γH)

∣∣∣∣ =

∣∣∣∣e2xHei 2yH + 1

(x+ i y) (e2xHei 2yH − 1)

∣∣∣∣

=

∣∣e2xHei 2yH + 1∣∣

|x+ i y| |e2xHei 2yH − 1|

=

∣∣e2xH − 1∣∣

|x+ i y| |e2xH + 1| ≤1

|x+ i y| ≤1

ǫn(3.12)

for any x ∈ R. (We used exp (i (2n+ 1) π) = −1 and

∣∣∣∣e2xH − 1

e2xH + 1

∣∣∣∣ ≤ 1

35

to show this.) For large |γ| we have

∣∣∣∣γ4 + u− ω2

γ tanh (γH)

∣∣∣∣ ≥∣∣γ4 + u

∣∣−∣∣∣∣

ω2

γ tanh (γH)

∣∣∣∣ .

Since the RHS of this inequality is positive from Eqn. (3.11) and Eqn. (3.12),

|w (γ)| ≤ 1

|γ4 + u| −∣∣∣ ω2

γ tanh(γH)

∣∣∣≤ 1

ǫ4n + C − ω2

ǫn

(3.13)

for any n ≥ N . Note that the same relationship holds for y = −ǫn.

For γ on the line segment ǫn − i ǫn to ǫn + i ǫn we use the fact that

∣∣e2xHei 2yH + 1∣∣

|x+ i y| |e2xHei 2yH − 1| ≤1

|x+ i y|1 +

∣∣e−2xH∣∣

1 − |e−2xH | ≤EN

ǫN

for any y, n ≥ N , where EN is defined as

1 +∣∣e−2xH

∣∣1 − |e−2xH | ≤

1 +∣∣e−2ǫNH

∣∣1 − |e−2ǫNH | = EN ,

1

|x+ i y| ≤1

ǫN.

From Eqn. (3.11) and the first line of Eqn. (3.13), we have

1

|γ4 + u| −∣∣∣ ω2

γ tanh(γH)

∣∣∣≤ 1

ǫ4N + C − ω2EN

ǫN

for any n ≥ N . The same proof can be applied for the line segment −ǫn − i ǫn to −ǫn + i ǫn.

We have proved that |w (γ)| is bounded on all sides of the contours Cn, n ≥ N where N is

chosen to be large so that the contours are a certain distance away from all the poles of w.

Hence, the expansion of w (γ) becomes, from w (0) = 0,

w (γ) =∑

q∈K

[R (q)

γ − q+R (q)

q

]=∑

q∈Kˆ

[2qR (q)

γ2 − q2+

2R (q)

q

]. (3.14)

Note that the summation on the first line is over all poles of w (γ). Note that R (q) =

−R (−q), since w (γ) is an even function and

− (γ − q) w (γ) = (−γ + q) w (−γ) = (γ + q) w (γ) .

Note that the term∑

2R (q) /q is zero. Indeed, expansion of the function w (γ) γ which

has the same analytic properties and poles as the function w and residues R (q) q at γ = q.

36

Hence, w (γ) γ is expanded as,

w (γ) γ =∑

q∈K

[qR (q)

γ − q+qR (q)

q

]=∑

q∈Kˆ

2γqR (q)

γ2 − q2.

The fact that∑

2R (q) /q is zero can also be confirmed by using the contour integration

of the function w (γ) /γ (Fig. (3.6) in section 3.5 shows this integration). The function

w (γ) /γ is an odd function and has the same poles as the function w (γ) with the residues

R (q) /q. Notice that γ = 0 is not a singular point of w (γ) /γ. Hence, the integration over

the real axis is zero and w (γ) /γ → 0 on the semi-arc with order of A−3 as A→ ∞.

3.3 Fundamental solution

In this section, we derive a fundamental solution for a localized forcing with a single fre-

quency applied on the ice sheet that is occupying the whole surface of the ocean. We use the

Fourier transform to obtain the fundamental solution which is expressed using an infinite

series expansion over all wave modes existing in a floating elastic plate. Then, we show the

complete analytical formulae of the coefficients of the modes and numerical computations

of those coefficients. The fundamental solution obtained here is normalized, that is, the

solution is expressed with dimensionless quantities by scaling the distance and time. With

our normalizing method, the BVP is simplified and expressed without any physical values,

which leads to the representation of the solution being insensitive to certain physical values

such as thickness of the ice. In the following subsections, we will show details of derivation

of the spatial Fourier transform of w (x, y), and then perform analytical integrations of the

inverse Fourier transform.

3.3.1 Mathematical model

As mentioned in the previous section, we will use non-dimensional variables and parameters.

To avoid confusion, we denote the dimensional variable and physical parameters with a bar

on top of them, i.e., (x, y, z, t), φ, w, pa and so on, and the dimensionless variables and

parameters unbarred.

Fig. (3.1) shows the setup of the BVP for the dynamics of an infinite plate floating on

water of depth H metres. We consider a localized forcing pa that may be either a point

load at the origin (x, y) = 0 or a line load on x = 0. We set the localized pressure, unit

force localized at the origin and unit force per length along the line y = 0 using the delta

function as follows:

pa (x, y, ω) =

δ (x, y) ,

δ (x) .(3.15)

37

pa

pw

Ice sheet

Water

Ocean floor z H=-

rz=0

Figure 3.1: Schematic drawing of forcing of an infinite plate.

At the surface −∞ < x, y < ∞, z = 0 the floating plate is described by the thin plate

equation (2.44) of w (x, y, ω) and φ (x, y, 0, ω)

[D∇4

x,y − mω2 + ρg]w + i ωρφ = pa.

In the water −∞ < x, y < ∞, −H < z < 0 the velocity potential φ satisfies Laplace’s

equation (2.9)

∇2x,y,zφ = 0

with the fixed boundary condition Eqn. (2.14) at z = −H

φz = 0.

For a point load, the solution is radially symmetric, thus we may denote the displacement

by wP (r) (or wP (r, ω) when the frequency dependence of the solution has to be emphasized)

where r =√x2 + y2 is the distance from the forcing point. For a line load on x = 0, the

solution is symmetric against the y-axis, thus we denote the displacement by wL (r) or

wL (r, ω) where r = |x|.

3.3.2 Non-dimensional formulation

Here, we rewrite the differential equations for w and φ with dimensionless quantities by

scaling distance and time by the characteristic length and time, respectively, defined by

(Fox [18])

lc =

(D

ρg

)1/4

, tc =

√lcg.

Recall that D = Eh3/12 (1 − ν2) and the dimension of the Young’s modulus is N m−2.

Because the reasons for our choice for these particular scaling factors will only become

clear after the solution is obtained, we will postpone the derivation and some properties

of the scaling until section 3.4. Formulation of dimensionless system of equations is called

38

non-dimensionalization or normalization. Non-dimensional variables are defined by

(x, y, z) =1

lc(x, y, z) , t =

t

tc, ω = tcω.

In terms of non-dimensional variables the plate equation becomes

(D

l4c∇4

x,y −m

t2cω2 + ρg

)w + i

ω

tcφ = pa

which is simplified by dividing both sides by ρglc to give

(∇4

x,y −m

ρlcω2 + 1

)w

lc+ iω

φ

lc√glc

=pa

ρglc.

Finally, we express the appropriate normalizing constants for the functions and physical

parameters

φ =φ

lc√glc, m =

m

ρlc, pa =

pa

ρglc, w =

w

lc, ω = ωtc, H =

H

lc,

to obtain the non-dimensional plate equation

[∇4 −mω2 + 1

]w + iωφ = pa. (3.16)

Laplace’s equation for the velocity potential and the bottom condition (2.42) remain the

same, i.e.,

∇2φ = 0 (3.17)

for −∞ < x, y <∞, −H < z < 0 and

φz = 0 at z = −H. (3.18)

The system of equations from Eqn. (3.16) to Eqn. (3.18) together with the radiation condi-

tion form the BVP, which we will solve for w and φ for a given ω.

3.3.3 Spatial Fourier transform

We solve the system Eqns. (3.16) to Eqns. (3.18) using the Fourier transform in (x, y)-plane

for point loading and on the x-axis for line load. We choose the Fourier transform with

respect to x and y defined as

φ (α, k, z) =

∫ ∞

−∞

∫ ∞

−∞

φ (x, y, z) ei(αx+ky)dxdy (3.19)

39

and the inverse Fourier transform defined as

φ (x, y, z) =1

4π2

∫ ∞

−∞

∫ ∞

−∞

φ (α, k, z) e− i(αx+ky)dαdk.

For the one-dimensional case, the definitions are

φ (α, z) =

∫ ∞

−∞

φ (x, z) eiαxdx, (3.20)

φ (x, z) =1

2π

∫ ∞

−∞

φ (α, z) e− iαxdα. (3.21)

We denote the spatial Fourier transform by using a hat over w and φ.

The Fourier transform of both sides of the Laplace’s equation (3.17) becomes an ordinary

differential equation with respect to z,

∂2φ

∂z2(α, k, z) −

(α2 + k2

)φ (α, k, z) = 0

with a solution

φ (α, k, z) = A (γ) eγz +B (γ) e−γz

where γ2 = α2 + k2 and A, B are unknown coefficients to be determined by the boundary

condition. We find that φ is a function only of the magnitude of the Fourier transform

variables, γ = ‖γ‖ = ||(α, k)||, hence we may now denote φ (α, k, z) by φ (γ, z). We can reach

the same conclusion using the fact that w (x, y) and φ (x, y, z) are functions of r =√x2 + y2

thus the Fourier transforms must be functions of γ =√α2 + k2.

We find the unknown coefficients A and B from the Fourier transformed ocean floor

condition that φz

∣∣∣z=−H

= 0 to be A (γ) = CeγH , and B (γ) = Ce−γH . Thus, we obtain the

depth dependence of the Fourier transform of the potential

φ (γ, z) = φ (γ, 0)cosh γ (z +H)

cosh γH. (3.22)

At the surface, z = 0, differentiating both sides of Eqn. (3.22) with respect to z, the vertical

component of the velocity is

φz (γ, 0) = φ (γ, 0) γ tanh γH. (3.23)

Using this relationship to substitute for φz in the non-dimensional form of the kinematic

condition, we find that

φ (γ, 0) =iωw (γ)

γ tanh γH. (3.24)

Thus, once we find w (γ) then we can find φ (γ, z) using Eqn. (3.23) and Eqn. (3.24).

40

The Fourier transform of the plate equation (3.16) is also an algebraic equation in the

parameter γ (γ4 + 1 −mω2

)w + iωφ = 1.

Hence, substituting Eqn. (3.24), we have the single algebraic equation for each spatial Fourier

component of w (γ4 + 1 −mω2 − ω2

γ tanh γH

)w = 1. (3.25)

Notice that we have used the fact that the Fourier transform of the delta function is 1.

We find that the spatial Fourier transform of the displacement of the ice sheet for the

localized forcing, both point and line, is

w (γ) =1

d (γ, ω)(3.26)

where

d (γ, ω) = γ4 + 1 −mω2 − ω2

γ tanh γH. (3.27)

We call the function d (γ, ω) the dispersion function, and the associated algebraic equation

d (γ, ω) = 0 the dispersion equation for waves propagating through an ice sheet. This

dispersion equation (and the Fourier transform Eqn. (3.26)) has been previously derived by

Kheisin ([32] chapter IV), Fox and Squire [21].

qT

-qT q1 q2 q3

Figure 3.2: Illustrates how to find the positions of the roots of the dispersion equation. Onthe right, the real roots ±qT and on the left the imaginary roots iq1, iq2, iq3, ..... Dottedcurves are tanh function on the left and tan function on the right. Solid curves are thecurves of Eqn. (3.28).

Our task now is to derive the inverse Fourier transform of Eqn. (3.26). We notice that

the roots of the dispersion equation for a fixed ω, are the poles of the function w (γ), which

are necessary for calculation of integrals involved in the inverse Fourier transform. The

roots of the dispersion equation are computed numerically using computer codes in MatLab

given by Fox and Chung [19]. The root-finder computer code is due to Fox and was initially

written to implement the mode matching technique in Fox and Squire [21]. Later, we will

41

use the observation that the dispersion equation d (γ, ω) is an even function of γ, and hence

if q is a root then so is −q. Fig. (3.2) shows the curves of the tan and tanh functions

intersecting the curves±ω2

γ (γ4 + 1 −mω2). (3.28)

and illustrates how the real and imaginary roots can be found. We notice that from observing

the curves in Fig. (3.2), there are a pair of two real roots ±qT (qT > 0) and an infinite number

of pure imaginary roots ± i qnn=1,2,..., (qn > 0). We notice that for typical cases mω2 ≤ 1,

an imaginary root i qn is found in each interval(n− 1

2

)π < γnH < nπ as shown in Fig. (3.2).

For fixed ω 6= 0, it is known (Fox and Squire [21], Chung and Fox [10]) that four complex

roots occur as plus and minus complex-conjugate pairs ±qD and ±q∗D (Re (qD) > 0 and

Im (qD) > 0).

We note that the dispersion equation represents a relationship between the spatial

wavenumbers γ and the radial frequency ω, which is how the name “dispersion” came

about.

We may also solve for the velocity potential at the surface of the water

φ (γ, 0) =iωw

γ tanh γH=

iω

γ tanh (γH) d (γ, ω)(3.29)

which is also a function of γ only and has the same poles as w (γ) does.

The Fourier transform of the differential equations can be interpreted as wave-like forcing

of the ice sheet, i.e.,

pa (x, y, t) = pa (α, k, ω) ei(ωt−αx−ky)

where pa (α, k, ω) is the amplitude of the wave-like forcing or a Fourier transform of the

wave-like force which we set to be one. Therefore, superposition or the inverse Fourier

transform of the solution under this wave-like force represents the response of a plate to a

localized force expressed by the delta function in Eqn. (3.15). Since the waves are radially

symmetric, pa can be written as a function of r,

pa (r, t) = pa (γ, ω)J0 (γr) eiωt

where pa (γ, ω) is the Fourier component of radially symmetric wave-like function.

3.3.4 The inverse Fourier transform

We calculate the displacement w (x, y) by performing the two dimensional inverse Fourier

transform of w (γ) in Eqn. (3.26) and, since w (γ) is radially symmetric, the inverse transform

42

may be written (Bracewell [6])

wP (r) =1

2π

∫ ∞

0

w (γ) γJ0 (γr) dγ (3.30)

where J0 is Bessel function and r is the distance from the point of forcing. The response to

line forcing is given by the inverse Fourier transform of w (γ) in the x-axis and since w (γ)

is an even function, this is

wL (r) =1

π

∫ ∞

0

w (γ) cos (γr) dγ (3.31)

where again r = |x| is the distance from the line of forcing. Note that the factors 1/2π and

1/π result from the form of the Fourier transform in two and one dimensional spaces that

we defined in Eqn. (3.19) and Eqn. (3.20).

The integrals of Eqn. (3.30) and Eqn. (3.31) are calculated using the singularities of w,

i.e., the roots of the dispersion equation. First, since w (γ) is an even fractional function

that equals zero when γ = 0 and is bounded in the whole plane except in regions around

its poles, we find that w (γ) can be expressed using the Mittag-Leffler expansion in section

3.2. Pairing each pole q with its negative counterpart −q, gives

w (γ) =∑

q∈Kˆ

2qR (q)

γ2 − q2(3.32)

where R (q) is the residue of w (γ) at γ = q. We denoted the set of roots of the dispersion

equation with positive imaginary part and a positive real root by Kˆ. Thus, this set is Kˆ =

qT, qD,−q∗D, i q1, i q2, i q3, · · · . Note that the rest of the roots of the dispersion equation

are the negative of the values in Kˆ. By substituting this expansion into the integrals in

Eqn. (3.30) and Eqn. (3.31), we are able to perform the integration and write each result

as a summation over the roots q ∈ Kˆ.

The residues R (q) can be calculated using the usual formula. Since each of the poles of

w (γ) is simple, the residue R (q) at a pole q can be found using the expression

R (q) =

[d

dγd (γ, ω)

∣∣∣∣γ=q

]−1

=

[4q3 + ω2

(qH + tanh qH − qH tanh2 qH

q2 tanh2 qH

)]−1

. (3.33)

As each pole q is a root of the dispersion equation, we may substitute tanh qH = ω2/ (q5 + uq),

where for brevity we have defined u = (1 −mω2). The residue may then be given as the

43

1 2 3 4

10−5

100

q

Mag

nitu

de o

f R(q

)

Figure 3.3: Graphs of |R (i q)| as function of real number q, with tanh-functions (solid line)and polynomial expression (dashed line).

rational function of the pole

R (q) =ω2q

ω2 (5q4 + u) +H[(q5 + uq)2 − ω4

] . (3.34)

This form avoids calculation of the hyperbolic tangent which becomes small at the imag-

inary roots. This causes numerical round-off problems and rapid variation in computing

Eqn. (3.33) since qnH tends to nπ as n becomes large, which makes tan qnH become small.

Fig. (3.3) shows the graphs of the two expressions, Eqn. (3.33) and Eqn. (3.34) of the

residue for imaginary argument. The imaginary roots i q1, i q2, i q3, ... are located where the

two curves in Fig. (3.3) coincide (the spiky parts of the solid curve). There are two such

points at a spike and the root is the one on the left. Eqn. (3.33), from the direct calculation,

is a rapidly varying function near the roots i qn∞n=1, hence a small numerical error in the

values of the roots will result in a large error in the residue. In contrast, Eqn. (3.34) gives

us a smooth function, and the resulting calculation of the residue is stable.

Using the identities (Abramowitz and Stegun [4] formula 11.4.44 with ν = 0, µ = 0,

z = − i q and a = r) ∫ ∞

0

γ

γ2 − q2J0 (γr) dγ = K0 (− i qr) (3.35)

for Im q > 0, r > 0, where K0 is a modified Bessel function and an identity between

the modified Bessel function, we notice that qT term of Eqn. (3.32) may pose a problem.

However, considering that qT = limεց0 (qT + i ε), we are able to apply Eqn. (3.35) to all

terms of Eqn. (3.32). Alternatively, we may put an additional imaginary term i βω, β > 0,

representing damping, to the dispersion equation

γ4 + iβω + 1 −mω2 − ω2

γ tanh γH

44

so that every element in Kˆ has positive imaginary part. Addition of the damping term

proportional to wt ensures that the inverse transform satisfies the radiation condition. We

omit the damping term here for the sake of algebraic simplicity.

We can calculate the integral of the inverse Fourier transform of w and, using the identity

between K0 and Hankel function of the first kind (Abramowitz and Stegun [4] formula 9.6.4),

K0 (ζ) = iπ2H

(1)0 (i ζ) for Re ζ ≥ 0, the displacement for point forcing may be written in the

equivalent forms

wP (r) =1

π

∑

q∈Kˆ

qR (q)K0 (− i qr)

=i

2

∑

q∈Kˆ

qR (q)H(1)0 (qr) (3.36)

where the subscript P of wP indicates the response to a point load.

The identity (Erdelyi [1] formula 1.2 (11) with x = γ, y = r, and a = − i q)

∫ ∞

0

cos (γr)

γ2 − q2dγ = − π

i 2qexp (i qr) (3.37)

for Im q > 0, r > 0, gives the surface displacement for line forcing

wL (r) = i∑

q∈Kˆ

R (q) exp (i qr) (3.38)

where the subscript L of wL indicates the response to a line load.

3.3.5 Modal expansion of the solutions

We have shown that the fundamental solution for finite water depth H is found by first

finding the roots in the upper-half plane, Kˆ, of the dispersion Eqn. (3.27). Numerical

calculation is achieved by truncating the sum after some finite number of roots. Straightfor-

ward computer code (in MatLab) to find the roots has been given by Fox and Chung [19].

After calculating the residue for each root given in Eqn. (3.34), the sum in Eqn. (3.36) can

be calculated by separating it into

wP (r) =i

2qTRTH

(1)0 (qTr) − Im

[qDRDH

(1)0 (qDr)

]+

1

π

∞∑

n=1

i qnRnK0 (qnr) (3.39)

We used the identities − i (−qD) = (− i qD)∗, R (q∗) = (R (q))∗ andH(1)0 (−ζ∗) = −

(H

(1)0 (ζ)

)∗,

and denote the residues for the poles in Kˆ by RT = R (qT), RD = R (qD), and Rn = R (i qn),

n ∈ N, respectively. Note that R (−γ∗D) → −R∗D as β → 0. Since, from Eqn. (3.34),

45

|qR (q)| ∝ |q|−8 as |q| increases, the sum may be terminated after a relatively small number

of terms without significant error, as shown in Fig. (3.4). We also notice that the smaller

the depth H , the fewer number of roots we need to achieve the desired accuracy, since for

a given n, qn increases as H becomes small.

100

101

102

10−8

10−6

10−4

10−2

Index of the evanescent mode n

Res

idue

R (

qn )

Figure 3.4: Log-log plot of the residues corresponding to the evanescent modes. +, * and♦ indicate for ω = 0.1, 1.0, 10 respectively. The water depth is H = 20π (deep water).

The surface displacement for line forcing is given by the sum

wL (r) = iRT exp (i qTr) − 2 Im [RD exp (i qDr)] + i

∞∑

n=1

Rn exp (−qnr) . (3.40)

We have written the solutions (3.39) and (3.40) in terms of the travelling, damped travelling

and evanescent mode in order to emphasize the behavior of each mode.

We consider here the energy flux due to the wave produced by the force at r = 0. From

the equation of motion (2.5) and v = ∇φ, the energy crossing a surface S in a time period

[t, t+ T ] is

ρ

∫ t+T

t

∫

S

φt (x, y, z, t)φn (x, y, z, t) dσdt.

For time-harmonic waves, i.e., φ = Re [φ exp (iωt)], and T = 2π/ω, the above equation

becomes

− i 2ρπ

∫

S

[φ∗φn − φφ∗n] dσ = 2ρπ Im

∫

S

φ∗φndσ. (3.41)

We notice that if S is the closed surface of domain V, then from Green’s theorem, we have

∫

S

[φ∗φn − φφ∗n] dσ =

∫

V

[φ∗∇2φ− φ∇2φ∗

]dV = 0, (3.42)

46

which states that there is no net energy propagation to infinity, i.e., the law of energy

conservation. Hence, only the travelling mode H(1)0 (qTr) for the point load and exp (iqTr)

for the line load carry the energy to infinity since the rest of the solution decays exponentially.

Note that the decaying rate of the Hankel function of a real variable is 1/√r and the integral

on the cylindrical surface gives dσ = rdrdθdz, thus we have non-zero energy flux for the

travelling mode of the point load response.

Since iRn in Eqn. (3.39) and Eqn. (3.40) is real for each n = 1, 2, · · · , the evanescent

modes in Eqn. (3.40) and (3.39) are always real. Hence, the only imaginary term in the

responses due to point or line load that give non-zero in Eqn. (3.41) is the coefficient of the

travelling wave. This corresponds to the travelling waves being the only modes that carry

energy away from the load. The damped-travelling and evanescent modes contribute motion

that is in phase with the forcing, while the travelling mode has a component in quadrature

to the forcing.

The velocity potential in the water can be found using Eqn. (3.22) to Eqn. (3.24), giving

φP (r, z) = −ω2

∑

q∈Kˆ

R (q)

sinh qHH

(1)0 (qr) cosh q (z +H)

for the point load and

φL (r, z) = −ω∑

q∈Kˆ

R (q)

q sinh qHexp (i qr) cosh q (z +H) .

for the line load.

Note that the zeros of d (γ) γ tanh γH are the same as those of the dispersion equation.

Thus the singularities of φ and w are the same. The term sinh qH is close to zero for

most imaginary roots so these expressions are not directly suitable for numerical compu-

tation. The substitution following Eqn. (3.33) may be used to give computationally stable

expressions.

Each term of wP and wL in Eqn. (3.39) and Eqn. (3.40) is a natural mode of a floating

ice sheet whose wavenumber is a root of the dispersion equation. We showed that the mode

of qT, the travelling mode carries the wave energy outwards, and the modes of qD and i qn,

n = 1, 2, ..., the damped travelling and evanescent modes, are exponentially decaying.

3.3.6 Summary of the analytic structure of w (γ)

Here we summarize and clarify the analytic properties of the complex valued function w (γ)

that have been discussed so far.

It may be worth reminding ourselves that the Mittag-Leffler expansion of w given by

Eqn. (3.14) has an extra term w (0) = 0 and is unique. Hence, even though a function such

47

as w (γ) + 1 has the same singularities and residues, the resulting series expansion will be

different. This uniqueness of the expansion guaranties the uniqueness of the solution derived

using the inverse Fourier transform of the series expansion form.

We consider the question of whether we can reconstruct the Fourier transform w from

the modified formula of residues given by Eqn. (3.34) and the positions of the roots given

by the dispersion equation d (γ) = 0. In addition to the residues and poles, we require that

w (0) = 0, to uniquely reconstruct w. The answer to the question is yes and no depending

on how the formulae are used. It is obvious that the function w (γ) for γ ∈ C can be

reconstructed from q ∈ Kˆ and R (q) (either Eqn. (3.33) or Eqn. (3.34)) as

w (γ) =∑

q∈Kˆ

2qR (q)

γ2 − q2.

However, we find the following

w (γ) 6= R (γ)∑

q∈Kˆ

2q

γ2 − q2(3.43)

despite the fact that near γ = q ∈ Kˆ the left and the right hand sides share the same

analytic property, i.e.,

∫

Cε(q)

w (γ) dγ =

∫

Cε(q)

R (γ)∑

q∈Kˆ

2q

γ2 − q2dγ

where Cε (q) is a circular contour of small radius ε around a pole q. The reason for (3.43)

is because the function R (γ) defined by Eqn. (3.33) and Eqn. (3.34) introduce zeros and

singularities of their own, which change the analytic properties of the right hand side of

Eqn. (3.43).

It may seem trivial that the formulae of the residues Eqn. (3.33) and Eqn. (3.34) are

valid only at the pole γ = q, nevertheless we have confirmed Eqn. (3.43).

3.4 Deep water solution

It has been mentioned that the imaginary roots lie in the interval qn ∈ ((n− 1/2)π/H, nπ/H)

as seen in Fig. (3.2). Furthermore qn become equally spaced nπ/H as the depth H becomes

large. Therefore the summation over the evanescent modes is taken at the equally spaced

points which will become closely placed as H tends to infinity. This inspires us to find an

integral expression of the summation when H = ∞.

We show here that the infinite summations over the evanescent mode of the solutions

take particularly simple forms when the water is very deep, i.e., in the limit H → ∞. In

48

the limit i qn form a continuum with equal density over the imaginary axis. If the residue

R (i qn) decreases proportional to 1/H , then the infinite sum over these roots may then

be calculated as an integral over the positive imaginary semi-axis. It will be shown that

for both types of forcing, the solutions are then given by a sum over special functions at

wavenumbers given by the roots of a fifth order polynomial. Furthermore, in the deep-water

case H ≈ ∞, the solution will be shown to be qualitatively unaffected by setting m = 0

for typical values of m for sea ice, i.e., smaller than 0.1. The governing non-dimensional

equations then contain no coefficients depending on the physical properties of the ice or

water.

The residue expressed by Eqn. (3.33) oscillates more rapidly as the water depth H tends

to infinity due to the tangent function, thus the formula becomes unsuitable for numerical

computation of the residue. On the other hand, the formula (3.34) is smooth and rapidly

decaying function as H becomes large. The residue at a root of the dispersion equation in

Eqn. (3.34) tends to

R (q) =1

H

ω2q1Hω2 (5q4 + u) + (q5 + uq)2 − ω4

(3.44)

→ 1

HQ (q)

as H → ∞, where

Q (q) =ω2q

(q5 + uq)2 − ω4. (3.45)

The sum over the imaginary roots in the response to point loading in Eqn. (3.39) is then

1

π

∞∑

n=1

i qnRnK0 (qnr) →1

π2

∞∑

n=1

inπ

HQ

(inπ

H

)K0

(nπHr) π

H

→ 1

π2

∫ ∞

0

i ξQ (i ξ)K0 (ξr) dq. (3.46)

Similarly, the sum over imaginary roots for line load in Eqn. (3.40) is

i∞∑

n=1

Rn exp (−qnr) →1

π

∫ ∞

0

iQ (i ξ) exp (−ξr) dξ. (3.47)

The integrals in Eqn. (3.46) and Eqn. (3.47) can be evaluated by first writing Q as

a sum over simple poles, as we did for the inverse Fourier transform. We notice that

Q (q) = (v (q) − v (−q)) /2 where

v (i ξ) =q

q5 + uq − ω2(3.48)

49

Then ω 6= 0, v (q) and v (−q) have no poles in common and hence poles of v are also poles

of Q. It follows that all poles of Q are plus and minus the poles of v. Since v and w (in the

limit H → ∞) coincide for arguments with positive real part, the poles of w with positive

real part, i.e. qT, qD, and q∗D, are also roots of v and hence Q. There are two further roots

of v with negative real part, which we denote qE and q∗E, with qE chosen to have positive

imaginary part. It can be shown that Im (qE) > 0 for any finite mass density m, and hence

qE is never real.

Real axis

Imaginary axis

*

Figure 3.5: Relative positions of the poles for the deep-water response. ∗, and ♦ indicateqT, qD (q∗D) and qE (q∗E).

Let Kv = qT, qD, q∗D, qE, q∗E , shown in Fig. (3.5), denote the set of poles of v. The

residue of v at a pole q ∈ Kv is

Rv (q) =q2

5ω2 − 4uq. (3.49)

Note that the residues at qT, qD and q∗D, are the same as defined in section 3.2.4. We

respectively denote these RT, RD and R∗D, as before, respectively. Write RE = Rv (qE) and

hence R∗E = Rv (q∗E). The residues of Q are 1/2 the residue of v.

The integral in Eqn. (3.46) may be evaluated using the fractional function expansion

i ξQ (i ξ) = −∑

q∈Kv

q2Rv (q)

ξ2 + q2(3.50)

and the integral (Abramowitz and Stegun [4] formula 11.4.47 with ν = 0, t = k, r = a),

∫ ∞

0

K0 (ξr)

(ξ2 + q2)dξ =

π2

4q[H0 (qr) − Y0 (qr)] ,

which holds for Re q > 0 since we always take r > 0. Here H0 is a Struve function of

zero order and its power expansion for numerical computation is shown in appendix A. For

50

notational brevity, we use the function h (rz) = H0 (rz) − Y0 (rz). Performing the integral

in Eqn. (3.46) and combining conjugate pair poles, Eqn. (3.39) for the response to point

load in the deep water limit can be written in terms of poles of v as,

wP (r) =i

2qTRTH

(1)0 (qTr) − Im

[qDRDH

(1)0 (qDr)

]

− qTRT

4h (qTr) −

1

2Re (qDRDh (qDr)) +

1

2Re (qEREh (−qEr)) . (3.51)

The pole −qE of Q has been used since Re (−qE) > 0. Note that only the first, travelling

wave term is imaginary, which corresponds to that mode being the only one that propagates

energy away from the point of forcing. The remaining terms give displacements that are in

phase with the forcing.

The integral in Eqn. (3.47) for line load may be found using the expansion

iQ (i ξ) =∑

q∈Kv

ξRv (q)

ξ2 + q2

and the integral (Abramowitz and Stegun [4] formula 5.2.13 with t = k/q and hence z = qr

and formula 5.2.7)

∫ ∞

0

ξ exp (−ξr)ξ2 + q2

dξ = −Ci (qr) cos (qr) − si (qr) sin (qr)

holding for Re (q) > 0 since r is positive real. Here Ci and si are cosine integral and

sine integral functions (see appendix A), respectively. As with the point-forcing case, the

notation and computation are simplified by defining the function g (qr) = −Ci (qr) cos (qr)−si (qr) sin (qr). Then, combining conjugate-pair poles, Eqn. (3.40) for the response to line

forcing in the deep water limit can be written as,

wL (r) = iRT exp (i qTr) − 2 Im [RD exp (i qDr)]

+RT

πg (qTr) +

2

πRe [RDg (qDr)] +

2

πRe [REg (−kEr)] . (3.52)

where, again, the pole −qE has been used.

We emphasize here that the denominator of the function v in Eqn. (3.48) is not meant

to be the dispersion function of the deep water problem, but it is only the partial expression

of the dispersion function. The full deep-water dispersion equation may be written as

γ4 −mω2 + 1 − ω2

γ sgn (Re γ)= 0 (3.53)

where sgn (Re γ) is sign-function of the real part of γ, which has continuous singularity

51

on the imaginary axis. We have obtained the deep-water solution without dealing with

this continuous singularity. The function γ sgn (Re γ) is often defined using the limit of√γ2 ± i ε2 as ε → 0, which is defined on a two-sheeted Riemann surface and denoted by

|γ|± because it is equal to |γ| on the real axis.

3.5 Computation of the solutions

3.5.1 Static load

We can find the displacement wP and wL for static loading by setting ω = 0 in Eqn. (3.27),

to give

d (γ) = γ4 + 1 (3.54)

and following the same procedure shown in section 3.2. The Fourier transform of the dis-

placement is

w (γ) =1

γ4 + 1

which has four poles, ±ei π/4, ±ei 3π/4, the set of poles in the upper half plane being Kˆ =ei π/4, ei 3π/4

. The residue of w (γ) at a pole q is

R (γ) =1

(γ4 + 1)′

∣∣∣∣γ=q

=1

4q3.

Hence, the solution is, from Eqn. (3.36)

wP (r) =1

4π

e− i π/2K0

(− i eiπ/4r

)+ e− i 3π/2K0

(− i ei 3π/4r

)

=i

4π

−K0

(e− i π/4r

)+K0

(eiπ/4r

)

= −kei (r)

2π(3.55)

where kei (ζ) is the Kelvin function (of zero order) and we have used the identity (Abramowitz

and Stegun [4] formulae 9.9.2 and 9.6.32)

i 2 kei (ζ) = K0

(ei π/4ζ

)−K0

(e− iπ/4ζ

).

The solution given by Eqn. (3.55) is the same solution derived by Wyman [51]. Using

Eqn. (3.38), we can derive the static line-loading solution

wL (r) =1

2exp

(−r√2

)

52

3.5.2 Deflection at the location of forcing

It appears that expression (3.39) has a singularity at the origin since the Hankel function and

the modified Bessel function behave as log-like function near the origin. Indeed, Strathdee,

Robinson and Haines [47] erroneously stated that the solution for a floating thin plate had

a singularity at the point of forcing. However, since the plate equation is fourth order and

Laplace’s equation is second order, we would expect the solution to be smooth everywhere,

including at r = 0. We show here that the summation is indeed bounded everywhere

and derive an expression for the displacement at the point of forcing that is convenient to

compute.

The modified Bessel function K0 (ζ) has the polynomial form

K0 (ζ) = − log

(ζ

2

)I0 (ζ) +

∞∑

l=0

(ζ/2)2l

(l!)2 ψ (l + 1) ,

where I0 and ψ are the modified Bessel function and the Psi function, respectively (Abramowitz

and Stegun [4] formulae 9.6.12 and 9.6.13, appendix A). For small |ζ |, K0 (ζ) ≈ − log ζ + c

where c = log 2 + ψ (1) since I0 (0) = 1. Hence, as r → 0, the infinite series of wP becomes

a series of log-function of r,

wP (r) → 1

π

∑

q∈Kˆ

qR (q) (− log (− i qr) + c)

= −1

π

∑

q∈Kˆ

qR (q) log (− i q) +c− log r

π

∑

q∈Kˆ

qR (q) . (3.56)

We notice that the second term becomes singular as r → 0.

Consider now a contour integration of the function w (γ) γ, anti-clockwise along the

contour shown in Fig. (3.6). The arc of radius A is chosen to avoid the poles on the

imaginary axis, and the arcs around the poles on the real-axis are taken to have small

radius. Since w (γ) γ is an odd function, the integral over the real axis, including the two

Re

Im

-qT

xx

x

x

x

x

iqn

qT

x

qD-qD

x

*

-A A

Figure 3.6: Contour used for integration with approximate pole positions shown as crosses.

53

small arcs, is zero. Further, as A → ∞, w (γ) γ tends to zero faster than A−2 on the semi-

circle of radius A, and the integral over the semi-circle tends to zero as A→ ∞. Hence the

integral over the whole contour tends to zero as A→ ∞. Since this limit equals a constant

multiplied by the sum of the residues of w (γ) γ at the poles enclosed within the contour,

the sum of residues of w (γ) γ at poles in the upper half plane is zero, i.e.,

0 =

∫

C

w (γ) γdγ = 2π i∑

q∈Kˆ

qR (q) .

We immediately see that the term multiplied by (c− log r) /π in Eqn. (3.56), which is the

singular part, is zero. Thus at r = 0 the complex displacement takes the finite value

wP (0) = −1

π

∑

q∈Kˆ

qR (q) log (− i q) = −1

π

∑

q∈Kˆ

qR (q) log (q) . (3.57)

Since qR (q) decreases as q−8n ∝ n−8 for the evanescent modes, relatively few terms are

required to evaluate this sum accurately. Again, the computation of the solution requires

fewer modes when the depth H is small because of the same reason mentioned earlier.

Fig. (3.7) shows the displacement at the point of forcing as a function of frequency and

for water depths H = 20π, 2π, and 0.2π. The depths are taken as multiples of 2π since that

is the wavelength of the travelling wave with unit non-dimensional wavenumber.

Water depths greater than 20π give visually identical deflections, so the curve for H =

20π may be taken as the deep-water solution. That is the solution for H = 2π is nearly

identical to the deep-water solution at all frequencies, so that H = 2π may be considered

deep for point load.

We notice by spliting Eqn. (3.57) into the real and imaginary parts,

wP (0) = −1

π

(qTRT log (qT) + 2 Re [qDRD log (− i qD)] +

∞∑

n=1

i qnRn log (qn)

)+ i

qTRT

2

that the imaginary part is just the coefficient of the travelling mode. This corresponds to

the travelling mode being the only mode that carries energy away from the point of forcing.

The displacement at the point of forcing when the ice is floating on deep water may be

found by evaluating the finite-depth solution in Eqn. (3.57) in the limit H → ∞ as described

in the previous section. The sum over evanescent modes is

−1

π

∞∑

n=1

i qnR (i qn) log (qn) → − 1

π2

∫ ∞

0

i qQ (i q) log (q) dq.

Using the expansion in Eqn. (3.50) and the integral identity (Erdelyi [2] section 14.2 (24)

54

0.1 1 100

0.05

0.1

0.15

0.2

|w P

(0)|

0.1 1 100

0.5

1

1.5

arg(

w P

(0))

Non−dimensional radial frequency ω

H=20π H=2π H=0.2π

Figure 3.7: Top to bottom shows the magnitude and argument part of the complex dis-placement at the point of forcing as a function of non-dimensional frequency. The non-dimensional water depths H = 2π × 10, 2π × 1, 2π × 0.1, are shown. In all cases we takem = 0.

with a = − i q, and y = i q) ∫ ∞

0

log x

x2 + q2dx =

π

2qlog q

for Re q > 0, we find that

wP (0) = iqTRT

2− qTRT

2πlog (qT)− 1

πRe (qDRD log (−qD))− 1

πRe (qERE log (−qE)) . (3.58)

This gives a very simple expression for finding the displacement at the point of forcing

when the water is deep. Hence, Eqn. (3.58) gives an efficient route to the result computed

numerically by Nevel [37].

Because the exponentials in Eqn. (3.40) are continuous everywhere, the displacement at

the line of forcing may found by setting r = 0 directly to give

wL (0) =∑

q∈Kˆ

iR (q) . (3.59)

Using this expression, the absolute value, and argument parts of the complex deflection,

plotted as a function of frequency, are shown in Fig. (3.8) for water depths of H = 20π, 2π,

55

and 0.2π.

0.1 1 100

0.2

0.4

|w L

(0)|

0.1 1 100

0.5

1

1.5

2

arg(

w L

(0))

Non−dimensional radial frequency ω

H=20π H=2π H=0.2π

Figure 3.8: Top to bottom shows the magnitude, argument part of the complex displacementon the line of forcing as a function of non-dimensional frequency for the non-dimensionalwater depths H = 2π × 10, 2π × 1, 2π × 0.1. We take m = 0.

We see little difference in Fig. (3.8) between the graphs for H = 2π and 20π, thus H

greater than 2π may be considered as deep for line forcing. However, at the frequency lower

than 1 the graphs show difference between H = 2π and 20π, hence H = 2π may not be

considered as deep at low frequency for line forcing.

Because the cosine integral has a log-like singularity at the origin, Eqn. (3.52) is not

directly suitable for computing the displacement on the line of forcing in the deep-water

limit. However, expanding Ci (qr) in terms of the log plus power series and using the

identity∑

q∈KvRv (q) = 0 allows us to write

wL (0) = iRT − RT

πlog (w) − 2

πRe [RDlog (−qD)] − 2

πRe [RElog (−qE)] (3.60)

which is easily evaluated to give the displacement on the line of forcing when the water is

deep.

56

3.5.3 Derivatives of the deflection

As mentioned, the modified Bessel function K0 and Hankel function H(1)0 have a log-like

singularity at r = 0. Hence depending on the software package used for numerical compu-

tation, the solutions may not be stable near the origin. In this section we show alternative

formulae for the derivatives of the solution that is stable near r = 0 using a power series

expansion of I0 given in Abramowitz and Stegun [4] formula 9.6.10 with ν = 0. We write

the deflection as the sum of the deflection at r = 0 and the remainder,

wP (r) = wP (0) +1

π

∑

q∈Kˆ

qR (q)

(∞∑

l=1

(−q2r2)l

4l (l!)2

(ψ (l + 1) − log

(− i qr

2

)))(3.61)

for all r. Note that near the origin the deflection behaves as (log (qr) + c) r2 and thus the

first derivative of the deflection near the origin is

d

dr

((log (qr) + c) r2

)= r (1 + 2 (log (qr) + c)) .

Therefore, we have w′P (0) = 0 as expected. It follows that the displacement function

obtained by the method above is regular everywhere.

Measurements of flexural response of ice sheets can be made using the strain gauge which

measures the curvature of the upper surface of the ice sheet. Measuring the strain will be

examined further in section 3.7. The second derivative with respect to r for each term in

Eqn. (3.61) with l ≥ 1 has the r-dependence

d2

dr2

((log (qr) + c) r2l

)= r2l−2

(log (qr)

(4l2 − 2l

)+ 4l2c+ 4l − 2lc− 1

)

which is non-zero at r = 0 only for the l = 1 term. Since∑

q∈Kˆ R (q) q3 = 1/2, evaluating

the integral of w (γ) γ3 on the contour shown in Fig. (3.6), the second derivative of the l = 1

term in Eqn. (3.61) can be written

d2

dr2wP (r) ≈ 1

8π

2 log

(r2

)+ 3 − ψ (2) +

∑

q∈Kˆ

q3R (q) log (− i q)

for small r. So we see that the strain has a singularity at the origin that behaves like log r

and is in phase with the forcing. Using formulae for the derivatives (recursive formulae in

57

Erdelyi [2] formula) of the Hankel function, we have the first and second derivatives,

d

drwP (r) =

∑

q∈Kˆ

q2R (q)H(1)1 (qr) , (3.62)

d2

dr2wP (r) =

∑

q∈Kˆ

q2R (q)

qH

(1)2 (qr) − 1

rH

(1)1 (qr)

. (3.63)

We may use simpler formulae to compute the deflection using asymptotic formulae of

Hankel function of the solutions. It is shown in Abramowitz and Stegun ([4] formula 9.2.3)

that

H(1)0 (ζ) ∼

√2

πζexp i (ζ − π/4)

for large |ζ |. Because the terms other than the travelling mode decay exponentially, selecting

just the term due to the travelling mode in Eqn. (3.51) gives the displacement at far field

as

wP (r) ∼√qTRT√2π

exp i (qTr + π/4)√r

(3.64)

for large r.

10−1

100

101

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Non−dimensional radial frequency

Coe

ffici

ent o

f far

fiel

d de

flect

ion

Figure 3.9: Curves of the coefficients of wP (solid) and wL (dash-dot) in the far field givenby Eqn. (3.64) and Eqn. (3.65) respectively.

Equivalently, we find from Eqn. (3.40) and Eqn. (3.52) that the deflection far from a line

load is

wL (x) ∼ RT exp i (qT |x| + π/2) (3.65)

for large |x|. Fig. (3.9) shows the coefficients of Eqn. (3.64) and Eqn. (3.65). The maximum

response of point load at the far field is achieved at ω = 0.81 and that of line forcing at

58

0 2 4 6 8 10 12 14 16

0

0.05

0.1

0.15

r

wP(

r)

0 2 4 6 8 10 12 14 16

0

0.05

0.1

0.15

r

wP(

r)

0 2 4 6 8 10 12 14 16

0

0.05

0.1

0.15

r

wP(

r)

| wP|

Re( wP)

Im( wP)

Figure 3.10: Amplitude and phase of the deep-water displacement function w (r, ω) of pointload at various non-dimensional frequencies ω = 0.2, 1.0 and 5.0 from the top respectively.

ω = 0.73.

All graphs of the solutions shown in Fig. (3.10) and Fig. (3.11) are generated using

MatLab, and the computer codes are direct implementation of the formulae reported here.

The special functions are computed using the built-in functions of MatLab, which are stable

enough for small r. The direct formula given by Eqn. (3.63) is also stable enough for small

r and graphs of the strain are shown in section 3.6. However it is also possible to compute

them using the power expansions of the special functions, although that computation is not

as fast as the built-in functions.

Curves in Fig. (3.10) show deflections for various non-dimensional radial frequencies

ω = 0.2, 1.0, 5.0 for the deep-water case. At low frequency ω = 0.2 the deflection is nearly

identical to the static solution having the imaginary part of the deflection nearly zero.

Curves in Fig. (3.11) show comparison of the deflection for a line-loading case between the

non-dimensional water depth H = 2π and ∞ at various non-dimensional radial frequencies.

59

0 5 10 15−0.4

0

0.4

r

wL

ω = 1.0

0 5 10 15−0.1

0

0.2

0.4

r

| wL|

ω = 0.2

H=∞ H=2π

0 5 10 15−0.1

0

0.2

0.4

r

wL

ω = 0.2

H=∞,Re part H=∞,Im part H=2π,Re part H=2π,Im part

0 5 10 15−0.1

0

0.1

r

wL

ω = 5.0

| wL|

Re( wL)

Im( wL)

Figure 3.11: Amplitude and phase of deep-water displacement function w (r, ω) for the lineload problem at various non-dimensional frequencies ω = 0.2, 1.0 and 5.0. The displacementat ω = 0.2 forH = 2π is plotted, since at a higher frequency, the response is indistinguishablebetween H = 2π and ∞.

60

The effects of the water depth can be seen only for the low frequency case as expected from

the deflection at the origin in Fig. (3.8).

3.6 Scaling of the solution

An advantage of scaling or non-dimensionalization of the solution obviously is simplification

of the system of equations. However, as shown by Fox [18], an appropriate scaling method

enables us to find a solution which represents all physical solutions for ranges of physical

parameters, so that we only need to obtain the non-dimensional solution and the charac-

teristic length in order to compute all physical solutions. We have not given theoretical

justifications for our particular choice of the characteristic length and time, other than the

fact that the scaling factors seemed well suited for the plate equation and the dispersion

equation. It may seem illogical to say that appropriate scaling factors can only be found

after the system of equations is solved and analytical solutions are derived. A properly

scaled solution gives us informations that are universal to all physical solutions, thus an

obvious application may be in scaled model experiments, where, for example, one might

interpret scaled tank experiments to the actual size measurements. It is shown by Fox [18]

that despite the fact that the scaling constants lc and tc are originally obtained from the

problem of static loading of an infinite plate, the same scaling relations hold for an ice sheet

of more general shape.

3.6.1 Scaled solutions and physical solutions

We again consider the solution for the static load shown in subsection 3.5.1. If we solve the

original system of differential equations, we find that the dispersion equation is

d (γ) = Dγ4 + ρg (3.66)

and the physical solution is

wP (r) = − kei (r)

2π√Dρg

= − kei (r)

2πρgl2c(3.67)

where r is the physical distance from the forcing point. Observing Eqn. (3.66) and Eqn. (3.67),

we find the appropriate non-dimensionalization constant lc to be (D/ρg)1/4 which of course

gave the non-dimensional solution (3.55) and then the relationship between the non-dimensional

and physical solutions. The conversion between the scaled and physical solutions is found

by back-stepping the non-dimensionalization and the Fourier transforms (in one and two di-

mensional space) shown in section 3.2. Using the notation with the bar for the dimensional

61

variables, we have the following conversion relationship for the unit point load, 1 Newton

wP (r) =1

ρgl2cwP (r) .

Note that wP (r) is displacement per Newton.

Again, considering the scaled and physical solutions of the static line-loading problem,

we find the conversion relation. The sum in Eqn. (3.60) gives the physical response to a

static line force as

wL (|x|) =1

2ρglcexp

(− |x|√2lc

)cos

( |x|√2lc

− π

4

).

Therefore, for the line load the physical displacement for the line forcing, we have

wL (|x|) =1

ρglcwL (|x|) .

The physical dimensions are completely absent in the static-load dispersion function

(3.54). Hence the roots of the dispersion equation are independent of any change in physical

parameters, such as ice thickness, water depth, or mass density. In other words, once the

formula (3.55) is obtained, the characteristic length lc is the only parameter that we need

to define the characteristics of an ice sheet. The expression in Eqn. (3.55) is the same as

the solution given by Wyman [51].

One may hope that the same characteristic length lc and normalizing scheme could be

applied to the dynamic ice sheet problem. Fortunately, we find that the same characteristic

length lc and time tc may be used to scale the system of equations for the range of ice

thickness h and forcing frequency ω that is relevant to the study of sea ice. The validity

of the non-dimensionalization of Eqn. (2.44) is based primarily on the dispersion equation

(3.53) for deep water. Fig. (3.12) shows the positions of the wave-numbers qT, qD and qE of

the normalized deep water solution for different ice thickness 0.1, 1.0 and 10 metres. We find

that the positions of the roots (wavenumbers) of the non-dimensional deep-water dispersion

equation remain virtually unchanged as ice thickness is varied.

By closer observation of Fig. (3.12), we find that up to ω ≈ 1 all the roots are nearly

identical for all the thickness of the ice sheet considered. Although, the value of the roots

other than qT vary slightly for higher frequencies, the position of the roots stays qualitatively

unchanged. Hence, we may conclude that our normalizing scheme is valid for the range of

ice thickness and frequency that are geophysically relevant.

It is obvious that the same conclusion can be made for finite water depth, since the depth

dependent term, tanh γH in the dispersion equation is dimensionless and independent of

changes in ice-thickness. Fig. (3.13) shows the positions of the roots when the water depth

affects the response of the ice sheet, H = 0.2π and π.

62

0.1 1 10

0.1

1

Non−dimensional radial frequency ω

Non

−di

men

sion

al w

ave

num

ber

qT

Im( qD)

−Re( qE)

Im( qE)

Re( qD)

Figure 3.12: Graph of the real and imaginary parts of the roots of the deep water dispersionequation agianst the non-dimensional radial frequency ω plotted in a loglog scale. Thethickness of the ice sheet is taken to be 0.1, 1 and 10 metres.

0.1 1 10

0.1

1

Non−dimensional radial frequency ω

Non

−dim

ensi

onal

wav

e nu

mbe

r

(a)

0.1 1 10

0.1

1

Non−dimensional radial frequency ω

Non

−dim

ensi

onal

wav

e nu

mbe

r

(b)

qT

Im( qD)

q1

Re( qD)

Figure 3.13: Graphs of the real and imaginary parts of the roots of the shallow-waterdispersion equation. (a) is when H = 0.2π and (b) is when H = π.

63

So far we have computed the deflection of the ice sheet with m = 0. This is justified

because for typical ice sheets, it is reasonable to say that m is smaller than 0.1 assuming

that ice thickness is in the order of metres and effective Young’s modulus is in the order

of 109 Pascals, and thus the characteristic length lc is typically larger than 10h3/4. Fur-

thermore m that is smaller than 0.1 has no noticeable effects on the response of the ice

sheet since the positions of the roots are virtually unchanged as shown in Fig.(3.14). Omit-

ting the mass density term, the resulting dispersion equation for the deep water problem

then becomes truly independent of any physical parameters, which is the objective of the

non-dimensionalization.

−3 −2 −1 0 1 2 3−0.5

0

0.5

1

1.5

2

2.5

3

Real part of roots

Imag

inar

y pa

rt o

f roo

ts

m=0 m=0.1

Figure 3.14: Positions of the roots of the non-dimensional dispersion equation when thenon-dimensional mass density m is set zero and 0.1. The radial frequency is ranging from0.1 to 10.

3.6.2 General scaling law of a floating ice sheet

We have seen that the scaling scheme that was originally based on the static loading of

an infinite plate can be extended to dynamic problems. It is shown by Fox [18] that our

scaling scheme can actually be extended to a plate of finite size and the scaling law for the

dynamics of an elastic floating plate (ice sheet) can be found in a more general context. It

can be shown that the representation of the fundamental solution by a modal expansion is

independent of the shape of the plate, as long as its shape is reasonably smooth, since the

dispersion function, and hence the denominator of the Fourier transform of w is unaffected

by the shape of the plate. This subsection shows a brief description of the work done by C.

Fox for the article [18], as it shows the relevance of the scaling regime in the general physical

situations.

64

Reviewing the method of solution of the infinite plate, we notice that the geometry of

the plate is unnecessary to obtain the dispersion function d (γ, ω), since it is determined

only by the plate equation.

We consider a smoothly shaped plate denoted by Ω, as in the previous chapter. Using

the same notations in chapter 2, we are able to find a solution of the Laplace’s equation in

V. Then the Fourier transform of the plate equation must integrated over the finite domain

Ω, which consists of the terms to be determined by the values of w and its derivatives at

the boundary of the plate ∂Ω.

ˆwΩ

(α, k

)=

∫

Ω

w (x, y) ei(αx+ky)dxdy

Then, using the inverse Fourier transform to calculate the displacement,

w (x, y) =1

4π2

∫ ∞

−∞

∫ ∞

−∞

ˆwΩ

(α, k

)e− i(αx+ky)dαdk. (3.68)

Note that w (x, y) calculated using Eqn. (3.68) is zero outside the ice sheet Ω.

Using the Fourier transform defined above and Green’s theorem, we find that the Fourier

transform of the plate equation becomes

(Dγ4 + ρg − mω2

)ˆwΩ

(α, k

)+ i ω ˆφΩ

(α, k, z

)= pa + b1

(α, k

)(3.69)

where b1 is function of(α, k

)that is determined by the boundary values of w and its

derivatives, which results Green’s theorem. The Fourier transform of Laplace’s equation in

Ω for −H < z < 0 becomes

∂2 ˆφΩ

∂z2− γ2 ˆφΩ = b2

(α, k

)(3.70)

where b2(α, k

)is again an inhomogeneous term that arises in Green’s theorem. We can

solve Eqn. (3.70), which is an ordinary differential equation, given the fixed ocean floor

condition and b2. Thus, we find the relation between ˆφΩ and its z derivative similar to that

in the previous sections,

∂ ˆφΩ

∂z

(α, k, 0

)= γ tanh

(γH) ˆφΩ

(α, k, 0

)+ b3

(α, k

)

where b3 depends on the functions b1 and b2 in Eqn. (3.69) and Eqn. (3.70). Therefore the

Fourier transform ˆw(α, k

)has the same denominator, the dispersion function

d (γ, ω) = Dγ4 + ρg − mω2 − ρω2

γ tanh γH

65

as the infinite plate solution,

ˆw(α, k

)=b(α, k

)

d (γ, ω).

The inverse Fourier transform of ˆw(α, k

)will then be dependent on the shape of the plate

and the boundary conditions given at the edge of the plate, which determine the function

b(α, k

). If we assume that the numerator b

(α, k

)has no singularities, in other words w and

its derivatives on the edge of the plate are well-behaved functions, then the inverse Fourier

transform of ˆw(α, k

)must again be expressed by the mode expansion over the roots of the

dispersion equation, Kˆ. Because our scaling is based on the behavior of the roots of the

dispersion equation, the scaling can be applied to an ice sheet of general shape.

For simple shapes of the plate, for example a circular disk, semi-infinite straight edged

plate, or rectangular plate under line load parallel to the edge of the plate, the function b

will depend only on γ, the amplitude of(α, k

). Hence, the inverse Fourier transform will

again then depend only on the roots of the dispersion equation, which can be effectively

scaled using the characteristic length and time. For an infinite plate, we have found that

b ≡ 1. We will see an example of b 6= 1, i.e., non-infinite plate, when the inverse Fourier

transform can be analytically calculated in the next chapter.

3.7 Determining characteristic length from field mea-

surements

We consider how the characteristic length lc, and hence the effective Young’s modulus, can

be determined from flexural motion of the ice sheet. As mentioned before, although the ice

sheet is modeled with a constant Young’s modulus, the value of E varies through the ice

sheet in reality mainly due to the temperature gradient from top to bottom. As a result,

we use a constant effective Young’s modulus as a substitute. A commonly used value of the

effective Young’s modulus is 5 × 109 Nm−2. However, it is not obvious how one can obtain

the actual Young’s modulus and then the effective Young’s modulus. We propose here a few

possible methods of measuring the effective Young’s modulus from field experiments using

the theoretical results in the previous sections.

3.7.1 Characteristic length

Fig. (3.15) shows a schematic drawing of how flexural waves in an ice sheet can be generated

using a hydraulic “thumper” that can lift a block of ice up and down at a prescribed

frequency. The amplitude of the oscillating force is calculated from the size of the ice block

and the length of the stroke of the thumper.

We consider methods of calculating the characteristic length in two cases, when the

66

water water

up

down

ice ice

Figure 3.15: Schematic of an experimented setup of a ‘thumper’. A block of ice is cut outand lifted up and down.

water is shallow, i.e., H = 0.2π, and when the water is deep, i.e., H larger than 2π. We

have seen that water depth larger than 2π can be regarded as deep in the previous sections.

Fig. (3.16) and Fig. (3.17) show the magnitude and phase of the normalized strain, which is

the second derivative of the non-dimensional displacement function, −wrr (r, ω), plotted as

a function of non-dimensional radian frequency ω and distance r from the point of forcing

in the case when the water depth is H = 0.2π. Then, the strain per Newton denoted by

S (r) is calculated by

S (r) = −h2wrr (r) = − h

2ρgl4cwrr (r)

= − ih

4ρgl4c

∑

q∈Kˆ

qR (q)(q2H

(1)2 (qr) − q

rH

(1)1 (qr)

). (3.71)

We used the formula for the derivative of Hankel function (Abramowitz and Stegun [4]).

The following graphs of the amplitude of the strain shows the strain scaled by h/ (2ρgl4c),

hence |S (r)| = |wrr (r)|.Fig. (3.18) and Fig. (3.19) are equivalent plots for water depth H = 2π. Note that

the plots of strain magnitude and phase have reversed distance axes in order to show the

structure of the curves. Fig. (3.16) and Fig. (3.18) show that the magnitude of strain at the

near field, r < 1, changes rapidly with r. Hence, because of the length of the strain gauge

itself and the physical size of the forcing device, it is impossible to measure the strain at a

point sufficiently accurately to fit to the graph of magnitude. If instead the strain gauges

are placed near r & 1, where the magnitude of strain changes little with respect to distance

from the forcing then, by sweeping the frequency, we should be able to find the frequency

where the dip in magnitude occurs and hence determine the characteristic frequency. A more

robust measure that does not require an accurate knowledge of the magnitude of forcing,

is to use the feature in Fig. (3.17) and Fig. (3.19) which show that for r & 1 the phase of

strain has a minimum value at ω ≈ 0.9 and the position of the minimum is an insensitive

function of distance. Hence, robust measurements can be made at r & 1 to find a frequency

where the dip in phase occurs to determine characteristic time and length.

67

0.1

0.5

1

1.5 0.1

1

10

0

0.05

0.1

0.15

0.2

Non−dimensionalfrequency ω

Non−dimensionaldistance r

Nor

mal

ized

str

ain

Figure 3.16: Magnitude of the normalized strain for shallow water H = 0.2π, as a functionof non-dimensional radial forcing fequency (log-scale) and non-dimensional distance, 0.1 ≤r ≤ 1.5, from the forcing point (in reversed axis for better view).

0.1

1

10

0.1

0.5

1

1.50

1

2

3

4

5

Non−dimensionalfrequency ω

Non−dimensionaldistance r

Pha

se o

f str

ain

Figure 3.17: Phase of the strain for shallow waterH = 0.2π, as a function of non-dimensionalradial forcing frequency (log-scale) and non-dimensional distance, 0.1 ≤ r ≤ 1.5, from theforcing point.

68

0.1

0.5

1

1.5 0.1

1

10

0

0.05

0.1

0.15

0.2

Non−dimensionalfrequency ω

Non−dimensionaldistance r

Nor

mal

ized

str

ain

Figure 3.18: Magnitude of the normalized strain for deep water, as a function of non-dimensional radial forcing fequency (log-scale) and non-dimensional distance, 0.1 ≤ r ≤ 1.5,from the forcing point (in reversed axis for better view).

0.1

1

10

0.1

0.5

1

1.50

1

2

3

4

5

Non−dimensionalfrequency ω

Non−dimensionaldistance r

Pha

se o

f str

ain

Figure 3.19: Phase of the strain for deep water, as a function of forcing frequency anddistance, 0.1 ≤ r ≤ 1.5, from the forcing point.

69

0.1 1 10

0.01

1

|w tt

(0)|

0.1 1 100

0.5

1

1.5

arg(

w tt

(0))

Non−dimensional radial frequency ω

H=2π H=0.2π

Figure 3.20: (a) Magnitude and (b) phase of normalized vertical acceleration at the pointof forcing, as a function of non-dimensional frequency.

Fig. (3.20) shows the magnitude and phase of the acceleration at the point of forcing as

a function of non-dimensional radial frequency ω. The acceleration is calculated from the

formula of the displacement at the point of forcing, Eqn. (3.57), −ω2w (0). We notice that

the both magnitude and phase of the acceleration are monotonically increasing functions

of ω. Hence, we may sweep frequency and measure the response to match the theoretical

results in Fig. (3.20), to find the characteristic length.

It may be difficult to measure the magnitude of the acceleration, since it requires an

accurate knowledge of the weight of the ice block. Therefore, it may be more robust to

make use of the graphed phase of the acceleration. In order to find the characteristic

length, we sweep frequency to locate the relative phase of 0.9 at which the shallow and deep

water phases coincide (at about ω = 0.9) so that water depth does not affect much. Let ω0

denote such frequency, then we find ω0 at which the measured relative phase is 0.9, i.e.,

arg (w (0, ω0)) = 0.9.

From ω0 we are then able to find the characteristic length using

lc =0.92

ω20

g. (3.72)

Another possible measurement method of the characteristic length is to use the tilt of

70

10−1

100

101

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Non−dimensional radial frequency ω

Sca

led

tilt

Figure 3.21: Sacled amplitude of tilt as a function non-dimensional radial frequency ω = ωtcby lifted ice-weight (103kg) devided by l3c and square root of non-dimensional distance

√r,

103kg/l3cm3/√r.

the ice sheet in far field calculated using an asymptotic expression

d

drw (r) ∼ − i q

3/2T RT√2πr

=− i√2πr

q7/2T

5ω2 − 4qT=

− i√2πr

q5/2T

5q4T + 1

.

We use the asymptotic formula of a Hankel function,

H(1)ν (r) ∼

√2

πrexp

[− i(r − ν

2− π

4

)].

Again we try to locate the physical radial frequency ω at which the maximum tilt occurs (at

unit non-dimensional radial frequency ω = 1) as seen in Fig. (3.21). Then, using Eqn. (3.72),

we find the characteristic length.

The methods of measuring the characteristic length proposed here assume that the forc-

ing is localized only on the ice sheet. However, in practice it may be difficult to generate

a pure surface-load on the ice because of the size of an ice-block which can be almost 2

metres tall. When the thumper is configured as in Fig. (3.15), the effects of water (about

1 cubic metre) which is pumped in and out by the ice-block cannot be ignored. There-

fore, if a thumper shown in Fig. (3.15) is used to generate flexural motions a mathematical

model that incorporates weight of an ice block and water-pumping action will be required

to analyze experiment data sets. This is ongoing research.

71

3.8 Summary

We may divide the content of this chapter into two parts. One that has to do with math-

ematical calculations involving integration, complex valued functions, to special functions,

and so on. Another that applies the analytical solutions to real geophysical studies of sea ice,

e.g. for the effective scaling of the solution and in-situ measurement of Young’s modulus.

The response of the ice sheet to a localized force is expressed by an infinite series of

natural wave modes of the ice sheet and the wavenumbers of the modes are the roots of the

dispersion equation which is a relationship between the forcing frequency and the wavenum-

bers. The dispersion equation is extended to the complex plane by analytic continuation,

which enables us to use the tools of contour integrals in the complex plane and commonly

available tables of integral transformations. The process of deriving the formulae of the

coefficients is tedious but a series of elementary calculations.

We are able to calculate the inverse transform analytically using the positions of the

roots of the dispersion equation in the complex plane. When the water depth is finite, the

solutions are expressed by infinite series of natural wave modes existing in an ice sheet.

For infinite water depth case, the finite water depth solutions can be directly extended to

derive a simple formula for the solution made up of five wave modes. Our derivation of the

deep-water solution avoids the use of analyticity properties of the dispersion equation for

deep-water problems. The formulae are simplified by non-dimensionalization of the equa-

tions and a consequent reduction of the number of physical parameters. We found that

although the characteristic length lc is originally based on the static solution, together with

the characteristic time tc, the dynamic system can also be effectively non-dimensionalized,

so that the resulting non-dimensional solutions are insensitive to changes of physical param-

eters.

Knowing the theoretical conversion relationship between the scaled and actual solutions

lets us find meaningful values of various parameters that can be interpreted to any physical

scale using the characteristic length and time. For example, we found that non-dimensional

water depth H = 2π can be considered as deep regardless of the wavelength and frequency

of surface waves, and that the maximum coupling between the forcing and the response

of the ice sheet happens just below unit non-dimensional frequency ω = 1. Furthermore,

our scaling scheme is shown to be applicable to more generally shaped ice sheet, hence the

values of the non-dimensional parameters, such as H = 2π and ω = 1, become important

for non-infinite ice sheets.

Section 3.6 proposes several possible methods for how the theoretical solution can be

used to obtain an effective Young’s modulus from an actual field experimental data set.

Experimental data show that existing strain gauges can be used to determine characteristic

length using the measurement schemes introduced in section 3.6.

All plots of displacement are computed using the software package MatLab on an Intel

72

Pentium III PC and all the graphs shown in this chapter are the results of the direct

implementation of the formulae given here. The number of roots used for the computation

is proportional to the water depth,

number of roots ≈(H

2π+ 2

)× 10,

which is about 20 to 100 for each given frequency. The number of roots given by the

formula above may be larger than the minimum number required, but finding the roots is

computationally inexpensive. The special functions other than Struve function and Ci and

si are computed by the built-in functions of MatLab. Struve function and Ci and si are

computed using the power expansion given in appendix A.

The idea from the dispersion equation to the series expansion Eqn. (3.32) was given to

me by my supervisor C. Fox. The computation of the Green’s function (except the roots of

the dispersion equation) for the finite and the infinite water depth cases are done by me.

73

Chapter 4

Wave propagation in semi-infinite

floating plates

In this chapter we study interaction between obliquely incident, monochromatic ocean waves

and a semi-infinite floating plate. We follow the same methods reported by Chung and Fox

[9, 10, 11] and Evans and Davies [17].

4.1 Background

This chapter presents a theoretical study of wave propagation in two adjoining floating plates

of different flexural rigidity. The primary aim of this chapter is to study the mathematical

model of a composite floating plate-like structure that is relatively homogeneous and large

compared to the wavelength of surface waves. An example of such a structure is fast-ice

sheets that abut across a pressure ridge.

During the winter months, large nearly featureless floating sea ice sheets form around

the coast of Antarctica. These ice sheets are called land-fast ice because they are attached

to land. Breakup and refreezing of land-fast ice occur due to ocean current and swell coming

from the open ocean as shown in Fig. (4.1). We are interested in the effects of ocean swell

on the ice sheets. As ocean waves arrive at the edge of the ice sheet some of the wave

energy is reflected and some propagates into the ice field causing bending motion in the

ice sheet, which can lead to breakup. Broken up ice sheets may re-freeze during calm

periods, creating a discontinuity in an apparently continuous piece of ice sheet. At these

discontiunities (sometimes referred as cracks), which may be open or re-frozen, propagating

wave ocean waves are again reflected and transmitted. In spring the ice eventually breaks

up completely with currents carrying the ice floes away from the coast where they melt,

leaving the coast free for the cycle to begin again the following winter.

For many years the formation and break-up of sea ice have interested not only geophysi-

cists but also mathematicians (Balmforth and Craster [5], Evans and Davies [17], Fox and

74

Figure 4.1: Sea meets the sea-ice in McMurdo Sound, about ten kilometers from Ross islandand Mt. Erebus. Photo courtesy of Colin Fox.

Squire [23], Gol’dshtein and Marchenko [24], Squire et al. [44]). However, despite a great

deal of idealization of the physical conditions, there have been few analytical solutions to

the boundary value problems. Our aim here is to calculate the reflection and transmission

of obliquely incident waves on a discontinuity in fast ice. Considering that a piece of ice

sheet is large compare to wavelength and relatively uniformly shaped, it is assumed that

the discontinuity is straight and infinitely long. In addition to these assumptions, we treat

an ice sheet as a thin elastic plate, which is widely accepted practice in modeling of fast ice.

The mathematical tool we use here is the well known Wiener-Hopf technique. The Wiener-

Hopf technique is commonly used in wave-guide problems in acoustic and electromagnetic

wave propagation when the boundary of the region of the sound or electromagnetic medium

is divided into two semi-infinite parts. There have been several studies that have applied

the Wiener-Hopf technique to calculate the transmission and reflection coefficients of an

elastic plate. Until recently only approximate or complex numerical methods for computing

these values have been available. We review those methods as well as detailing methods

of solution based on the Wiener-Hopf method. The latter, while following a complicated

analysis, has recently been found to lead to very simple methods for computation (Chung

and Fox [9, 10])

The earliest method including the elastic response of the ice was given by Hendrikson and

Webb [26] and subsequently Wadhams [50] who corrected minor errors in the earlier work.

These authors used an incomplete set of modes and hence gave approximate solutions. Their

method was to match travelling and damped travelling waves by satisfying continuity and

boundary conditions on the surface at the edge of the ice cover. Subsequent comparison with

exact solutions in Fox and Squire [21] showed that these solutions correctly predicted the

75

general trend for complete reflection at very short periods through to complete transmission

at very long periods. However these solutions contain erroneous features in the region of

partial reflection and also do not conserve energy – the latter problem being extreme at

short periods.

Fox and Squire [21] computed the reflection and transmission coefficients by solving the

mathematical model exactly. They used the complete set of modes to express solutions

with the coefficients found by matching through the water column beneath the edge of the

ice sheet. The matching was performed numerically and led to a large system of equations

that became unwieldy at short periods or large depths. Later this solution was extended to

obliquely incident waves by Fox and Squire [23] using the same basic method.

More than thirty years ago, Evans and Davies [17] had formally solved the mathematical

model using the Wiener-Hopf method. That method solves for the Fourier transform of the

solution in each half plane, i.e., over the region of open water and the ice-covered region.

Until recently the solution given by Evans and Davies was thought to be unsuited for actual

computation because the required inverse Fourier transform was too difficult. Indeed Evans

and Davies stated this opinion in their report. That belief, coupled with the deceptively

complicated calculations in the Wiener-Hopf analysis caused this solution to be over looked

for many years. Two routes for taking that analysis through to stable computation have

recently been found. In [5], Balmforth and Craster showed how the Wiener-Hopf analysis

for a range of ice-sheet models could be made more straightforward by a formal application

of the method with inverse transforms calculated by stable quadrature. Fox and Squire

[22] reports that the thin plate and the thick models give the same response for the wave-

ice interaction problem. We will outline a second route developed by the present author

in which the factorizations required in the Wiener-Hopf solution are written explicitly in

terms of the wave-numbers of modes, and solutions are calculated as stable sums over these

modes. Our advance over Evans and Davies is largely through a few modifications to the

formulae and being able to find the roots of the dispersion equations. Furthermore, this

method has a simple extension to the deep water case by using the asymptotic behaviour

of the coefficients to evaluate the expansion over the evanescent modes via an integration

over the imaginary axis.

The deep water problem has also been studied by Gol’dshtein and Marchenko [24] also

using a Wiener-Hopf technique. They analyzed the asymptotic case when the rigidity of the

ice tends to zero, concluding that the reflection becomes zero in that limit. Extension of the

same method was used by Marchenko [34] to solve for wave propagation near a transition

between different thicknesses of ice covers. Again, no explicit formula for the solution was

given.

Chung and Fox [10] showed that the solutions obtained by Evans and Davies [17] could

in fact be computed without numerical computation of integral transforms by finding the

zeros of the dispersion functions. By following and extending the method by Evans and

76

Davies [17] we show that we can deal with various edge conditions without going back to

the original differential equations. We also show an extension of the solutions for infinitely

deep water in section 4.6, which is a different approach from those by Balmforth and Craster

[5], Gol’dshtein and Marchenko [24] and Squire and Dixon [43].

4.2 Methods of solution

We formulate BVPs of two semi-infinite plate interaction using the differential equations

that are introduced in chapter 2, then introduce mathematical method which we will employ

to solve the BVPs. The solution is expressed by an infinite series of the modes which have

seen in chapter 3. There have been a number of methods developed by various researchers

to find the coefficients of the modes. We briefly introduce summary of two of these methods

by Fox and Squire [21] and Wadhams et al. [50].

The line-loading response shown in Fig. (3.11) indicates that the effects of the evanescent

modes are negligible at r > 10 (10 times the characteristic length). Although there are many

physical characteristics to be considered, we propose that the length of the edge and width

of the ice sheet must be at least larger than 10lc (characteristic length) when we use the

semi-infinite ice sheet model.

4.2.1 Mathematical formulation

Ice sheet

Water

Ocean floor z=-H

xz=0

Ice sheet

Incoming Plane wave

Figure 4.2: Side-view of the two semi-infintie ice sheets. The origin is placed at the transitionand a plane wave is incident from x = −∞.

We consider the dynamics of two semi-infinite homogeneous thin ice sheets joined at x =

0 as shown in Fig. (4.2). The case when the thickness of the ice sheet on the left in Fig. (4.2)

is zero, i.e., the case of free ocean surface and a semi-infinite ice sheet is also included in

this mathematical model. A plane wave of radial frequency ω is coming from x = −∞at an angle θ to the x-axis. We assume that the amplitude and frequency of the incident

wave are small and low enough that the water can be assumed to be incompressible and

irrotational, and the ice sheets can be modelled as an elastic thin plate. Then, the vertical

displacement of the ice sheets, w (x, y, ω), and the velocity potential of water, φ (x, y, z, ω)

77

satisfy the following partial differential equations (Evans and Davies [17], Fox and Squire

[23]). Two plate equations for two different flexural rigidity, D1 and D2, mass density m1

and m2

pw (x, y) = D1∇4w (x, y) −m1ω2w (x, y) , (4.1)

pw (x, y) = D2∇4w (x, y) −m2ω2w (x, y) , (4.2)

in x < 0 and x > 0 respectively. From chapter 2, the velocity potential and displacement at

z = 0 satisfy linearized kinematic condition (2.39) and linearized Bernoulli’s equation (2.40)

iωw = φz,

i ρωφ+ ρgw + pw = 0.

In the water the velocity potential satisfies Laplace’s Eqn. (2.43) and the fixed bottom

condition (2.42) at z = −H,

∇2φ = 0,

φz|z=−H = 0. (4.3)

Note that the subscript j = 1, 2 on D and m indicate values corresponding to two domains

x < 0 and x > 0, respectively. Here mj = ρihj where hj is the thickness of the ice sheets.

The flexural rigidity for each plate is calculated using Dj = Eh3j/12 (1 − ν2), j = 1, 2.

We do not include effects of dissipation in the ice sheet, instead we assume that solutions

satisfy the radiation condition at x = ∞, that is, there exist no source of waves at x = ∞.

Since the incident wave is assumed to be a plane wave, for a given wavenumber of the

incident wave, say λ, then the wavenumber in the y direction is k = λ sin θ. The incident

wave is assumed to be harmonic in time with a single radial frequency ω, hence from the

linearity of the system of equations with respect to φ and w, the solutions can be expressed

as

φ (x, y, z, t) = Re [φ (x, z) exp i (ky + ωt)] ,

w (x, y, t) = Re [w (x) exp i (ky + ωt)]

where φ (x, z) and w (x) are complex valued functions. Notice that we use the same notations

φ and w since there is no risk of confusion. The equations from Eqn. (4.1) to Eqn. (4.3) for

78

φ (x, z) are

(Dj

(∂2

∂x2− k2

)2

+ ρg −mjω2

)φz − ρω2φ = 0, j = 1, 2, z = 0 (4.4)

(∂2

∂x2+

∂2

∂z2− k2

)φ = 0, −H < z < 0, (4.5)

φz = 0, z = −H. (4.6)

Note that we deal with the velocity potential φ and φz rather than w on the surface. In

chapter 3, our primary interest was in the flexural response of the plate on the surface,

i.e., w (x, y, t). However in this chapter we deal directly with Laplace’s equation in the two

regions and its Fourier transform, thus we solve the equations for φ and φz at the surface.

In the following section, analytical solutions of the boundary value problem defined by

Eqn. (4.4), Eqn. (4.5) and Eqn. (4.6) in x > 0 and x < 0 will be derived using the Wiener-

Hopf technique based on ocean and wave-ice interaction problem reported in Evans and

Davies [17].

4.2.2 Mode matching by Fox and Squire

Fox and Squire [21] expressed waves in the open ocean, φ1, and ice covered sea, φ2, by

an infinite series of modes which form a complete solution of Laplace’s equation with the

boundary conditions of open water and ice sheet,

φ1 (x, z) = aTeiλx coshλ (z +H) +

∑ane

λnx cosλn (z +H) , (4.7)

φ2 (x, z) = bTeiµx coshµ (z +H) + bDe

iµDx coshµD (z +H)

+ b′De−iµ∗

Dx coshµ∗

D (z +H) +∑

bne−µnx cosµn (z +H) . (4.8)

Here λ and λn, n = 1, 2, ... are positive real wave numbers, aT and an are the corresponding

coefficients, µ and µn are real , µD is complex with positive real and imaginary parts and

bT, bn and bD are corresponding coefficients. Fox and Squire [21] found the coefficients by

minimizing the error function,

ǫ = c1

∫ 0

−H

|φ1 − φ2|2 dz + c2

∫ 0

−H

∣∣∣∣∂φ1

∂x− ∂φ2

∂x

∣∣∣∣2

dz

+ c3

∣∣∣∣∂2

∂x2

∂φ2

∂z

∣∣∣∣2

+

∣∣∣∣∂3

∂x3

∂φ2

∂z

∣∣∣∣2

where c1, c2 and c3 are the positive Lagrange multipliers. Notice that in the right hand side

of the equation, the first and second terms enforce the continuity of the solution while the

third term penalizes only misfit in the natural boundary conditions. The minimum of the

79

error function (which is zero) occurs for coefficients which give the solution, for any choice

of the Lagrange multipliers. Fox and Squire performed the minimization by numerically

solving the normal equations written in terms of the unknown coefficients. They often

required up to a hundred modes to achieve the minimum with a reasonable precision. Fox

and Squire [23] later extended this procedure to treat obliquely incident waves by treating

the boundary conditions as “hard” constraints so the error function represented the misfit

in continuity only. While this simplified the numerical procedure, it remains too unwieldy

for general use.

4.2.3 Approximation by Wadhams

The method presented by Wadhams effectively assumed that the coefficients an, bD, b′D,

and bn, in the expansions in Eqn. (4.7) and Eqn. (4.8), are zero, therefore omitting the

evanescent modes in both water and ice sheet. As mentioned above, this method gives

reflection and transmission coefficients that are correct in the simple regimes of extreme

period and wavelength, but are in error for periods of geophysical significance (Fox and

Squire [21]). In particular, this approximation does not predict characteristic features of

the strain response near the edge of shore-fast sea ice. An example is the feature observed

by Squire and others [45] during field measurements made in McMurdo Sound, Antarctica,

which shows that the surface strain of the ice is not a simple exponentially decaying function

of distance from the edge of the ice sheet.

This solution continues to be referred to without recognition of its inaccuracies (Wad-

hams and others [49]), perhaps because of its mathematical simplicity.

4.3 The Wiener-Hopf technique

4.3.1 Theoretical background

We first give a brief explanation of the mathematics behind the Wiener-Hopf technique. We

define the Fourier transform along the x-axis by Eqn. (3.20). The Fourier transform, as we

have seen in chapter 3, can be used to derive a solution to partial differential equations in

an infinite domain with free boundary conditions or conditions at r = ∞. The Wiener-Hopf

technique is an extension of the Fourier transform method to semi-infinite domains of simple

geometry, such as those with a straight or circular boundary. In the Wiener-Hopf technique

the variable α is extended into the complex plane so that the Fourier transform φ (α, z) may

have singularities on the complex plane depending on the integrability of φ (x, z).

Consider a function ψ (x) of x ∈ R that is bounded except at a finite number of points

and has the asymptotic property |ψ (x)| ≤ A exp (δ−x) as x→ ∞ and |ψ (x)| ≤ B exp (δ+x)

as x → −∞. If δ− < δ+, the Fourier transform of ψ (x) exp (−δx) for δ− < δ < δ+ can be

80

obtained using the usual definition by Eqn. (3.20) for real variable ε,

F (ε) =

∫ ∞

−∞

ψ (x) e−δxei εxdx.

Then, the integral above defines the Fourier transform in the complex plane and the function

ψ (α) defined as

ψ (α) =

∫ ∞

−∞

ψ (x) eiαxdx (4.9)

is an analytic function of α = ε + i δ, regular in δ− < δ < δ+. Using the usual inverse

transform, we have for α = ε+ i δ in δ− < δ < δ+

1

2π

∫ ∞+i δ

−∞+i δ

∫ ∞

−∞

ψ (ξ) eiαξdξ

e− iαxdα

=1

2πe−δx

∫ ∞

−∞

∫ ∞

−∞

(ψ (ξ) eδξ

)ei εξdξ

e− i εxdε

= e−δx(ψ (x) eδx

)= ψ (x) .

Note that in the second line we change the variable from α to ε. Thus the inverse Fourier

transform is obtained by

ψ (x) =1

2π

∫ ∞+i δ

−∞+i δ

ψ (α) e− i αxdx (4.10)

where δ− < δ < δ+. An immediate consequence of this is that if a function ψ (x) satisfies

|ψ (x)| ≤ A exp (δ−x) as x→ ∞ then the Fourier transform in the half space

ψ+ (α) =

∫ ∞

0

ψ (x) eiαxdx

is an analytic function of α and regular in δ− < δ. Also the function can be recovered by

ψ (x) =1

2π

∫ ∞+i δ

−∞+i δ

ψ+ (α) e− iαxdα

as x→ −∞, where ψ is zero in x < 0. The equivalent relation holds for ψ defined in x < 0

satisfying |ψ (x)| ≤ B exp (δ+x) as x → −∞, then the Fourier transform ψ− is regular in

δ < δ+.

Conversely, suppose that ψ (α) is regular in the strip defined by δ− < δ < δ+ and tends

to zero uniformly as |α| → ∞ in the strip. If ψ is defined as a solution of the equation

ψ (x) =1

2π

∫ ∞+i δ

−∞+i δ

ψ (β) e− iβxdβ (4.11)

81

then for a given α = ε+ i δ, δ− < c < δ < d < δ+

I =1

2π

∫ ∞

−∞

∫ ∞+i δ

−∞+i δ

ψ (β) e− iβxdβ

eiαxdx

=1

2π

∫ 0

−∞

∫ ∞+i c

−∞+i c

ψ (β) e− iβxdβ

eiαxdx+

1

2π

∫ ∞

0

∫ ∞+i d

−∞+i d

ψ (β) e− i βxdα

ei αxdx

since ψ is regular in the strip and Im (α− β) < 0 for δ < d and Im (α− β) > 0 for c < δ,

which makes each split integral convergent. Then, using Cauchy’s integral theorem and that

ψ → 0 as |α| → ∞ in the strip, we have

I = − 1

i 2π

∫ ∞+i d

−∞+i d

ψ (β)

β − αdβ +

1

i 2π

∫ ∞+i c

−∞+i c

ψ (β)

β − αdβ

=1

i 2π

∫

C

ψ (β)

β − αdβ = ψ (α)

where C is a rectangular contour formed by four points (±∞ + i c) and (±∞ + i d). There-

fore, the function ψ can be obtained using Eqn. (4.9).

Detailed discussion of the analyticity of complex valued functions that are defined by

integral transforms can be found in sections 1.3 and 1.4 of (Noble [38]) and chapter 7 of

(Carrier, Krook and Pearson [7]).

We apply the Fourier transform to Eqn. (4.4) and Eqn. (4.5) in x < 0 and x > 0 and

obtain algebraic expressions of the Fourier transform of φ (x, 0). The Fourier transforms of

φ (x, 0) in x < 0 and x > 0 are defined as

Φ− (α, z) =

∫ 0

−∞

φ (x, z) eiαxdx and Φ+ (α, z) =

∫ ∞

0

φ (x, z) ei αxdx. (4.12)

Notice that the superscript ‘+’ and ‘−’ correspond to the integral domain. The radiation

conditions introduced in section 2.3 restrict the amplitude of φ (x, z) to stay finite as |x| → ∞because of the absence of the dissipation. It follows that Φ− (α, z) and Φ+ (α, z) are regular

in Imα < 0 and Imα > 0, respectively.

It is possible to find the inverse transform of the sum of functions Φ = Φ− + Φ+ using

the inverse formula (4.10) if the two functions share a strip of their analyticity in which a

integral path −∞ < ε <∞ can be taken. The Wiener-Hopf technique usually involves the

spliting of complex valued functions into a product of two regular functions in the lower

and upper half planes and then the application of Liouville’s theorem, which states that a

function that is bounded and analytic in the whole plane is constant everywhere. A corollary

of Liouville’s theorem is that a function which is asymptotically o (αn+1) as |α| → ∞ must

be a polynomial of n’th order.

We will show two ways of solving the given boundary value problems in this chapter.

82

First in this section, we figure out the domains of regularity of the functions of complex

variable defined by integrals (4.12), thus we are able to calculate the inverse that has the

appropriate asymptotic behaviour. Secondly in section 4.7, we find the asymptotic behaviour

of the solution from the physical conditions, thus we already know the domains in which

the Fourier transforms are regular and are able to calculate the inverse transform.

4.3.2 Weierstrass’s factor theorem

As mentioned above, we will require splitting a ratio of two functions of a complex variable

in α-plane. We here remind ourselves of Weierstrass’s factor theorem ([7] section 2.9) which

can be proved using the Mittag-Leffler theorem described in section 3.2.

Let H (α) denote a function that is analytic in the whole α-plane (except possibly at

infinity) and has zeros of first order at a0, a1, a2, ..., and no zero is located at the origin.

Consider the Mittag-Leffler expansion of the logarithmic derivative of H (α), i.e.,

d logH (α)

dα=

1

H (α)

dH (α)

dα

=d logH (0)

dα+

∞∑

n=0

[1

α− an

+1

an

].

Integrating both sides in [0, α] we have

logH (α) = logH (0) + αd logH (0)

dα+

∞∑

n=0

[log

(1 − α

an

)+α

an

].

Therefore, the expression for H (α) is

H (α) = H (0) exp

[αd logH (0)

dα

] ∞∏

n=0

(1 − α

an

)eα/an .

If H (α) is even, then dH (0) /dα = 0 and −an is a zero if an is a zero. Then we have

the simpler expression

H (α) = H (0)

∞∏

n=0

(1 − α2

a2n

).

4.3.3 Derivation of the Wiener-Hopf equation

We derive algebraic expressions for Φ± (α, z) using integral transforms (Eqn. (4.12)) on

Eqn. (4.4) and Eqn. (4.5). The Fourier transforms of Eqn. (4.5) according to the definition

83

given by Eqn. (4.12) gives

∂2

∂z2−(α2 + k2

)Φ± (α, z) = ±iαφ (0, z) − φx (0, z) .

Hence, the solutions of the above ordinary differential equations with the Fourier transform

of condition (4.6),

Φ±z (α,−H) = 0,

can be written as

Φ± (α, z) = Φ± (α, 0)cosh γ (z +H)

cosh γH± g (α, z) (4.13)

where γ =√α2 + k2 and g (α, z) is a function determined by iαφ (0, z) − φx (0, z),

g (α, z) =hz (α,−H)

γ(tanh γH cosh γ (z +H) − sinh γ (z +H))

+ h (α, z)

(1 − cosh γ (z +H)

cosh γH

),

h (α, z) =

∫ z sinh γ (z − t)

γφx (0, t) − iαφ (0, t) dt.

Note that Re γ > 0 when Reα > 0 and Re γ < 0 when Reα < 0. We have, by differentiating

both sides of Eqn. (4.13) with respect to z at z = 0

Φ±z (α, 0) = Φ± (α, 0) γ tanh γH ± gz (α, 0) (4.14)

where Φ±z (α, 0) denotes the z-derivative. We apply the integral transform to Eqn. (4.4) in

x < 0 and x > 0,

D1γ

4 −m1ω2 + ρg

Φ−

z (α, 0) − ρω2Φ− (α, 0) + P1 (α) = 0, (4.15)D2γ

4 −m2ω2 + ρg

Φ+

z (α, 0) − ρω2Φ+ (α, 0) − P2 (α) = 0, (4.16)

where

Pj (α) = Dj

[cj3 − i cj2α−

(α+ 2k2

) (cj1 − i cj0α

)], j = 1, 2,

c1i =

(∂

∂x

)i

φz

∣∣∣∣∣x=0−,z=0

, c2i =

(∂

∂x

)i

φz

∣∣∣∣∣x=0+,z=0

, i = 0, 1, 2, 3.

From Eqn. (4.14), Eqn. (4.15) and Eqn. (4.16) we have

f1 (γ) Φ−z (α, 0) + C1 (α) = 0 (4.17)

f2 (γ) Φ+z (α, 0) + C2 (α) = 0 (4.18)

84

where

fj (γ) = Djγ4 −mjω

2 + ρg − ρω2

γ tanh γH, j = 1, 2,

C1 (α) = −ρω2gz (α, 0)

γ tanh γH+ P1 (α) , C2 (α) =

ρω2gz (α, 0)

γ tanh γH− P2 (α) .

As we have seen in chapter 3, functions f1 and f2 are called dispersion functions and

the zeros of these functions are the primary tools in our method of deriving the solutions.

Notice that the dispersion functions have the same form as the one given in chapter 3 and

the reason for this is given in section 3.5.2 with the general scaling consideration.

Functions Φ−z (α, 0), and Φ+

z (α, 0) are defined in Imα < 0 and Imα > 0, respectively.

However they can be extended in the whole plane defined by Eqn. (4.17) and Eqn. (4.18) via

analytic continuation. Eqn. (4.17) and Eqn. (4.18) show that the singularities of Φ−z and Φ+

z

are determined by the positions of the zeros of f1 and f2, since gz (α, 0) is bounded and zeros

of γ tanh γH are not the singularities of Φ±z . We denote sets of singularities corresponding

to zeros of f1 and f2 by K1 and K2 respectively

Kj =α ∈ C | fj (γ) = 0, α =

√γ2 − k2 either Imα > 0 or α > 0 for α ∈ R

.

Fig. (4.3a, b) show the relative positions of the singularities. From Eqn. (4.17) and Eqn. (4.18)

Re

Im

l-l

Re

Im

-l

x

x

x

x-m

(a) (b)

it

-it

x x

Figure 4.3: Locations (not to scale) of the singularities which determine Φ−z (figure (a)) and

Φ+z (figure(b)). Thick arrow at − i τ in (a) and at i τ in (b) shows the integral path for the

inverse Fourier transform. Figures (a) and (b) illustrate how the negative real singularity−λ of Φ−

z is moved to become a singularity of Φ+z .

and using the Mittag-Leffler theorem ( [7] section 2.9), functions Φ±z can be expressed by a

series of fractional functions that contribute to the solutions. Thus, we have series expan-

sions of Φ−z and Φ+

z

Φ−z (α, 0) =

Q1 (−λ)

α+ λ+∑

q∈K1

Q1 (q)

α− q, Φ+

z (α, 0) =∑

q∈K2

Q2 (q)

α + q,

85

where λ is a positive real singularity of Φ−z and Q1, Q2 are coefficient functions yet to be

determined. Note that Φ−z (α, 0) has an additional term corresponding to −λ because of

the incident wave. The solution φ (x, 0), x < 0 is then obtained using the inverse Fourier

transform taken over the line shown in Fig. (4.3a)

φz (x, 0) =1

2π

∫ ∞−i τ

−∞−i τ

Φ−z e

− i αxdα = iQ1 (−λ) eiλx +∑

q∈K1

iQ1 (q) e− i qx (4.19)

where τ is an infinitesimally small positive real number. Note that k = λ sin θ. Similarly,

we obtain φ (x, 0) for x > 0 by taking the integration path shown in Fig. (4.3b), then we

have

φz (x, 0) =1

2π

∫ ∞+i τ

−∞+i τ

Φ+z e

− iαxdα = −∑

q∈K2

iQ2 (q) ei qx.

The Wiener-Hopf technique enables us to calculate coefficients Q1 and Q2 without know-

ing functions C1, C2, or φx (0, z) − iαφ (0, z). It requires the domains of analyticity of

Eqn. (4.17) and Eqn. (4.18) to have a common strip of analyticity which they do not have

right now. We create such a strip by shifting a singularity of Φ−z in Eqn. (4.17) to Φ+

z in

Eqn. (4.18) (we can also create a strip by moving a singularity of Φ+z , and more than one

of the singularities can be moved). Here, we shift −λ as shown in Fig. (4.3a), so that the

common strip of analyticity denoted by D is created on the real axis, which passes above

the two negative real singularities and below the two positive real singularities. We denote

the domain above and including D by D+ and below and including D by D−. Hence, the

zeros of f1 and f2 belong to either D+ or D−.

Let Ψ−z be a function created by subtracting a singularity from function Φ−

z . Then

Ψ−z (α, 0) is regular in D−. Since the removed singularity term makes no contribution to the

solution, from Eqn. (4.17), Ψ−z satisfies

f1 (γ) Ψ−z (α, 0) + C1 (α) = 0. (4.20)

Eqn. (4.18) becomes, as a result of modifying function Φ+z to a function denoted by Ψ+

z with

an additional singularity term,

f2 (γ) Ψ+z (α, 0) − f2 (λ′)Q1 (−λ)

α + λ+ C2 (α) = 0. (4.21)

Our aim now is to find a formula for

Ψz (α, 0) = Ψ−z (α, 0) + Ψ+

z (α, 0)

in α ∈ D so that its inverse Fourier transform can be calculated.

86

Adding both sides of Eqn. (4.20) and Eqn. (4.21) gives the Wiener-Hopf equation

f1 (γ) Ψ−z (α, 0) + f2 (γ) Ψ+

z (α, 0) − f2 (λ′)Q1 (−λ)

α + λ+ C (α) = 0 (4.22)

where C (α) = C1 (α) − C2 (α). This equation can alternatively be written as

f2 (γ)[f (γ)Ψ+

z (α, 0) − f2(λ′)Q1(−λ)α+λ

+ C (α)]

= −f1 (γ)[f (γ) Ψ−

z (α, 0) + f2(λ′)Q1(−λ)α+λ

− C (α)] (4.23)

where f (γ) = f2 (γ) − f1 (γ) .

We now modify Eqn. (4.23) so that the right and left hand sides of the equation become

regular in D− and D+ respectively. Using Weierstrass’s factor theorem given in the previous

subsection, the ratio f2/f1 can be factorized into infinite products of polynomials (1 − α/q),

q ∈ K1 and K2. Hence, using a regular non-zero function K (α) in D+,

K (α) =

(∏

q∈K1

q′

q + α

)(∏

q∈K2

q + α

q′

)(4.24)

where q′ =√q2 + k2, then we have

f2

f1= K (α)K (−α) .

Note that the factorization is done in the α-plane, hence functions f1 and f2 are here seen

as functions of α and we are actually factorizing

f2 (γ) γ sinh γH

f1 (γ) γ sinh γH

in order to satisfy the conditions given in the previous subsection. Then Eqn. (4.23) can be

rewritten as

K (α)[f (γ) Ψ+

z + C]−(K (α) − 1

K (λ)

)f2 (λ′)Q1 (−λ)

α + λ

= − 1

K (−α)

[f (γ)Ψ−

z − C]−(

1

K (−α)− 1

K (λ)

)f2 (λ′)Q1 (−λ)

α + λ. (4.25)

Note that the infinite products in Eqn. (4.24) converge in the order of q−5 as |q| becomes

large, thus numerical computation of K (α) does not pose any difficulties.

The left hand side of Eqn. (4.25) is regular in D+ and the right hand side is regular in D−.

Notice that a function is added to both sides of the equation to make the right hand side of

the equation regular in D−. The left hand side of Eqn. (4.25) is o (α4) as |α| → ∞ in D+,

87

since Ψ+z → 0 and K (α) = O (1) as |α| → ∞ in D+. The right hand side of Eqn. (4.25) has

the equivalent analytic properties in D−. Liouville’s theorem (Carrier, Krook and Pearson

[7] section 2.4) tells us that there exists a function, which we denote J (α), uniquely defined

by Eqn. (4.25), and function J (α) is a polynomial of degree three in the whole plane. Hence

J (α) = d0 + d1α + d2α2 + d3α

3.

Equating Eqn. (4.25) for Ψz gives

Ψz (α, 0) =−F (α)

K (α) f1 (γ)or − K (−α)F (α)

f2 (γ)(4.26)

where

F (α) = J (α) − Q1 (−λ) f2 (λ′)

(α + λ)K (λ).

Notice that procedure from Eqn. (4.23) to Eqn. (4.25) eliminates the need for calculating

constant C in Eqn. (4.25).

For x < 0 we close the integral contour in D+, and put the incident wave back, then we

have

φz (x, 0) = iQ1 (−λ) eiλx −∑

q∈K1

iF (q) q′R1 (q′)

qK (q)e− i qx, (4.27)

where R1 (q′) is a residue of [f1 (γ)]−1 at γ = q′

R1 (q′) =

(df1 (γ)

dγ

∣∣∣∣γ=q′

)−1

=

5D1q

′3 +b1q′

+H

q′

((D1q

′5 + b1q′)

2 − (ρω2)2

ρω2

)−1

. (4.28)

We used b1 = −m1ω2 + ρg and f1 (q′) = 0 to simplify the formula. Displacement w (x) can

be obtained by multiplying Eqn. (4.27) by − i /ω. Notice that the formula for the residue

is again expressed by a polynomial using the dispersion equation as shown in section 3.3.4,

which gives us a stable numerical computation of the solutions.

The velocity potential φ (x, z) can be obtained using Eqn. (4.13) and Eqn. (4.14),

φ (x, z) =iQ1 (−λ) cosh λ′ (z +H)

λ′ sinh λ′Heiλx −

∑

q∈K1

iF (q)R1 (q′) cosh q′ (z +H)

qK (q) sinh q′He− i qx

where λ′ =√λ2 + k2.

For x > 0, the functions φz (x, 0) and φ (x, z) are obtained by closing the integral contour

88

in D−,

φz (x, 0) = −∑

q∈K2

iK (q)F (−q) q′R2 (q′)

qei qx, (4.29)

φ (x, z) = −∑

q∈K2

iK (q)F (−q)R2 (q′) cosh q′ (z +H)

q sinh q′Hei qx,

where R2 is a residue of [f2 (γ)]−1 and its formula can be obtained by replacing the subscript

1 with 2 in Eqn. (4.28). Notice that since Rj ∼ O (q−9), j = 1, 2, the coefficients of φz of

Eqn. (4.29) decay as O (q−6) as |q| becomes large, so the displacement is bounded up to

the fourth x-derivatives. In a physical sense, the biharmonic term of the plate equation for

the vertical displacement is associated with the strain energy due to bending of the plate as

explained in chapter 2. Hence, up to fourth derivative of the displacement function should

be bounded, as has been confirmed. The coefficients of φ, have an extra 1/q′ tanh q′H term

which is O (q4), hence the coefficients decay as O (q−2) as |q| becomes large. Therefore, φ is

bounded everywhere including at x = 0.

Shifting a singularity of one function to the other is equivalent to subtracting an incident

wave from both functions then solving the boundary value problem for the scattered field

as in [5]. As mentioned, any one of the singularities can be shifted as long as it creates a

common strip of analyticity for the newly created functions. We chose −λ because of the

convenience of the symmetry in locations of the singularities. The method of subtracting

either incoming or transmitting wave requires the Fourier transform be performed twice, first

to express the solution with a series expansion, and second to solve the system of equations

for the newly created functions. Thus, we find the method of shifting a singularity shown

here is advantageous to other methods since it needs the Fourier transform only once to

obtain the Wiener-Hopf equation.

The polynomial J (α) is yet to be determined. In the following section the coefficients

of J (α) will be determined from conditions at x = 0±, −∞ < y <∞, z = 0.

4.4 Determination of J (α) from the transition condi-

tions

The four coefficients of J (α) are determined by physical conditions at x = 0, which we call

transition conditions. The transition conditions are expressed by displacement w, slope of

the ice sheet wx, effective shear force

B1w = Dj

wxxx − k2 (2 − ν)wx

(4.30)

89

and bending moment

B2w = Dj

wxx − k2νw

(4.31)

at x = 0+ for j = 1 and x = 0− for j = 2 (Shames and Dym [41] section 6.3). Since

each of the possible transitions conditions is a linear equation with respect to w, the set of

transition conditions can be expressed by algebraic equations of vector d = (d0, d1, d2, d3)t

made of the coefficients of polynomial J . We write

w|x=0± = A± · d + B±, wx|x=0± = C± · d + D±,

B1wx|x=0± = E± · d + F±, B2wx|x=0± = G± · d + H±

where A±, C±, E± and G± are row vectors, and B±, D±, F± and H± are scalar values that

are calculated from Eqns. (4.27, 4.29, 4.30, 4.31), and · denotes the vector inner product.

For computer codes, it may be convenient to compute the x-derivatives of w at x = 0±, then

to express the elements of each term. Let X1 (m,n) and X2 (m,n) be matrices of n rows m

columns corresponding to the x-derivatives of w (0−) and w (0+) respectively. Then, the

elements of the matrices are

X1 (m,n) = −∑

q∈K1

i qn−2q′R1 (q′) (− i q)m−1

K (q),

X2 (m,n) =∑

q∈K2

iK (q) (−q)n−2 q′R2 (q′) (i q)m−1 ,

We then can compute A+ = X2 (1, :), A− = X1 (1, :), C+ = X2 (2, :), C− = X1 (2, :), and

E+ = D2

X2 (4, :) − k2 (2 − ν)X2 (2, :)

,

E− = D1

X1 (4, :) − k2 (2 − ν)X1 (2, :)

,

G+ = D2

(X2 (3, :) − k2νX2 (3, :)

),

G− = D1

(X1 (3, :) − k2νX1 (3, :)

),

where : indicates all the elements in the row as used in MatLab. The scalar terms are,

Y1 (m) = iQ1 (−λ)

(iλ)m−1 +

∑

q∈K1

q′R1 (q′) f2 (λ) (− i q)m−1

q (λ+ q)K (q)K (λ)

,

Y2 (m) =∑

q∈K2

iK (q) q′R2 (q′) f2 (λ)Q1 (−λ) (i q)m−1

q (λ− q)K (λ).

Therefore, B+ = Y2 (1), B− = Y1 (1), D+ = Y2 (2), D− = Y1 (2),

F+ = D2 Y2 (4) − k2 (2 − ν) Y2 (2), F− = D1 Y1 (4) − k2 (2 − ν) Y1 (2),H+ = D2 Y2 (3) − k2νY2 (1), and H− = D1 Y1 (3) − k2νY1 (1).

90

From chapter 2, we know that there must be four transition conditions where two plates

interact, which are either specified values or continuity conditions at x = 0+ and x = 0−.

The following sets out the sets of conditions that have been studied in the past.

water water

(a) (b)

Figure 4.4: Magnified view of two examples of transition conditions, (a) shows freely movingtransition and (b) shows joined transition.

Condition 1 (Dock problem) D1 = m1 = 0, w|x=0+ = 0. This is a generalization of

the usual dock problem that has D2 = ∞.

Condition 2 (Ocean wave-ice interaction) D1 = m1 = 0, B1wx|x=0+ = 0 and

B2wx|x=0+ = 0. Plane ocean waves are incident into a semi-infinite ice sheet on the right

hand side of the ocean surface.

Condition 3 (Ice-ice interaction, open crack) B1wx|x=0± = 0 and B2wx|x=0± = 0.

(See Fig. (4.4a).) The ocean surface is covered by two ice sheets that are joined at x = 0,

and the edge of each ice sheet is free to move.

Condition 4 (Ice-ice interaction, continuous joint) See Fig. (4.4b).

B1wx|x=0− = B1wx|x=0+ , B2wx|x=0− = B2wx|x=0+ ,

w|x=0− = w|x=0+ , wx|x=0− = wx|x=0+ .

The ocean surface is covered by two ice sheets that are joined at x = 0, and the transition

of the ice sheets is frozen solid. A difference in thickness or rigidity of the ice sheets causes

the reflection of the plane waves at the transition.

The solutions for all of the above conditions 1 to 4 can be found by simply finding the

polynomial J (α). In the following subsections from 4.4.1 to 4.4.4 we show how the solutions

for the conditions 1 to 4 can be obtained.

4.4.1 Dock problem

When x < 0 is free surface and x > 0 is covered by rigid plate, the boundary value problem

is called a dock problem (Roos [40] section 3.10). Now the surface condition for x > 0 is

w (x) = φz (x, 0) = 0. Thus we solve the Wiener-Hopf equations for Φ± (α, 0) instead of

91

Φ±z (α, 0). We have a Wiener-Hopf equation

f1 (γ) Φ− (α, 0) + f2 (γ)Φ+ (α, 0) = 0.

Since m1 = 0 and D1 = 0, the dispersion functions become

f1 (γ) = ρgγ tanh γH − ρω2, f2 (γ) = γ tanh γH

The dispersion function f1 has two real zeros and an infinite number of pure imaginary

zeros. The dispersion function f2 has zeros on the imaginary axis, at inπ/H , n = 0, 1, 2....

For this case there is no need to shift −λ since there are only imaginary wave numbers

in x > 0. However, by following the calculation in section 4.4, we find that J (α) = d0

and Eqn. (4.27) and Eqn. (4.29) remain unchanged except now the formulae are for φ (x, 0)

instead of φz (x, 0). Hence, the velocity potentials for x < 0 and x > 0 are

φ (x, z) = − g

ω2

∑

q∈K1∪−λ

i d0q′R1 (q′)

qK (q)e− i qx cosh q′ (z +H)

cosh q′H,

φ (x, z) = −∑

q∈K2

i d0K (q)

qHei qx cosh q′ (z +H)

cosh q′H,

respectively. The constant d0 can be determined by the amplitude of the incident wave.

Note that q ∈ K2 are i√n2π2/H2 + k2, n = 0, 1, 2... and R2 (q′) is

R2 (q′) =1

q′H.

4.4.2 Ocean wave and ice sheet

We consider a free surface (D1 = 0 and m1 = 0) for x < 0 and an ice sheet at the surface

for x > 0. Then the dispersion equations are

f1 (γ) = ρg − ρω2

γ tanh γH,

f2 (γ) = D2γ4 −m2ω

2 + ρg − ρω2

γ tanh γH.

The dispersion function f1 has two real zeros and infinite number of pure imaginary zeros.

By following the calculations in section 4.4 we find that J (α) = d0 + d1α. Hence, the

transition conditions matrix is now 2 × 2, so we have

[E+

G+

]d +

(F+

H+

)=

(0

0

)(4.32)

92

where the first term is a matrix made of row vectors E+ and G+.

If bending and shear are specified on the edge of the ice instead of the free edge conditions,

then we have [E+

G+

]d +

(F+

H+

)=

(B1w|x=0+

B2w|x=0+

). (4.33)

The elements of the matrices and vector are calculated using the formulae shown in the

beginning of this section (4.4.)

A brief history of the wave-ice interaction problem can be found in papers by Balm-

forth and Craster [5], Chung and Fox [10], Evans and Davies [17], Fox and Squire [23] and

Gol’dshtein and Marchenko [24] (infinite depth case).

The wave-ice interaction problem is an example where b1 and b2 are non-zero in Eqn. (3.69)

and Eqn. (3.70) in subsection 3.6.2, i.e., the solutions are dependent on the inhomogeneous

boundary terms. Later in section 4.7, we will study the non-dimensional solutions of the

wave-ice interaction problem together with a comparison between the simplified method

shown here and a conventional way of applying the Wiener-Hopf technique to the wave-ice

interaction problem.

4.4.3 Open crack problem

We consider the case when the transition is free to move (open crack). Vector d is computed

from

E−

E+

G−

G+

d +

F−

F+

H−

H+

=

0

0

0

0

. (4.34)

The elements of the matrix and vector in Eqn. (4.34) can be calculate using the formulae

in the beginning of this section (4.4). If the thickness of the two ice sheet is equal, i.e.,

D1 = D2, and transition is free to move, then we have one dispersion function

f1 (γ) = f2 (γ) = D1γ4 −m1ω

2 + ρg − ρω2

γ tanh γH.

Note that there is no need for factorization shown in Eqn. (4.24) in section 4.3.3 since,

K = 1. The sets of wave numbers K1 and K2 are the same. The incident wave does not

automatically appear in this case since the wave number for an incident wave and travelling

93

wave are the same. Hence, Eqn. (4.29) must be changed to

φz (x, 0) =

(I − iF (−λ)λ′R2 (λ′)K (λ)

λ

)ei λx

−∑

q∈K1\λ

iF (−q) q′R2 (q′)K (q)

qei qx.

An alternative method is shown by Squire and Dixon [43] using a Green’s function

satisfying the plate equation at z = 0 to derive analytical solutions for the case of a normally

incident wave.

4.4.4 Two semi-infinite ice sheets

If the transition is connected (so the surface of the ice sheet at the transition is smooth), d

can be obtained from

A− −A+

C− − C+

E− − E+

G− − G+

d +

B− − B+

D− −D+

F− − F+

H− −H+

=

0

0

0

0

. (4.35)

The elements of the matrix and vector in Eqn. (4.35) can be calculated using the formulae

in the beginning of this section (4.4).

The conditions at x = 0 can be expressed by a matrix made of row vectors A±, C±, E±,

G± and a vector made of B±, D±, F±, H±. Hence, we may say that problems in subsections

4.1 to 4.4 are now reduced to algebraic Eqns. (4.32, 4.33, 4.34, 4.35) and sets of zeros of

the dispersion functions, K1 and K2.

4.5 Reflection and transmission coefficients

In this section we show the numerical computation of the solution in the form of the reflection

coefficients.

As |x| → ∞ only the oscillating waves of the displacement are significant, that is,

w (x) → Tei µx as x→ ∞w (x) → Iei λx +Re− iλx as x→ −∞. (4.36)

We have a relation between the transmission and reflection coefficients, which is called a

power-flow relation (Evans and Davies [17], Fox and Squire [23]) derived from the energy

conservation law given by Eqn. (3.41) and Eqn. (3.42). The transmission and reflection

94

coefficients satisfy

sT 2 + R2 = 1 (4.37)

where T = |T | / |I| and R = |R| / |I| are the ratios between the magnitude of the displace-

ment of the transmitted and reflected waves and the amplitude of incident wave. Notice

that Q1 (−λ) in Eqn. (4.19) is found in all the coefficients in Eqn. (4.27) and Eqn. (4.29).

Hence the calculation of T and R can become simple by setting Q1 (−λ) to be ω so that

the amplitude of the incident wave becomes |I| = 1 in Eqn. (4.36). The formulae for the

transmission and reflection coefficients are

T = |T | =

∣∣∣∣µ′F (−µ)R2 (µ′)K (µ)

µ

∣∣∣∣ ,

R = |R| =

∣∣∣∣λ′F (λ)R1 (λ′)

λK (λ)

∣∣∣∣ .

The multiplying factor s is (see appendix C)

s =Re (µ)λ′2 sinh 2λ′H

Re (λ)µ′2 sinh 2µ′H

2µ′H (D2µ′4 + b2) + (5D2µ

′4 + b2) sinh 2µ′H

2λ′H (D1λ′4 + b1) + (5D1λ′4 + b1) sinh 2λ′H

The power-flow relation (4.37) holds for all transition conditions that do not introduce any

potential energy to the system. Note that when D1 = 0 and m1 = 0 the formula for the

factor s is reduced to the one shown in [23] for ocean wave-ice interaction problem.

We used Eqn. (4.37) to check the accuracy of the numerical computation shown in

Fig. (4.5) to Fig. (4.8). We set the water depth H = 10 m. The reflection coefficient R is

plotted as a function of incident wave radial frequency ω and angle in degree. The frequency

axis is logarithmic and the incident angle is in degrees. In terms of numerical computation,

the deeper the water is, the more roots are needed to achieve accurate solutions and the

reflection coefficients.

Fig. (4.5) shows the reflection coefficient for the wave-ice interaction problem in subsec-

tion 4.4.2. The top flat surface of the graphs (except Fig. (4.6)) represents the region of

100 percent reflection of the wave energy occurs. Thus, the edge of the top surface gives

the curve of the critical incident angle versus the radial incident wave frequency. Fig. (4.6)

shows the reflection coefficient for the open crack problem of two ice sheets with the same

thickness described in subsection 4.3. Fig. (4.7) and Fig. (4.8) show the reflection coefficient

for an open crack and a continuous joint in subsections 4.4.3 and 4.4.4. Since, the critical

angle is determined only by the real roots of the dispersion equations, the regions of the

total reflection are the same in Fig. (4.7) and Fig. (4.8). Comparison between Fig. (4.7) and

Fig. (4.8) confirms an obvious physical fact that waves transmit better if the ice sheets are

frozen together. However, at an incident angle greater than 40 degrees, the transition condi-

tions make no difference to the wave propagation across the transition. Studying Fig. (4.6)

95

0.1

1

10

020

4060

80

0

0.2

0.4

0.6

0.8

1

frequency (radian/sec)incident angle (degree)

refle

ctio

n co

effic

ient

Figure 4.5: Three dimensional plot of the reflection coefficient for wave-ice interaction prob-lem when water depth H = 10 m and ice thickness h2 = 1.0 m.

0.1

1

10

020

4060

800

0.2

0.4

0.6

0.8

1

frequency (radian/sec)incident angle (degree)

refle

ctio

n co

effic

ient

Figure 4.6: Three dimensional plot of the reflection coefficients for open crack problem whenH = 10 m, h1 = h2 = 1.0 m.

96

0.1

1

10

0

20

40

60

80

0

0.2

0.4

0.6

0.8

1

frequency (radian/sec)Incident angle (degree)

Ref

lect

ion

coef

ficie

nt

Figure 4.7: Three dimensional plot of the reflection coefficients when the thicknesses of icesheets are h1 = 0.25 m and h2 = 1.0 m. The water depth is 10 m. The ice sheets are free tomove at the transtion.

0.1

1

10

0

20

40

60

80

0

0.2

0.4

0.6

0.8

1

frequency (radian/sec)Incident angle (degree)

Ref

lect

ion

coef

ficie

nt

Figure 4.8: Three dimensional plot of the reflection coefficients when the thicknesses of icesheets are h1 = 0.25 m and h2 = 1.0 m. The water depth is 10 m. The transtion is connected.

97

and Fig. (4.7) tells us that the region of the total reflection diminishes as h1 tends to h2.

When h1 = h2, non-zero wave transmission occurs at all incident angles. Furthermore at

each incident angle there exists a clear cut-off radial frequency, say ωθ, i.e., for ω < ωθ, Ris nearly zero.

4.6 Deep water solution

We extend the method shown in the previous section to deep water solutions using the fact

that the infinite summations over the pure imaginary zeros can be replaced by integrals as

H → ∞. We here show the tedious but straightforward calculation.

As depicted in Fig. (3.2), the pure imaginary zeros of f1 (γ) and f2 (γ) on the γ-plane,

denoted by i γn and i δn, n ∈ N, respectively become equally spaced,

γn → nπ

H, δn → nπ

Has H → ∞.

The infinite products in function K over the pure imaginary zeros can be alternatively

written as∞∏

n=1

√1 + k2

δ2n

− iαδn√

1 + k2

γ2n

− i αγn

Taking the log and using an approximation log (1 + x) ≈ x for small x, we have

∞∑

n=1

log

√1 + k2

δ2n

− iαδn√

1 + k2

γ2n

− i αγn

→∞∑

n=1

(1 − γn

δn

)(iα− k2

2√

γn+k2

)

√γn + k2 − iα

. (4.38)

From the relative positions of γn and δn expressed by curves of functions

tan γnH,−ρω2

γn (D1γ4n + b1)

, tan δnH,−ρω2

δn (D2δ4n + b2)

,

we find that from appendix B

1 − γn

δn→ 1

Hv (ξn) ,

v (ξn) =−ρω2

(f2 (ξn) − f1 (ξn)

)t2 (ξn)

ξ3nf2 (i ξn) f2 (− i ξn) t1 (ξn)

98

where ξn = nπ/H , n = 1, 2, ... and fj and tj (γ) are

fj (γ) = Djγ4 + ρg −mjω

2 − ρω2

γ,

tj (γ) = Djγ4 + ρg −mjω

2, j = 1, 2.

Now, Eqn. (4.38) becomes an integration over the real axis as H → ∞, which is denoted by

κ (α)

κ (α) =1

π

∫ ∞

0

v (ξ)iα− k2

2√

ξ2+k2

√ξ2 + k2 − iα

dξ.

Hence, taking exponential of κ and multiplying the rest of the singularity terms, and then

we have the factorizing function

K (α) = eκ(α)

∏

q∈µ,−µ∗

D,µD

q + α

q′

∏

q∈λ,λD,−λ∗

D

q + α

q′

where λD and µD are complex wave numbers corresponding to the complex zeros of f1 and

f2 in the first quadrant respectively.

When k = 0 (normally incident waves) we can calculate the integral of function κ

analytically using Stieltjes transform of a fractional function [[1] formula 14.2.(3)],

∫ ∞

0

1

ξ2 + q2

1

ξ − iαdξ =

1

α− q2

[i πα

2q+ log

− iα

q

],

where Re q > 0 and |arg (− iα)| < π. Since v (0) is bounded for ω 6= 0, κ can be expressed

as a summation over the poles of the integrand. We have

κ (α) =1

π

∑

q

∫ ∞

0

2qvr (q)

ξ2 + q2

iα

ξ − iαdξ =

i

π

∑

q

αqvr (q)

α− q2

[i πα

2q+ log

− iα

q

]

where vr (q) is residue of the integrand v (i q) at its poles q in the upper half plane. The

summation over γn in Eqn. (4.27) becomes

∞∑

n=1

iF (iαn) γnR1 (i γn)

αnK (iαn)eαnx → i

π

∫ ∞

0

F (i ξ′) ξG1 (i ξ)

ξ′K (i ξ′)eξ′xdξ (4.39)

99

where ξ′ =√ξ2 − k2 and function G1 is a limit of residue R1 when H → ∞,

R1 (i γn) → G1 (i γn)

H,

G1 (i γn) =1

2

(1

f1 (i γn)− 1

f1 (− i γn)

).

Hence, φz (x, 0) for x < 0 is

φz (x, 0) = Iei λx −∑

q∈λD,−λ∗

D

iF (q) q′R1 (q′)

qK (q)e− i qx

− i

π

∫ ∞

0

F (i ξ′) ξG1 (i ξ)

ξ′K (i ξ′)eξ′xdξ.

Similarly, φz (x, 0) for x > 0 is

φz (x, 0) = −∑

q∈µ,µD,−µ∗

D

iK (q)F (−q) q′R2 (q′)

qei qx

− i

π

∫ ∞

0

K (i ξ′)F (− i ξ′) ξG1 (i ξ)

ξ′e−ξ′xdξ

where

Rj (q′) =q′2

4Djq′5 + ρω2 sgn (Re q′).

Although the integrals in the solution have to be computed numerically, the derivation

procedure of extending the finite solution seems natural and straightforward compared to

the method used in Balmforth and Craster [5] and Gol’dshtein and Marchenko [24].

4.7 Scaled solution for wave-ice interaction

The scaling used in chapter 3 can be applied for the wave-ice interaction problem described

in subsection 4.4.2, since the free surface condition given by Eqn. (4.4) when D1 = m1 = 0

becomes dimensionless when scaled using lc and tc. In this section we again derive the

Wiener-Hopf equation and present a conventional way of applying the Wiener-Hopf tech-

nique shown by Chung and Fox [10, 9], which is a modified version of the method originally

developed by Evans and Davies [17]. Hence, the strip of common analyticity will be found

from the asymptotic behavior of the transmitted waves, which we know from the property

of the mathematical model we have chosen to use.

100

4.7.1 Derivation of the Wiener-Hopf Equation

Since the solutions of Helmholtz equation (4.5) can be obtained by the separation of vari-

ables, suppose that a solution φ (x, z) can be expressed in the separation-of-variables form,

exp (± iαx) exp (±γz). Then, combining the exp (γz) and exp (−γz) terms, and Eqn. (4.3)

gives a form of solution

exp (± iαx) cosh γ (z +H) . (4.40)

The complex parameters (α, k, γ) satisfy the dispersion equations obtained by substituting

Eqn. (4.40) into the free surface condition and plate equation for φ (x, z),

ω2φ− φz = 0, for x < 0, z = 0, (4.41)(

∂

∂x2− k2

)2

−mω2 + 1

φz − ω2φ = 0, for x > 0, z = 0. (4.42)

Hence, substituting Eqn. (4.40) to Eqn. (4.41) and Eqn. (4.42) for φ, we find the dispersion

equations for (α, k, γ)

γ2 = α2 + k2, (4.43)

fsea (γ) = ω2 cosh γH − γ sinh γH = 0, (4.44)

fice (γ) = ω2 cosh γH −(γ4 + 1 −mω2

)γ sinh γH = 0. (4.45)

Note that Eqn. (4.43) and Eqn. (4.44) are in a slightly different form to the same dispersion

equations that we have seen in the previous sections. It is shown by Lawrie and Abrahams

[33] that the functions given by Eqn. (4.40) for γ and α satisfying Eqns. (4.43,4.44,4.45)

form an orthogonal basis of the solutions of Laplace’s equation with open sea and ice sheet

conditions on the surface.

Re

Im

-lxx

-m

x

x

x

x

iln

x

-imn

mD

mD

x*x

x

x

-iln

imn

m

x

-mDx *

x-mD

-ikD

l

D+

D-

Figure 4.9: Schematics of the positions (not to scale) of the wavenumbers and the domaisof analyticity D, D+ and D− on the α plane.

101

The modes due to complex wave numbers µD and µ∗D exponentially decay faster than

e−ky, since Reµ′D < Imµ′

D, so Reµ′2D < 0 and Re (k2 − µ′2

D) > k2. Thus, the damped

travelling mode (µD-term) decays faster than exp (−kx), i.e.,

exp (iµDx) = exp

(−x√k2 − µ′2

D

)< A exp (−kx) .

Hence, only when µ′ > k, that is µ is real, does a wave propagate through the ice sheet.

When µ′T < k, all the wave modes in x > 0 are exponentially decaying. Since k = λ′ sin θ,

when the incident angle θ is greater than a critical angle θT at which µ′ = λ′ sin θ there is no

wave propagation through the ice sheet. It follows that the derivation of the Weiner-Hopf

equation by Evans and Davies [17] is based on the following fact

φ (x, z) =

O (1) as x→ −∞,

T ei µx coshµ′ (z +H) +O(e−kx

)as x→ ∞,

(4.46)

hence function ψ (x, z) which is defined as

ψ (x, z) = φ (x, z) − Tei µx coshµ′ (z +H)

is

ψ (x, z) =

O (1) as x→ −∞

O(e−kx

)as x→ ∞

.

Hence, the Fourier transform of ψ (x, z) denoted by Ψ (α, z) converges in a strip-like domain

D = α ∈ C : −k < Imα < 0 shown in Fig. (4.9) and is a regular function of α ∈ D. Note

that we again use the same notation D as in the previous sections for the common strip

of analyticity which has finite width k this time. Hence, ψ (x, z) can be obtained using

Eqn. (4.10) in the strip D.

We now rewrite the system of equations for the function ψ (x, z), and then follow the

same method of the previous sections. The Fourier transform of Laplace’s equation and the

bottom condition for −∞ < x <∞ are

(∂2

∂z2− γ2

)Ψ (α, z) = 0,

∂

∂zΨ (α,−H) = 0. (4.47)

Thus, a solution of Eqn. (4.47) can again be expressed as

Ψ (α, z) = Ψ (α, 0)cosh γ (z +H)

cosh γH, (4.48)

and also

Ψz (α, 0) = Ψ (α, 0)γ tanh γH, (4.49)

102

for α ∈ D. Note that from Eqn. (4.48) and Eqn. (4.49), Ψ (α, z) can be obtained from

Ψz (α, 0). We denote the half space Fourier transform of ψ (x, 0) and ψz (x, 0) defined

by Eqn. (4.12) by Ψ± (α) and Ψ±z (α) respectively. Then, Ψ+ and Ψ+

z are regular in

D+ = α ∈ C : −k < Imα, and Ψ−z and Ψ−

z are regular in D− = α ∈ C : Imα < 0 (see

Fig. (4.9)).

We find from Eqn. (4.41) that the non-dimensional free-surface condition for ψ is

ψz − ω2ψ − Teiµx(µ′4 −mω2

)µ′ sinh µ′H = 0. (4.50)

Note that we used Eqn. (4.45) to simplify the sinh-term. Thus, the transform of Eqn. (4.50)

becomes

ω2Ψ− (α) = Ψ−z (α) − iAT

α+ µ, α ∈ D− (4.51)

where

A = −(µ′4 −mω2

)µ′ sinh µ′H

Note that it is assumed that ψz is integrable and ψ is bounded for x ≤ 0.

Similarly, the transform of Eqn. (4.42) gives

ω2Ψ+ =(γ4 + 1 −mω2

)Ψ+

z −(c3 − i c2α−

(α2 + 2k2

)(c1 − i c0α)

)(4.52)

=(γ4 + 1 −mω2

)Ψ+

z −M3, α ∈ D+

where the four constants, cj , j = 0, 1, 2, 3 are the derivatives of ψz at x = 0+ with respect

to x, i.e.,

cj =

(∂

∂x

)j

ψz (x, 0)

∣∣∣∣∣x=0+

.

It is clear from Eqn. (4.52) that up to ∂3x-derivative of ψz is O (exp−kx) as x → ∞. It is

assumed that ∂4xψz is integrable for x ≥ 0.

Adding both sides of Eqn. (4.51) and Eqn. (4.52) gives a typical Wiener-Hopf equation

fice (γ) Ψ+z (α) + fsea (γ) Ψ−

z (α) + C (α) = 0, α ∈ D (4.53)

where

C (α) =

(iAT

α+ µ+M3 (α)

)γ sinh γH.

Note that Eqn. (4.49) is used to transform Ψ to Ψ−z and functions in Eqn. (4.53) are regular

and non-zero in strip D and Ψ±z are regular in D± respectively.

The solution of Eqn. (4.53), Ψ−z (α), can be obtained by decomposing fice/fsea into two

functions as K+/K−, i.e.,fice

fsea=K+

K−

103

where K± are regular non-zero in D± respectively. Note that the above decomposition is

done in the α-plane, so that the dispersion function fice and fsea are seen as functions on the

α-plane here. The decomposition is again done by expressing fice/fsea by infinite products

of polynomials of roots using Weierstrass’s factor theorem, hence K± are

K+ (α) =∏

q∈S−

q′

q − α

∏

q∈I−

q − α

q′,

K− (α) =∏

q∈S+

q − α

q′

∏

q∈I+

q′

q − α,

where q′ =√q2 + k2. The sets S± consist of the zeros of fsea in D±, and I± the zeros of fice

in D±, respectively. Note that unlike in subsection 4.3.2, the roots in the complex plane are

not divided symmetrically here, which makes the factorization slightly more complicated

than it was in Eqn. (4.24). The factorization shown here is slightly different from the one

in the previous sections because the singularities in the upper and lower half planes are not

symmetric. It is clear that K± are indeed regular non-zero in D± respectively.

Notice that

λn = πn/H +O(n−1), µn = πn/H +O

(n−5)

as n → ∞. The infinite products of K+ over the evanescent modes can alternatively be

expressed as

∞∏

n=1

√1 + k2

µ2n

− i αµn√

1 + k2

λ2n

− iαλn

=

∞∏

n=1

(λn

µn

) ∞∏

n=1

õ2

n + k2 − iα√λ2

n + k2 − iα

Now λn/µn = 1 +O (n−2) so that∏∞

n=1 λn/µn converges. Also

∞∏

n=1

õ2

n + k2 − iα√λ2

n + k2 − iα=

∞∏

n=1

(1 + gn) , (4.54)

where

gn (α) =

õ2

n + k2 −√λ2

n + k2

√λ2

n + k2 − iαfor n = 1, 2, 3....

Clearly gn (α) → 0 as |α| → ∞, α ∈ D+ and |gn (α)| = O (n−2). Hence, the infinite products

(4.54) tends to one as |α| → ∞ in D+. Similarly, the infinite products in K− are also O (1).

Hence,

K+ (α) = O (α2) as |α| → ∞, α ∈ D+,

K− (α) = O (α−2) as |α| → ∞, α ∈ D−.

Using the identity

fice − fsea = −(γ4 −mω2

)γ sinh γH,

104

Eqn. (4.53) becomes

fice

(γ4 −mω2

)Ψ+

z − iAT

α+ µ−M3 (α)

= −fsea

(γ4 −mω2

)Ψ−

z +iAT

α + µ+M3 (α)

.

Thus, substituting K+/K− for fice/fsea gives

K+ (γ4 −mω2)Ψ+z −M3 (α) − iAT (K+(α)−K+(−µ))

α+µ

= −K− (γ4 −mω2) Ψ−z +M3 (α) − iAT (K−(α)−K+(−µ))

α+µ.

(4.55)

Note that − iATK+(−µ)α+µ

is added to both sides in order to avoid α = −µ becoming a singularity

in D+.

From Eqn. (4.52), the left hand side of Eqn. (4.55) is

K+

ω2Ψ+ − Ψ+

z

− iAT (K+ (α) −K+ (−µ))

α + µ.

Since ψ and ψz are bounded for x ≥ 0, Ψ+,Ψ+z → 0 as |α| → ∞ in D+. It has been shown

that K+ (α) = O (α2) as |α| → ∞ in D+, and M3 = O (α3). Thus the left hand side of

Eqn. (4.55) is o (α2) as |α| → ∞ in D+. Similarly, the right hand side of Eqn. (4.55) is also

o (α2) (see [38]).

Both the right and the left hand sides of Eqn. (4.55) are analytic in D. Thus by analytic

continuation, Eqn. (4.55) defines a function J (α) that is regular in the whole plane. Fur-

thermore, by Liouville’s theorem, J (α) is a polynomial of degree one, i.e., J (α) = a1α+a0.

Hence, equating the both sides of Eqn. (4.55) to J and solving for Ψ±z gives

(γ4 −mω2

)Ψz (α) =

(J (α) − iATK+ (−µ)

α+ µ

)(1

K+ (α)− 1

K− (α)

). (4.56)

From Eqn. (4.48) and Eqn. (4.49)

Ψ (α, z) =F (α) cosh γ (z +H)

K+ (α) dsea (γ) cosh γHor

F (α) cosh γ (z +H)

K− (α) dice (γ) cosh γH(4.57)

where

F (α) = a1α + a0 −iATK+ (−µ)

α + µ,

dsea (γ) = ω2 − γ tanh γH,

dice (γ) = ω2 −(γ4 + 1 −mω2

)γ tanh γH.

105

It is clear that Ψ (α) is O (α−2) as |α| → ∞. Thus, the inverse transform can be calculated

by the contour integration over a semi-arc in either D+ or D−.

4.7.2 Determination of the solutions

Since Ψ (α, z) is singular only at the roots of dsea (γ) and dice (γ) and α = −µ, Ψ (α, z)

can be expanded by fractional functions of the sets of wavenumbers S = S+ ∪ S−, I =

I+ ∪ I− and α = −µ by the Mittag-Leffler theorem in [7],

Ψ (α, z) =

F (α)

K+ (α)

∑q∈S

cosh q′ (z +H)

cosh q′H

q′Rsea (q′)

q (α− q)F (α)

K− (α)

∑q∈I

cosh q′ (z +H)

cosh q′H

q′Rice (q′)

q (α− q)

where the residues Rsea and Rice of 1/dsea and 1/dice are

Rsea (q) =[− tanh qH − qH

(1 − tanh2 qH

)]−1,

Rice (q) =[qH(q4 + 1 −mω2

) (tanh2 qH − 1

)−(5q4 + 1 −mω2

)tanh qH

]−1.

Then, the inverse Fourier transform of Ψ (α, z) can obtained using the contour integration

over the residues of the integrand in either D− or D+.

For x < 0 we take the contour in D+, then

ψ (x, z) = −Teiµx cosh µ′ (z +H)

+∑

q∈S+

iF (q) q′Rsea (q′)

qK+ (q)e− i qx cosh q′ (z +H)

cosh q′H(4.58)

And for x > 0, the contour is taken in D−. Hence

ψ (x, z) = −∑

q∈I−

iF (q) q′Rice (q′)

qK− (q)e− i qx cosh q′ (z +H)

cosh q′H. (4.59)

Note that the coefficients of the solutions are O (q−2) as |q| → ∞, thus infinite summations

in Eqn. (4.58) and (4.59) converge for any x and z.

Solutions (4.58) and (4.59) contain two unknown constants a1 and a0. Substituting

Eqn. (4.59) to Condition 2, given in section 4.4, determines these constants.

Rewriting Condition 2 for ψ (x, z) gives

ψzxx − νk2ψz = Tµ′ (µ2 + νk2) sinh µ′H

ψzxxx − (2 − ν) k2ψzx = iTµ′µ (µ2 + (2 − ν) k2) sinh µ′H(4.60)

106

at x = 0+, z = 0. Substituting Eqn. (4.59) to Eqn. (4.60) gives two equations for a0 and a1.

The derivatives of ψ can be expressed as

(∂

∂x

)n

ψz (0+, 0) = An ·(a0

a1

)+ bn, n = 0, 1, 2, 3,

where An, n = 0, 1, 2, 3 are row vectors with m’th element

(An)m =∑

q∈I−

(− i q)n qm−1G (q, q′) , m = 1, 2.

The scalar term bn is

bn = − iATK+ (−µ)∑

q∈I−

(− i q)nG (q, q′)

q + µ

for n = 0, 1, 2, 3 where

G (q, q′) =− i q′2Rice (q′)

qK− (q)tanh q′H.

Hence, from Eqn. (4.60), coefficients a0 and a1 satisfy

[A2 − νk2A0

A3 − (2 − ν) k2A1

](a0

a1

)= Tµ′ sinh µ′H

(µ2 + νk2

iµ (µ2 + (2 − ν) k2)

)

−(

b2 − νk2b0

b3 − (2 − ν) k2b1

). (4.61)

Note that solutions expressed by Eqn. (4.58) and Eqn. (4.59) are derived without using

the natural boundary conditions (4.60). The natural boundary conditions are expressed

simply by Eqn. (4.61).

The formulae for the solution contain many exponentials and since as ω or H become

large, sinh, cosh and tanh become very sensitive to numerical errors in the roots and the

formulae are not suitable for numerical computation. However, it is possible to remove the

exponentials using Eqn. (4.44) and Eqn. (4.45).

The two constants a1 and a0 of function, J (α), are multiplied by the coefficient T . Thus,

the coefficients in Eqn. (4.58) and Eqn. (4.59) have T in front of them and every term of the

function ψ (x, z) is multiplied by T , so T can be set as T = 1/ coshµ′H to make formulae

simple.

107

Equating Eqn. (4.58) and Eqn. (4.59) for the tanh function gives

tanh q′H =

ω2

q′ (q′4 −mω2 + 1), q′ ∈ I

ω2

q′, q′ ∈ S

(4.62)

so that AT and the first vector of the right hand side of Eqn. (4.61) become

−ω2 (µ′4 −mω2)

µ′4 −mω2 + 1, and

ω2

µ′4 −mω2 + 1

[µ2 + νk2

iµ (µ2 + (2 − ν) k2)

]

respectively. The tanh functions in the formulae for Rsea, Rice, and G can also be removed,

Rsea (q′) =−q′

ω2 +H (q′2 − ω4),

Rice (q′) =−q′ (q′4 −mω2 + 1)

Hq′2 (q′4 −mω2 + 1)2 − ω4

+ ω2 (5q′4 −mω2 + 1)

,

G (q, q′) =− i q′Rice (q′)

qK− (q)

ω2

q′4 −mω2 + 1.

Notice thatG is O (q′−7) as |q′| → ∞ so that the infinite summations in Eqn. (4.61) converge.

All coefficients of the various modes have been expressed by polynomials in the roots of

the dispersion equations for the free surface and the ice sheet. By examining the order of

the coefficients in the solution, boundedness of the derivatives of the solution can be shown.

In the expression for Eqn. (4.58) ψz (x, 0) for x ≤ 0 the summation over iλn becomes,

from Eqn. (4.58)∞∑

n=1

iF (iλn)λ′nRsea (iλ′n)

λnK+ (iλn)eλnxλ′n tanλ′nH.

Hence the coefficients in the summation are O (λ−2n ) from the identity (4.62). Similarly,

in the expression for ψz (x, 0) for x ≥ 0 the coefficients of the summation are found to

be O (µ−6n ). Thus each derivative up to the fourth derivative of ψz (x, 0) with respect to

x is bounded for any x ≥ 0. Hence, the assumptions made for the Fourier transform of

Eqn. (4.51) and (4.52) are justified.

Only the integrability of ψz (x, 0) , x ≤ 0 and ψzxxxx (x, 0) , x ≥ 0 were required and

Evans and Davies [17] claimed that those two functions have log like singularity at x = 0 by

studying the transform functions Ψ (α, 0) for |α| → ∞. However, the solution expressed by

the polynomials of the roots of the dispersion equations revealed that ψz (x, 0) , x ≤ 0 and

ψzxxxx (x, 0) , x ≥ 0 are bounded.

108

4.7.3 Computation of the reflection and transmission coefficients

The formulae for the reflection, R, and transmission coefficients, T , and the multiplying

factor s for the normalized system are

s =Re (µ)λ′2 sinh 2λ′H

Re (λ)µ′ sinh 2µ′H

2µ′H (µ′4 −mω2 + 1) + (5µ′4 −mω2 + 1) sinh 2µ′H

2λ′H + sinh 2λ′H,

T 2 =µ′2 sinh2 µ′H

λ′2 sinh2 λ′H

|T |2

|I|2, R2 =

|R|2

|I|2.

The above formulae are slightly different from those in section 4.5 due to the scaling and the

definition of the amplitude of the incident, reflected and transmitted waves. This formula

is used as an independent check of the numerical solution presented in this paper. The

transmission coefficient T and the reflection coefficients R are again expressed without the

exponentials.

0.1

1

10

020

4060

80

0

0.2

0.4

0.6

0.8

1

non−dimensional frequencyIncident angle (degree)

Ref

lect

ion

coef

ficie

nt

Figure 4.10: Three dimensional plot of the reflection coefficient as a function of incidentwave non-dmensional radial frequency ω and incident angle in degree. The non-dimensionalwater depth is set H = 2π.

Now T = 1/ coshµH , and the incident wave amplitude |I| can be written as a polyno-

mials of λT over coshλTH ,

|I| =

∣∣∣∣F (−λ)λ′R− (λ′)

−λK+ (−λ) cosh λ′H

∣∣∣∣ =

∣∣∣I∣∣∣

coshλ′H

where ∣∣∣I∣∣∣ =

∣∣∣∣F (−λ)λ′R− (λ′)

−λK+ (−λ)

∣∣∣∣ .

109

Thus, the transmission coefficient can be simplified as

|T | =

∣∣∣∣µ′ tanhµ′H

λ′ tanhλ′H

∣∣∣∣1∣∣∣I∣∣∣

=1

|µ′4 −mω2 + 1|1∣∣∣I∣∣∣. (4.63)

Also, since |K+ (λ) /K+ (−λ)| = 1, the reflection coefficient can be simply computed by

R =

∣∣∣∣F (λ)

F (−λ)

∣∣∣∣ , (4.64)

which again does not contain any exponentials. Fig. (4.10) shows the reflection coefficient

for various incoming angles and non-dimensional wave frequencies when non-dimensional

water depth is H = 2π. Note the difference between Fig. (4.10) and Fig. (4.5) in the low

frequency region of the graphs. The curve of the critical angle reaches 90 in Fig. (4.10)

whereas it asymptotically approaches to 90 as ω becomes small in Fig. (4.5). When the

sea is deep, ocean waves of a wider range of frequency (lower frequency) and incident angle

(steeper incident angle) are permitted to travel into ice sheet than when the sea is shallow.

We used a typical scaled mass density m = 0.06 for the characteristic length lc = 16 to

generate the curves in Fig. (4.10) of the reflection coefficient.

We here again consider setting m = 0 as in chapter 3. Fig. (4.11) shows the curves of the

reflection coefficient at various incident angle and mass density. Up to θ = 60 the curves of

the reflection coefficient for m = 0, 0.05 and 0.1 are nearly identical. However, setting the

mass density m = 0 has an smoothing effect on the curves, and removes the zero reflection

frequency that occurs just before the total reflection frequency at a higher incident angles.

Hence, at low frequencies setting m = 0, steep incident angle may not provide acceptable

accuracy for computing the reflection coefficient.

The formulae of the reflection and transmission coefficients given by Eqn. (4.63) and

Eqn. (4.64) become free of physical parameters by eliminating the one remaining physical

parameter m. Then the resulting formulae represent the reflection and transmission coeffi-

cients for an ice sheet of any thickness and rigidity. For example, the non-dimensional radial

frequency at which the total reflection occurs as shown in Fig. (4.11) can be converted to

physical frequency using only the characteristic length and time of the ice sheet.

4.8 Summary

We have derived analytical formulae for the reflection and transmission coefficients of simple

harmonic waves in two semi-infinite elastic plates on the surface the water. The Wiener-

Hopf technique was used to obtain all the coefficients in a natural mode expansion of the

velocity potential of the water. Application of the Wiener-Hopf technique to deal with

the interaction between ocean waves and an ice sheet was originally developed by Evans

110

10−1

100

101

0

0.2

0.4

0.6

0.8

1Incident angle 0o

ω

Ref

lect

ion

coef

ficie

nt

10−1

100

101

0

0.2

0.4

0.6

0.8

1Incident angle 30o

ω

Ref

lect

ion

coef

ficie

nt

10−1

100

101

0

0.2

0.4

0.6

0.8

1Incident angle 60o

ω

Ref

lect

ion

coef

ficie

nt

10−1

100

101

0

0.2

0.4

0.6

0.8

1Incident angle 85o

ω

Ref

lect

ion

coef

ficie

nt

Figure 4.11: The reflection coefficients for wave-ice interaction as a function of non-dimensional radial frequency ω for various values of incident angle θ = 0 , 30 , 60 , and 85

and mass density m = 0 (solid line), m = 0.05 (dotted line) and m = 0.1 (dashed-dottedline). The non-dimensional water depth is 2π. The ω-axis is in log scale.

and Davies [17], and then later the method was modified by Chung and Fox [10] to take

advantage of better understanding of the properties of the roots of the dispersion equations,

as shown in section 4.7.

The original method of solution first shown by Evans and Davies [17] is based on knowing

that the solutions can be written as an infinite series of various modes, namely travelling,

damped travelling and evanescent modes. Hence asymptotic behaviour of each mode and

the solutions in x < 0 and x > 0 are known. Having this information about the solutions

enabled us to subtract the travelling wave from the solutions in order to create a strip of

analyticity in which the modified functions are regular. We are then able to construct a

Wiener-Hopf equation in the strip, which gives us the Fourier transform of the solution that

can be inverted to calculate the solutions in x < 0 and x > 0 using contour integration on

the complex plane.

In contrast, the method of solution shown in sections 4.2 to 4.4 is more direct. The

radiation condition and integrability of the solution are available prior to applying the

111

Fourier transform to the system of equations in x < 0 and x > 0. Hence, a common strip

of analyticity is created after the Fourier transform of the solution is performed. In other

words, the strip of analyticity is not created from the asymptotic behaviour of the solution

but directly from observing the positions of the singularities of the complex valued functions,

which can easily be manipulated by shifting a singularity (perhaps more than one) from one

function to the other as shown in section 4.3. It can be said that the application of the

Wiener-Hopf technique in section 4.7 uses the Fourier transform twice, first to find the

wavenumber of each mode so that its asymptotic behaviour (Eqn. (4.46)) can be found, and

second to derive the Wiener-Hopf equation (Eqn. (4.53)) and the subsequent factorization

of the dispersion functions. Sections from 4.2 to 4.4 show a stream-lined application of the

Wiener-Hopf technique, which requires the Fourier transform to be performed just once,

when we only deal with the Fourier transforms as functions of a complex variable.

A less obvious but mathematically significant fact when dealing with the interaction of

semi-infinite ice sheets (elastic plates) using the Wiener-Hopf technique is how the method

incorporates the transition conditions into the solutions. As we construct the Wiener-Hopf

equation for Ψ+z and Ψ−

z , which consist of boundary value term C (α) in Eqn. (4.23) and

M3 (α) in Eqn. (4.52), we find that the conditions at the edge or the transition are not

enough to determine C and M3. We are able eliminate P and M3 from the solutions by

inverting Ψz instead of dealing with Ψ+z and Ψ−

z separately as in [5]. It is both stream-lined

and computationally practical to find the coefficients of J (α) in Eqn. (4.26) and Eqn. (4.56)

using the algebraic equations in section 4.4 and Eqn. (4.61), only because analytical formulae

of each coefficient of the solutions can be calculated.

Another notable feature of the Wiener-Hopf technique, particularly in section 4.7, is the

apparent omission of the continuity conditions of φ and φx for −H < z < 0 described in

section 2.1. These continuity conditions throughout the depth of the water −H < z < 0

were necessary to solve Laplace’s equation. However, the use of the continuity conditions

is apparent in the Fourier transforms of Laplace’s equation in subsection 4.3.2 as we derive

the Wiener-Hopf equation (4.22), φ (0, z) and φx (0, z) are canceled as we calculate C (α) =

C1 (α) − C2 (α).

In chapter 3, the scaling regime has been shown to be effective for the plates that are not

infinite. Section 4.7 gives an example in which the scaling method using the characteristic

length and time is applicable to a plates that of semi-infinite size.

In terms of computing the solutions, once the positions of the roots of the dispersion

equations are known, turning the formulae of the solutions to computer codes that produce

the curves of the reflection coefficients is a straightforward process because of the fast con-

vergence of functions that appear in the formulae and the absence of numerical integration.

For example, 40 zeros of f1 and f2 at each ω are needed to generate curves of reflection

coefficients shown in section 4.5 when the water depth is 10 m, and 200 zeros of fsea and fice

are used to draw figures in section 4.7 when the non-dimensional water depth is 2π which is

112

about 100m to 200m in actual depth, depending on lc. All the transition conditions given in

section 4.4 are dealt with using the same computer code, only changing the matrices which

are given in subsections from 4.4.1 to 4.4.4. The software package MatLab was used for

the numerical computation. The roots of the dispersion equations are computed using the

MatLab codes written by Colin Fox. On an Intel Pentium III PC, it took about 5 to 10

minutes to draw 3-D curves of the reflection coefficients such as in Fig. (4.7).

113

Chapter 5

The Wiener-Hopf technique and

Boundary integral equations

This chapter takes a boundary integral equation (BIE) approach to the dynamics of an ice

sheet, which is different to the Fourier transform methods that we have seen so far. This

chapter complements chapter 4 to give a better understanding of the Wiener-Hopf technique.

We discuss theoretical relationship between the Fourier transform methods in chapters 3 and

4 and the BIE. A possible method of solution for a finite ice sheet is discussed.

5.1 Background

The fundamental solutions and the Wiener-Hopf technique studied in the previous chapters

deal with the given PDEs in the complex plane (frequency domain). A more common

approach to a boundary value problem with a complicated boundary shape is the boundary

element method (BEM) or boundary integral equation (BIE) method, which deals with the

PDEs in the real space variable domain. Examples of numerical methods for calculating the

hydro-elastic response for floating objects other than an infinite or semi-infinite plate can

be found in Kaleff [30] and Meylan [35]. In this chapter, we consider relationships between

the BEM and the Wiener-Hopf technique, and interpret the Fourier integral method in the

complex plane in the physical space. Furthermore, from this relationship we simplify the

BIE that have to be solved in order to obtain the solution of the boundary value problem.

A primary purpose of this chapter is to put the Wiener-Hopf technique in the context of

integral equations which have direct connection to the physical world as opposed to the

modes in frequency domain in which the Wiener-Hopf technique is used.

The most interesting feature of the Wiener-Hopf technique in section 4.7 may be that the

formula of Φz (α), the Fourier transform of the solution can be represented by a polynomial

J (α) = a1+a2α. Here only the edge conditions at x = 0+ are required to determine the two

constants of J (α), despite the boundary value problem being formulated in the two dimen-

114

sional body of water as shown in Fig. (4.2). We notice that the inverse Fourier transform

which involves J (α) is equivalent to a differential operator, i.e., a1 + a2∂x. Although, the

Wiener-Hopf technique uses the Fourier transform performed in the complex plane, each

manipulation of the functions involved has corresponding procedures in the real physical

variable space. Hence we should be able to reach the same representation of the solution, a

linear summation of a special solution and its derivative, using an integral equation method

in the real physical variable space.

5.2 Formulation of BIE

In chapters 3, we derived the formulae for the surface displacement of an ice sheet and

free surface, w (x, y), and the velocity potential of the water, φ (x, y, z), using the Fourier

transform of the system of PDEs. The solutions are the response due to a localized forcing,

i.e., Dirac delta function δ (x, y). We call such a solution a fundamental solution.

The solution of the boundary value problem can be expressed using Green’s theorem

and a fundamental solution of Laplace’s equation denoted by GI (r, ρ, z) and GW (r, ρ, z) for

ice covered water and free surface, respectively, where r = (x, y), ρ = (ξ, η) in R2. Then,

GI (r, ρ, z) and GW (r, ρ, z) satisfy Laplace’s equation,

∇2r,zGI (r, ρ, z) = ∇2

ρ,zGI (r, ρ, z) = ∇2r,zGW (r, ρ, z) = ∇2

ρ,zGW (r, ρ, z) = 0, (5.1)

for r, ρ ∈ R2, −H < z < 0. The plate equation for GI (r, ρ, 0) and GIz (r, ρ, 0) for r, ρ ∈ R2

are (∇4

ρ −mω2 + 1)GIz (r, ρ,0) − ω2GI (r, ρ,0) = LIGI (r, ρ,0) = −δ (r − ρ) , (5.2)

where LI denotes the differential operator defined by the left hand side of the equation.

GWz denotes the z derivative of GW (r, ρ, z) at surface z = 0. Note that we are using the

non-dimensional equation introduced in chapter 3. For the free surface, we have for r, ρ ∈ R2

GWz (r, ρ, 0) − ω2GW (r, ρ,0) = LWGW (r, ρ,x) = −δ (r− ρ) , (5.3)

where LW is a differential operator defined by the left hand side of the equation. The source

or forcing in Eqn. (5.1) and Eqn. (5.2) is placed only at the surface since the fundamental

solution derived in chapter 3 will be used to represent the solution at the surface. An

example of using a Green’s function for Laplace’s equation with the free surface is shown

by Meylan [36] to compute flexural motions of a very large floating structure.

We also assume that the fundamental solutions satisfy the fixed bottom conditions

GIz (r, ρ,−H) = GWz (r, ρ,−H) = 0.

115

Since GI (r, ρ, z) and GW (r, ρ, z) are each a radially symmetric function in R2 as seen

in chapter 3, we may write GI (r, ρ, z) = GI (r − ρ, z) and GW (r, ρ, z) = GW (r − ρ, z).

We note that Gz (r, ρ, 0) corresponds to the time derivative of the solutions of the system

of equations given in sections 3.3 and 4.3. Thus Gz (r, 0) is equivalent to φz (r, z) / iω in

chapters 3 and 4. We omit the term iω to avoid the clutter. We know from chapter 3 that

the fundamental solutions can be expressed by infinite summation of the natural modes of

ice sheet and free surface denoted by KˆI and Kˆ

W respectively, that is

GIz (r − ρ, 0) =∑

q∈KˆI

qRI (q)H(1)0 (q |r − ρ|) ,

GWz (r − ρ, 0) =∑

q∈KˆW

qRW (q)H(1)0 (q |r − ρ|) ,

where RI and RW are the residues corresponding to the dispersion equations for the ice

sheet and free surface respectively. The set of roots KˆI is the same as Kˆ in chapter 3, and

KˆW consists of roots with positive imaginary part together with the positive real root.

We recall the notations for the domain and subdomains in which the boundary value

problem of a finite floating ice sheet is formulated in chapter 2. The domains V and Vc

denote the body of water in three dimensional space whose surface is covered by Ω and Ωc,

ice-covered and free surface respectively. We denoted the wall of V by Vs. We do not give

any particular notation for the ocean floor, since its contribution to the solutions is zero

due to the fixed surface condition. The velocity potential in each domain, φI (r, z) for r ∈ Ω

and φW (r, z) for r ∈ Ωc, −H < z < 0, can be represented using Green’s theorem and the

fundamental solutions on the boundary ∂V and ∂Vc

0 =

∫

V

[GI (r − ρ,z)∇2

ρ,zφI (ρ, z) − φI (ρ, z)∇2ρ,zGI (r − ρ,z)

]dτρ,z

=

∫

∂V

[∂φI (ρ, z)

∂nρ,zGI (r − ρ,z) − φI (ρ, z)

∂GI (r − ρ,z)

∂nρ,z

]dσρ,z, (5.4)

for r ∈ Ω and similarly

0 =

∫

∂Vc

[∂φW (ρ, z)

∂nρ,zGW (r− ρ,z) − φW (ρ, z)

∂GW (r − ρ,z)

∂nρ,z

]dσρ,z (5.5)

for r ∈ Ωc, where φI and φW are the solutions of Laplace’s equation and satisfying the

surface conditions in the ice sheet and free surface,

LIφI (r, 0) = 0 for r ∈ Ω,

LWφW (r, 0) = 0 for r ∈ Ωc.

Note that ∂/∂nρ denotes the normal derivative on the respective boundary with respect

116

to (ρ, z). We denoted the boundary of the domains V and Vc shown in Fig. (2.2) by ∂Vand ∂Vc. Therefore, splitting the integration on ∂V to Ω and Vs (∂Vc to Ωc and Vs) and

considering the zero contribution from the ocean floor, we have the identity of boundary

integrations

∫

Ω

[φIGIz − φIzGI] dσρ =

∫

Vs

[∂φI

∂nρ,z

GI − φI∂GI

∂nρ,z

]dσρ,z, (5.6)

∫

Ωc

[φWGWz − φWzGW] dσρ = −∫

Vs

[∂φW

∂nρ,zGW − φW

∂GW

∂nρ,z

]dσρ,z + I (φW) , (5.7)

where I (φW) is a term due to an incident wave from infinity. Note that the normal derivative

on Vs in Eqn. (5.7) is outward from Ω. From appendix A and Eqns. (5.2, 5.6, 5.7), φIz (r, 0)

for r ∈ Ω can be expressed as

φIz (r, 0) =

∫

Ω

GIzLIφI (ρ, 0) − φIzLIGI (r − ρ,0) dρ

=

∫

∂Ω

(B1φIz)GIz − (B2φIz)

∂GIz

∂nρ

− φIzB1GIz +∂φIz

∂nρ

B2GIz

dsρ

− ω2

∫

Ω

[φI (ρ, 0)GIz (r − ρ, 0) − φIz (ρ, 0)GI (r − ρ, 0)] dρ

=

∫

∂Ω

(B1φIz)GIz − (B2φIz)

∂GIz

∂nρ− φIzB1GIz +

∂φIz

∂nρB2GIz

dsρ

− ω2

∫

Vs

[∂φI

∂nρ,zGI − φI

∂GI

∂nρ,z

]dσρ,z (5.8)

where B1 and B2 are the boundary differential operators defined in chapter 2. Each term

of the integration over ∂Ω can be considered as integral operator defined by the derivatives

of GIz and we denote these boundary integrals by

V1 (b) (r) =

∫

∂Ω

b (ρ)GIz (r − ρ, 0) dsρ,

V2 (b) (r) =

∫

∂Ω

b (ρ)∂GIz (r − ρ, 0)

∂nρdsρ,

V3 (b) (r) =

∫

∂Ω

b (ρ)B2ρGIz (r − ρ, 0) dsρ,

V4 (b) (r) =

∫

∂Ω

b (ρ)B1ρGIz (r − ρ, 0) dsρ.

Then, the solution (5.8) can be simply written as

φIz (r, 0) =4∑

i=1

Vi (Fi) (r) − ω2

∫

Vs

[∂φI

∂nρ

GI − φI∂GI

∂nρ

]dσρ, (5.9)

117

where

B1φIz (r, 0)|∂Ω = F1 (r) ,

B2φIz (r, 0)|∂Ω = −F2 (r) ,

∂φIz (r, 0)

∂n

∣∣∣∣∂Ω

= F3 (r) ,

φIz (r, 0)|∂Ω1= −F4 (r) .

The boundary integrals (or operators) denoted by Vi, i = 1, 2, 3, 4 are commonly called

single- (or simple), double-, triple- and quadruple-layer potentials of bi-harmonic differential

operator ∇4. When there is no hydrodynamics involved, the bending motion of a plate can be

expressed using the layer potentials composed of a fundamental solution of the bi-harmonic

equation, which is called the layer potential representation.

From Eqn. (5.3) and Eqn. (5.7), the free surface solution φWz (r, 0) for r ∈ Ωc can be

expressed as

φWz (r, 0) =

∫

Ωc

GWz

[φWz − ω2φW

]− φWz

[GWz − ω2GW

]dρ

= −ω2

∫

Ωc

φWGWz − φWzGW dρ

= ω2

∫

Vs

[∂φW

∂nρ,zGW − φW

∂GW

∂nρ,z

]dσρ,z + I (φW) . (5.10)

Therefore, if the boundary values at ∂Ω, Fi, i = 1, 2, 3, 4, and the values φI (φW) and φIn

(φWn) at Vs are given, we are able to compute the solutions. However, as shown in chapter

2 we need only two edge conditions at ∂Ω and the natural continuity conditions at Vs. Here,

we assume that the effective shear force intensity B1φIz and the bending moment B2φIz are

given. The theory of BEM explained by Chen and Zhou [8] tells us that the boundary values

of the solutions φWz and φIz at the boundary ∂Ω and ∂Ωc can be expressed as

12φIz (r, 0) =

∑4i=1 Vi (Fi) (r) − ω2

∫Vs [φInGI − φIGIn] dσρ,z

12φWz (r, 0) = ω2

∫Vs [φWnGW − φWGWn] dσρ + I (φW) .

12B1φIz (r, 0) =

∑4i=1B1Vi (Fi) (r) − ω2B1

∫Vs [φInGI − φIGIn] dσρ,z

12B2φIz (r, 0) =

∑4i=1B2Vi (Fi) (r) − ω2B2

∫Vs [φInGI − φIGIn] dσρ,z

(5.11)

Since, φI, φW, φIn and φWn are continuous at Vs, the rest of the boundary values φI (φW)

and φIn (φWn) at Vs can be obtained by solving the above system of BIEs. Note that

Eqns. (5.11) are defined on ∂Ω and we used the free surface condition to convert φWz (r, 0)

to φW (r, 0). Then, the rest of the boundary values can be determined by the usual BIEs of

φIz (see Chen and Zhou [8]).

The system of BIEs defined by Eqn. (5.11) are defined on ∂Ω and the wall of the water

118

column domain Vs. Therefore, the numerical solution of the BIEs requires digitizing the

functions in Eqn. (5.11) on Vs, which is a three dimensional surface as opposed to a curve

∂Ω on the two dimensional plane.

BIEs without the second integral term in Eqn. (5.11) can commonly be found in many

text books on BEM or plate dynamics. We show that BIEs (5.11) can be reduced to BIEs

only on the edge of the plate ∂Ω using an argument inspired by the Wiener-Hopf technique.

The Wiener-Hopf technique enabled us to derive the analytical solutions of two-plate inter-

action and water wave-ice interaction problems only from the transition conditions at the

joint of the two plates on the surface. In the following section, we study the relationship

between the Fourier transforms in the Wiener-Hopf technique and the BIE method.

5.3 Semi-infinite plates

We consider the simplest case of wave-ice interaction problem when a plane wave is normally

incident, i.e., θ = k = 0 in chapter 4. We simplify Eqn. (5.8) and Eqn. (5.10) for the semi-

infinite ice sheet for x > 0 and for x < 0 using the fact that the shear force intensity and the

bending moment are the third and second derivatives with respect to ξ at ξ = 0+. Hence

we have

φIz (x, 0) = [φIzξξξGIz − φIzξξGIzξ − φIzGIzξξξ + φIzξGIzξξ]ξ=0+ (5.12)

+ ω2

∫ 0

−H

[φI (0, z)GIξ (x, z) − φIξ (0, z)GI (x, z)] dz,

for x > 0 and

φWz (x, 0) = −ω2

∫ 0

−H

[φW (0, z)GWξ (x, z) − φWξ (0, z)GW (x, z)] dz + I (φW) (5.13)

for x < 0. We denote the fundamental solutions by GI (x− ξ, z) and GW (x− ξ, z) since

the boundary value problem is reduced to a two-dimensional problem. We note that the

transmitted wave is included in the fundamental solution GI, since we use the model in

section 4.3 rather than that in section 4.7.

The Wiener-Hopf technique inspires us to find the solutions without solving the BIEs

on Vs, i.e., x = 0, −H < z < 0. We recall the Fourier transform of the velocity potential

obtained from the Wiener-Hopf equation,

Ψz (α, 0) =

F (α)

f1 (α)K+ (α),

F (α)

f2 (α)K− (α),

119

where the top expression is used to calculate the solution for x < 0 and the bottom one for

x > 0. The denominators of the functions above may be rewritten as

Ψ±z (α) =

F (α)

f+1 (α) f−

2 (α),

F (α)

f−1 (α) f+

2 (α),

(5.14)

where f±1 (α) and f±

2 (α) are split functions of f1 (α) and f2 (α) respectively. The superscripts

+ and − indicate the regular functions in the upper and lower half planes, D+ and D−

respectively, and then

f1 (α) = f+1 (α) f−

1 (α) ,

f2 (α) = f+2 (α) f−

2 (α) .

We note that since f±i , i = 1, 2 are split functions of each dispersion function rather than the

ratio of the two as seen in chapter 4, a slight modification of the factorization is required.

We first factorize fiγ sinh γH , which is analytic in the whole plane and has zeros at the roots

of the dispersion equations. Thus the factorization is already shown in chapter 4. Then we

factorize γ sinh γH

γ sinh γH = γ2

∞∏

n=1

(1 + i

γH

nπ

)(1 − i

γH

nπ

).

In order to split the above formula into two regular non-zero functions in the upper and

lower half planes, we split the second order zero at the origin into two first order zeros in

the upper half and the lower half plane.

We notice that since the fundamental solutions GI and GW at z = 0, ξ = 0 are the inverse

Fourier transforms of the inverse of the dispersion functions, they can be represented by

functions that are defined on the half spaces, x < 0 and x > 0. Thus

GIz (x, 0) =

∫ ∞

−∞

e− iαx

f+2 (α) f−

2 (α)dx

= S ′I (x, 0) ∗ S ′

I (−x, 0) =

∫ ∞

max(0,x)

S ′I (x

′, 0)S ′I (x

′ − x, 0) dx′,

where S ′I is the inverse Fourier transform of 1/f+

1 which is a z-derivative of SI at z = 0,

defined in x > 0, zero in x < 0 and ∗ denotes convolution. Similarly, we have the

120

fundamental solution of the free surface

GWz (x, z) =

∫ ∞

−∞

e− iαx

f+1 (α) f−

1 (α)dx

= S ′W (x, 0) ∗ S ′

W (−x, 0) =

∫ ∞

max(0,x)

S ′W (x′, 0)S ′

W (x′ − x, 0) dx′,

where S ′W is the inverse Fourier transform of 1/f+

1 and the prime indicates the z-derivative

at z = 0, defined in x > 0, zero in x < 0. From the line-loading case of chapter 3, we

know the explicit expression of GIz (x, 0) and GWz (x, 0),

GIz (x, 0) =∑

q∈KˆI

RI (q) exp i q |x| ,

GWz (x, 0) =∑

q∈KˆW

RW (q) exp i q |x| .

The split fundamental solutions SI and SW can be represented by infinite series of the natural

modes of the ice sheet and free water surface since the zeros of the split functions b+1 and b+2are the same as those of the dispersion functions in the lower half planes. Thus the integral

contour of the inverse Fourier transform is closed in the lower half plane.

We are able to construct the fundamental solution which satisfies both surface conditions

using the usual convolution of the fundamental solution in each region. Let G (x, 0) denote

the convolution of SI (x, 0) and SW (−x, 0),

G (x, 0) =

∫ ∞

max(0,x)

SI (x′, 0)SW (x′ − x, 0) dx′,

Gz (x, 0) =

∫ ∞

max(0,x)

S ′I (x

′, 0)S ′W (x′ − x, 0) dx′,

We can compute G (x, z) for −H < z < 0 using convolution

G (x, z) = G (x, 0) ∗ P (x, z)

where P (x, z) is the inverse Fourier transform of cosh γ (z +H) / cosh γH . Then, it is

obvious by construction that G (x, z) is again a solution of Laplace’s equation and simulta-

neously satisfies the homogeneous plate equation for x > 0 and the free surface equation for

x < 0. Since, GWz (x, 0) and ∂4xGIz (x, 0) have a singularity at the origin, the split function

SWz (x, 0) and the derivative ∂4xSIz (x, 0) have a singularity at the origin, which give the delta

function on the right hand side of Eqn. (5.2) and Eqn. (5.3) at the origin. Therefore, we

may replace GI and GW in Eqn. (5.12) and Eqn. (5.13) with one function G. Furthermore,

G (x, z) is continuous at x = 0,−H < z < 0, since it is defined by the convolution of the

121

two continuous functions that satisfy Laplace’ equation.

Following the Wiener-Hopf technique, we notice that the solution can be written using

a linear sum of the fundamental solution G (x, z) and Gx (x, z), i.e., the function φIz (x, 0)

can be expressed in the following form

a1Gz (x, 0) + a2Gxz (x, 0) (5.15)

where the two constants a1 and a2 can be determined from the edge conditions of the plate,

since F (α) in Eqn. (5.14) consists of the first order polynomial J (α) and multiplication by

α corresponds to the first order derivative.

We are able to combine the boundary integral at x = 0 using the continuity of the

fundamental solution and the velocity potential. From Eqn. (5.10) for x > 0, Eqn. (5.13)

becomes

0 = ω2

∫ 0

−H

φW (0, z)Gξ (x, z) − φWξ (0, z)G (x, z) dz + I (φW) ,

then Eqn. (5.12) becomes

φIz (x, 0) = [φIzξξξGz − φIzξξGzξ − φIzGzξξξ + φIzξGzξξ]ξ=0+ + I (φW) .

Since, the x-derivatives of Gz satisfy the system of equations and there are only two edge

conditions to satisfy, the representation of the solution can be reduced to

φIz (x, 0) = a1Gz (x, 0) + a2Gzξ (x, 0) + I (φW) (5.16)

with two unknown constants a1 and a2. This is the same form of the solution given by the

Wiener-Hopf technique in Eqn. (5.15). Note that due to the symmetry of the fundamental

solution, the function G satisfies

Gzξ (x, 0) = Gzξ (x− ξ, 0)|ξ=0 = Gzx (x, 0) .

The two constants are determined by following simultaneous equation

d3

dx3a1Gz (x, 0) + a2Gzx (x, 0) + I (φW)x=0+ = g1,

d2

dx2a1Gz (x, 0) + a2Gzx (x, 0) + I (φW)x=0+ = g2,

(5.17)

where g1 and g2 are given edge conditions. Note that derivatives of Gz and Gzx at x = 0+

are bounded, thus the jump conditions or 12

seen in Eqn. (5.11), which usually appear in

BEM are not required here.

122

Similarly for x < 0, we have

φWz (x, 0) = I (φW) + a1Gz (x, 0) + a2Gzξ (x, 0) .

We need to derive the fundamental solutions GIz and GWz only for ξ = 0, since the final

representation of the solutions requires only Gz (x, 0) and its derivatives with respect to x

or ξ at ξ = 0+.

It is trivial to see that for the semi-infinite ice sheet case the two constants (a1, a2) are

uniquely determined by the simultaneous equation (5.17). Therefore the expression given

by Eqn. (5.16) is the unique solution to the interaction between plane water wave and a

semi-infinite ice sheet.

5.4 Finite plate

In chapter 4 and the previous section, the splitting of the fundamental solutions could be

performed analytically by inspection due to the simple geometry of the boundaries, which

enables analytical calculation of the Fourier transform. We may extend the argument of

simplification of the BIEs using a fundamental solution constructed using the convolution

of split fundamental solutions for a finite floating plate.

We construct a fundamental solution G (r, ρ, 0) and Gz (r, ρ, 0) which satisfy both sur-

face conditions in Ω and Ωc using the integral of the two split functions of GI (r, ρ, 0) and

GW (r, ρ, 0) defined as

GI (r − ρ, 0) =

∫SI1 (r′, ρ, 0)SI2 (r − r′, ρ, 0) dr′, (5.18)

GW (r − ρ, 0) =

∫SW1 (r − r′, ρ, 0)SW2 (r′, ρ, 0) dr′ (5.19)

where SIi and SWi, i = 1, 2, are solutions of the plate and free surface equation respectively,

that is,

LISI1 (r − ρ, 0) = LWSW1 (r− ρ, 0) = 0 for r ∈ Ω, ρ ∈ ∂Ω,

LISI2 (r − ρ, 0) = LWSW2 (r− ρ, 0) = 0 for r ∈ Ωc, ρ ∈ ∂Ω.

The split functions SI1 and SW1 are zero in Ωc, and SI2 and SW2 are zero in Ω,

SI1 (r − ρ, 0) = SW1 (r − ρ, 0) = 0 for r ∈ Ωc, ρ ∈ ∂Ω,

SI2 (r − ρ, 0) = SW2 (r − ρ, 0) = 0 for r ∈ Ω, ρ ∈ ∂Ω.

123

Therefore, the integrals in Eqn. (5.18) and Eqn. (5.19) are over

r′ ∈ Ω and r − r′ ∈ Ωc,

r′ ∈ Ωc and r − r′ ∈ Ω,

respectively. Then, the new fundamental solutions of ice sheet and free surface regions

G (r, ρ, 0) and Gz (r, ρ, 0) can be represented as

G (r, 0) =

∫

Ω

SI1 (r′, 0)SW2 (r − r′, 0) dr′, (5.20)

Gz (r, 0) =

∫

Ω

S ′I1 (r′, 0)S ′

W2 (r − r′, 0) dr′. (5.21)

We need not find the fundamental solution for all ρ ∈ R2 since G (r − ρ, 0) only for ρ ∈ ∂Ω

is required for the representation of the solutions, as seen in section 5.2.

It is obvious that we cannot find the split functions from the fractional decompositions

as shown in the previous section because the functions are defined on the two dimensional

plane. Hence, they must be computed numerically solving a system of matrix equations

given by Eqn. (5.18) and Eqn. (5.19) at each point ρ ∈ ∂Ω. Since, the functions themselves

have no boundary conditions associated with the shape of the plate, we have for a given

ρ ∈ ∂Ω,

S ′I1 (r − ρ, 0) =

∑q∈Kˆ

I

c1 (q)H(1)0 (q |r − ρ|) for r ∈ Ω,

0 for r ∈ Ωc,

S ′I2 (r − ρ, 0) =

0 for r ∈ Ω,∑

q∈KˆI

c2 (q)H(1)0 (q |r − ρ|) for r ∈ Ωc.

Therefore, we now have to find the coefficients c1 (q)q∈KˆI

and c2 (q)q∈KˆI

from an al-

gebraic version of Eqn. (5.18) at a given point r in Ω. The coefficients of S ′W1 (r − ρ, 0)

and S ′W1 (r − ρ, 0) can be found using the same expression. There may be many numerical

computational problems that must be addressed for the practical implementation of the

theory. However, we will consider only the theoretical aspects of the method, as our pri-

mary purpose is to find the simple representation of the surface response using boundary

integrals.

We can from Eqn. (5.8) and continuity of G (r − ρ, z) for ρ ∈ ∂Ω, −H < z < 0, express

the solution φIz (r, 0) for r ∈ Ω as

φIz (r, 0) =

4∑

i=1

Vi [Fi] (r) − I (φW) (5.22)

using the layer potential representation introduced in section 5.2. Therefore, the solution

124

represented by Eqn. (5.22) uses only the conditions on the edge of the ice sheet on the

surface. It may be possible to reduce the boundary integral parts in Eqn. (5.22) to a linear

sum of the two-layer-representation, as seen in section 5.3,

φIz (r, 0) =

∫

∂Ω

[b1 (ρ)Gz (r − ρ, 0) + b2 (ρ)

∂Gz

∂nρ(r − ρ, 0)

]dsρ + I (φW) (5.23)

where functions b1 and b2 are defined on the boundary ∂Ω, which must be determined from

the boundary conditions. And similarly, we have for r ∈ Ωc

φWz (r, 0) =

∫

∂Ω

[b1 (ρ)Gz (r − ρ, 0) + b2 (ρ)

∂Gz

∂nρ(r − ρ, 0)

]dsρ + I (φW) .

It is natural to assume that the unknown functions b1 and b2 can be uniquely determined

from the two edge conditions, since the response of the ice sheet is uniquely determined by

the two edge conditions. However, the uniqueness of such expression is not obvious and too

technical and lengthy to discuss here. In the following, we only give a brief justification of

Eqn. (5.23).

A fundamental solution E (r, ρ) to the biharmonic equation

∇4ρE (r, ρ) = −δ (r − ρ)

is

E (r, ρ) = − 1

8π|r − ρ|2 log |r − ρ| .

We notice from the discussion on the derivatives of the fundamental solution in section 3.4

that Gz (r, ρ, 0) has the same behavior as E (r, ρ) near |r− ρ| = 0, r ∈ Ω. Therefore, we

may use the theories developed for the simple biharmonic equation to manipulate the BIE

given in Eqn. (5.22) and justify the representation by Eqn. (5.23).

The bottom two BIEs are due to the jump conditions of the triple- and quadruple-layer

potential,

B1V1 (b) (r) =1

2b (r) +

∫

∂Ω

[B1rGz (r − ρ, 0)] b (ρ) dsρ,

B2V2 (b) (r) = −1

2b (r) +

∫

∂Ω

[B2r

∂Gz (r− ρ, 0)

∂nρ

]b (ρ) dsρ.

We intuitively find that on the boundary, the solution is expressed by Eqn. (5.11) because

the boundary integral receives half of the contribution of the delta function and that is

possible only if the above jump conditions are satisfied. The jump condition for the simple-

and double-layer potential can be proven using the same argument as for Laplace’s equation.

From section 8.3 of (Chen and Zhou [8]), we know that B1Gz (r − ρ, 0) and B2Gzn (r − ρ, 0)

125

near the boundary behave as |r − ρ|−1. All other combinations of the boundary derivatives

are either bounded or have a singularity that depends only on the term θ − ϕ, where

θ = arg r and ϕ = arg ρ. Hence, unless the limit is taken tangentially on ∂Ω, the derivatives

of the layer potentials are continuous at ∂Ω. We do not explore the detailed proof of

the jump conditions of the layer-potentials and the uniqueness of the representation in

Eqn. (5.23), since the rigorous mathematical proof is very technical and outside the scope

of this monograph.

All the integral operators are defined in Sobolev spaces of appropriate indices according

to smoothness and integrability of the functions involved. However we here omit the details

of the indices of Sobolev spaces and assume that the shape of Ω and the boundary conditions

are smooth enough to meet the conditions for the definitions of the integral operators shown

in this chapter.

5.5 Summary

Formulation of the BIEs of the dynamics of an elastic pale floating in an ocean wave field is

presented in this chapter. The effects of the hydrodynamics add an extra boundary integral

term to the representation of the solutions given in Eqn. (5.9), which arises from integration

over the depth of the water. The motivation to interpret the Wiener-Hopf technique in

terms of functions of physical variables rather than the Fourier transforms is to use the

explicit formulae of the fundamental solutions obtained in chapter 3 for other than an

infinite ice sheet. It is based on the knowledge that the elastic response of any ice sheet

may be described using the natural modes of ice sheet. The Wiener-Hopf technique showed

us that the solutions of wave-ice interaction for a semi-infinite ice sheet can be expressed by

a simple linear summation of a special solution and its first order derivative on the surface.

This leads us to extend the Fourier transform method to an integral equation method for

more general boundary shapes, and then to find a representation of the solutions by a linear

summation of boundary integrals over the edge of the ice sheet ∂Ω, rather than the whole

boundary of the domain, the wall of the water column Vs.

First, we showed that the concept of the Wiener-Hopf technique could be applied to the

BIE method to reduce the number of BIEs that must be solved for the boundary values

in the water −H < z < 0 for semi-infinite ice sheet. The system of BIEs are defined on

the edge of the plate using the two split fundamental solutions of the ice-covered and free

surface regions, instead of the entire wall of the water column region Vs. We were able to

reach the same formula for the solution, a linear summation of a solution and its derivative,

using the two different methods: the Wiener-Hopf technique and BIE method. We note

that although BIEs are formulated over the three dimensional domain V that includes the

body of water, we use the fundamental solution on the surface rather than that of Laplace’s

126

equation. We found that the manipulation of the functions in the frequency domain or

spatial variable domain are equivalent. We have not explored the details of how to find

the split fundamental solutions for the general shape of ice sheets, since they are numerical

computational procedure and it is out of the scope of this monograph. We have not given a

rigorous mathematical proof of the regularity of the integral operators defined from the new

fundamental solution Gz (r, ρ, 0). It is not certain how effective the theory in this chapter

would be in terms of numerical computation of the solution. The computation of the split

fundamental solutions SI1, SI2, SW1, SW2 and the convolution required to construct G may

pose difficulties.

127

Chapter 6

Conclusions and review

We conclude this monograph by considering the boundary value problems studied in chapters

2 to 5 in connection with each other, briefly reviewing the content of each chapter.

The primary focus of this mongraph is the analytical solution of idealized mathematical

models (as opposed to numerical solutions of closer-to-reality models) of the wave propaga-

tion in floating ice sheets. The mathematical model introduced in chapter 2 focuses on the

facts that a large piece of ice sheet behaves as a thin elastic plate and the ocean may be

modelled as incompressible, irrotational fluid satisfying Laplace’s equation. The mathemat-

ical tools used to find the solutions are the classical Fourier transform of a complex variable,

which involve various special functions and contour integrations of analytic functions.

Once the singularity of the complex function

w (γ) =1

d (γ)

are found, w (γ) can be inverted by hand using the fractional expansion,

w (γ) =∑

q∈Kˆ

2qR (q)

γ2 − q2

where R (q) is a numerically well behaved formula. Furthermore, even when the numerator

of w is not unity, such as the case shown in chapter 4,

Φ−z (α) =

C1 (α)

f1 (γ (α)), Φ+

z (α) =C2 (α)

f2 (γ (α))

can again be inverted by hand using the seemingly complicated Wiener-Hopf method, which

eventually gives numerically computable formulae for the coefficients of the mode expan-

sion of the solutions. Then, we are able to produce the curves of the reflection coefficient

for various transition conditions and ice thickness as shown in section 4.5. The reflection

coefficient is chosen to show the numerical computation since we are primarily interested in

128

the amount of energy penetrating across the discontinuity between two ice sheets.

We realized that each mode of the waves with the wavenumber q in real variable space

exp i qx (or H(1)0 (qr)) is equivalent to a fractional function

1

α− q,

(or

1

γ2 − q2

)

in the complex plane, which has a simple pole at γ = q giving the corresponding wavenum-

ber. By observing the Fourier transform of the solutions, it becomes obvious that the modes

of the waves that exist in either an infinite or a semi-infinite ice sheet are determined only by

the zeros of the unchanging dispersion function d (γ). Fortunately, the Fourier type integra-

tions of the fractional functions and special functions have been well studied and analytical

calculation can be found in many tables of integral transforms.

The two immediate consequences of having analytical formulae are deep-water solutions

and non-dimensionalization by characteristic length and characteristic time

lc =

(D

ρg

)1/4

, tc =

√lcg.

From the analytical solutions, we are able to find the deep-water solution by taking the limit

of the finite depth solution, H → ∞, without using the deep-water dispersion equation

which has continuous singularity, or branch cuts on the imaginary axis. Hence, finding

the deep-water solution simply requires computing the five complex roots of the fifth order

polynomial (γ4 + 1

)γ = ω2,

which could be found using a single line command of root-finder, roots in MatLab. In

short, we may say that the complete description of an ice sheet is given by (qT, qD, qE) (the

roots of the dispersion equation) and (lc, tc). Furthermore, since the scaled roots remain

unchanged for any ice sheet, (lc, tc) describes all the characteristics of the ice sheet.

The effectiveness of our non-dimensionalization scheme became obvious only after the

solutions were calculated and shown to be dependent only on the dispersion equation d (γ) =

0, and consequently the positions of the roots. We were able to eliminate not only the

physical parameters D, ρ and g but also make the mass density m become insensitive to

the ice thickness, m ∝ h1/4.

We found that the spliting involved in the Wiener-Hopf technique,

f2

f1=K+

K−,

129

is equivalent to the decomposition of the fundamental solution of each region

GI = SI1 ∗ SI2, GW = SW1 ∗ SW2.

Thus, the new fundamental solution which simultaneously satisfies the surface conditions of

the two regions, ice cover and free surface could be created from the convolution of the two

halves of the fundamental solutions

G = SI1 ∗ SW2.

We found that the Wiener-Hopf technique is just another expression of boundary integral

representation of the solution. Therefore, the Wiener-Hopf technique may be said to a

reduced version of splitting the fundamental solutions in the one dimensional space. Hence,

the solution expressed as

φIz (x, 0) = a1Gz (x, 0) + a2Gzx (x, 0) − I (φW)

in one-dimensional space can be extended to the boundary integral representation of the

solutions

φIz (r, 0) =

∫

∂Ω1

[b1 (ρ)Gz (r, ρ, 0) + b2 (ρ)

∂Gz (r, ρ, 0)

∂nρ

]dσρ − I (φW) .

It may be said that Liouville’s theorem in the Wiener-Hopf technique gives us the min-

imum number of terms (or derivatives of a fundamental solution) that are required for a

representation of a unique solution. Finding the minimum number of terms in the case of

an ice sheet of a general shape needs a lot of mathematical tools in the theory of integral

operators. In the complex plane the differentiability is expressed by a simple polynomial.

Thus the order of the polynomial in the solution multiplied by the split function in the

complex plane gives us required number of derivatives of the split function in the real space

to express the solution.

We review the content of each chapter.

Chapter 2. We showed how the differential equations of water, waves, ice sheet, forces

applied to it are derived and the use of them is justified in the context of sea ice dynamics.

Laplace’s equation and its boundary conditions were derived from the law of conservation

of mass and equilibrium of force on the surface of the ocean. The plate equation and its

boundary conditions were derived from conservation of the total strain energy in the ice

sheet.

130

Chapter 3. Analytical formulae of the response of an infinite ice sheet to a localized

forcing are derived. We found that the solutions were represented by an infinite series of

natural modes of the ice sheet and the coefficients of the modes are dependent only on the

dispersion equation which is an algebraic equation of wavenumbers and frequency derived

from the Fourier transform of the system of differential equations. The solution is further

simplified for the deep-water case, which require finding five roots of a fifth order polynomial.

Non-dimensionalization using the characteristic length and time reduced the solution to a

single formula that is insensitive to ice thickness. Hence converted to physical solution for

any ice thickness within the range of geophysical interest.

Chapter 4. As a natural extension to an infinite ice sheet, the interaction of two semi-

infinite ice sheets, including the case where one becomes open water, is studied. The Wiener-

Hopf technique is used to find analytical formulae of the coefficients of the mode expansion

of the waves. The coefficients are again only dependent on the positions of the roots of the

dispersion equations of the two semi-infinite regions. The non-dimensionalization scheme

used in chapter 3 was again applied to wave-ice interaction problem, which was proven to

be effective as predicted in chapter 3. It is interesting to note that characteristic length of a

stationary ice sheet can be effective for an oscillating ice sheet and the resulting characteris-

tics of the solutions are determined not by the nature of the waves in the ice sheet, such as

wavelength and frequency, but by the physical properties of the ice sheet, such as Young’s

modulus and ice thickness of the ice sheet.

Chapter 5. We made a connection between the Fourier transform method and boundary

integral method of solution of two semi-infinite space problem. Then we were able to simplify

the boundary integral equations which were required for the solutions of the boundary value

problem. The Wiener-Hopf technique is given a different perspective from the variable space

using the boundary integral representation of the solutions to the boundary value problem.

Furthermore, the fact that the solutions are expressed by a linear summation of a special

solution and its first derivative for the semi-infinite plate case is applied to a general finite

plate. As a result the response of a finite floating plate can be represented by a linear

summation of single- and double-layer potentials which are constructed from the newly

found fundamental solution.

131

Appendix A

Integrals and Special functions

A.1 Calculations of bending and shear

Repeated use of Green’s theorem

∫

Ω

v div (∇w) dr =

∫

∂Ω

v∇w · n dσ −∫

Ω

∇v · ∇w dr,

gives us that

∫

Ω

[v∇4w − w∇4v

]dr =

∫

∂Ω

[v(∇2w

)n− w

(∇2v

)n− vn∇2w + wn∇2v

]dσ. (A.1)

From Eqn. (2.31) in chapter 2, the bending moment and shear force intensity are

B2w = ∇2w + (1 − ν)(2nxnywxy − n2

ywxx − n2xwyy

)vn,

B1w =(∇2w

)n− (1 − ν) v

∂

∂s

[(n2

x − n2y

)wxy − nxny (wxx − wyy)

].

Hence, using the integral by parts and identities of the normal, tangential derivatives,

vn = nxvx + nyvy, vs = −nyvx + nxvy,

132

we have

∫

∂Ω

[v(∇2w

)n− vn∇2w

]dσ

=

∫

∂Ω

vB1w − vB2w + (1 − ν) v

∂

∂s

[(n2

x − n2y

)wxy − nxny (wxx − wyy)

]

+ (1 − ν)(2nxnywxy − n2

ywxx − n2xwyy

)vn

dσ

=

∫

∂Ω

vB1w − vB2w − (1 − ν) vs

[(n2

x − n2y

)wxy − nxny (wxx − wyy)

]

+ (1 − ν)(2nxnywxy − n2

ywxx − n2xwyy

)vn

dσ

=

∫

∂Ω

[vB1w − vB2w + (1 − ν) (nywxyvx − nxwyyvx − nxwxyvy − nywyyvy)] dσ

Therefore, Eqn. (A.1) can be written using only the natural boundary condition terms,

∫

Ω

[v∇4w − w∇4v

]dr =

∫

∂Ω

[vB1w − wB1v − vB2w + wB2v] dσ.

Note that we used the integral identities

∫

Ω

wxyvxydr =

∫

∂Ω

nywxyvxdσ −∫

Ω

wxyyvxdr

=

∫

∂Ω

[nywxyvx − nxwyyvx] dσ +

∫

Ω

wyyvxxdr,∫

Ω

wxyvxydr =

∫

∂Ω

nxwxyvydσ −∫

Ω

wxxyvydr

=

∫

∂Ω

[nxwxyvx − nywxxvy] dσ +

∫

Ω

wxxvyydr.

A.2 Special functions

We here list some of the identities and series expansions of the special functions used in this

mongraph. The identities of Hankel function and modified Bessel function are

H(1)ν (ζ) = Jν (ζ) + iYν (ζ) ,

Iν (ζ) =

e−1

2π iJν

(ζe

1

2π i)

for −π < arg ζ ≤ π2

e3

2π iJν

(ζe−

3

2π i)

for π2< arg ζ ≤ π

.

The derivatives of Hankel function are

(1

z

d

dζ

)k (z−kH(1)

ν (ζ))

= (−1)k ζ−ν−kH(1)ν+k (ζ) .

133

Struve function of the deep water solution for a point loading is computed using the following

power expansion [4] formula 12.1.3

Hν (ζ) =∞∑

n=0

(−1)n (ζ/2)2n+µ+1

Γ (n+ 3/2) Γ (ν + n+ 3/2)

where Γ is a Gamma function which is computed using a MatLab built-in function. For

large value of ζ we have a more stable expansion [4] formula 12.1.30

H0 (ζ) − Yo (ζ) ∼ 2

π

[1

z− 1

z3+

1 · 32

z5− 1 · 32 · 52

z7+ · · ·

]

for |arg ζ | < π. Ci and si functions are computed using following expansion

Ci (ζ) = γ + log ζ +

∞∑

n=1

(−1)n ζ2n

2n (2n)!,

Si (ζ) =

∞∑

n=1

(−1)n ζ2n+1

(2n+ 1) (2n + 1)!,

si (ζ) = Si (ζ) − π

2,

from [4] formulas 5.2.16, 5.2.14 and 5.2.5 respectively. Note that in some literature Ci, Si

and si are defined differently.

134

Appendix B

Deep water solution

Here, we show that

1 − γn

δn→ 1

Hv(πnH

), n = 1, 2, 3, ... (B.1)

which was used in section 4.6.

tangHdn gntan'dnH

y1

y2

Figure B.1: Illustration of the relative positions of the curves which determine the imaginaryroots γn and δn, n = 1, 2, 3, ....

Fig. (B.1) shows the relative positions of the curves which are used to find the imaginary

roots γn and δn. We notice by observing Fig. (B.1) that

(γn − δn) tan′ γnH ≈ y1 (ξn) − y2 (ξn)

→

(f1 (ξn) − f2 (ξn)

)ξ4n

ξ2t1 (ξn) t2 (ξn)

where

y1 (ξn) =−ρω2

ξnt1 (ξn), j = 1, 2.

135

Hence, we have

1 − γn

δn≈ 1

H

ρω2

δn (1 + tan2 δnH)

(f2 (ξn) − f1 (ξn)

)ξ4n

ξ2t1 (ξn) t2 (ξn)

which is the desired formula (B.1). We know that the derivative of tan at the roots can be

written by an algebraic expression of γn using the dispersion equation,

(tan γH)′∣∣γ=δn

= H(1 + tan2 δnH

)= H

1 +

( −ρω2

δn (D1δ4n −m1ω2 + ρg)

)2

= Hf2 (i δn) f2 (− i δn)

t22 (δn).

136

Appendix C

Relationship between R and T

We find that from Eqn. (3.41) for a closed surface S,

Im

∫

S

φφ∗ndσ = 0.

Let S be a rectangular contour in the (x, z)-plane defined by four points (±x0, 0) and

(±x0,−H), then above identity becomes

0 = Im

[∫ 0

−H

φφ∗x|x=x0

dz +

∫ x0

−x0

φφ∗z|z=0 dx+

∫ 0

−H

φφ∗x|x=−x0

dz

]. (C.1)

We evaluate the each integration separately noting that as x0 tends to infinity, only the

modes of real wavenumbers survive, that is

φ (x, z) ∼ Teiµx coshµ′ (z +H)

µ′ sinhµ′Hfor x > 0,

φ (x, z) ∼(Ieiλx +Re− i λx

) coshλ′ (z +H)

λ′ sinh λ′Hfor x < 0

where T , R and I are complex value amplitude of tranvelling, reflection and incindent waves.

The first integration in Eqn. (C.1) is

Im

∫ 0

−H

φφ∗x|x=x0

dz = −∫ 0

−H

µ |T |2 cosh2 µ′ (z +H)

µ′2 sinh2 µ′Hdz

=−Re [µ] |T |2

µ′2 sinh2 µ′H

(H

2+

sinh 2µ′H

4µ′

). (C.2)

Similarily, the third integration of Eqn. (C.1) is

Im

∫ 0

−H

φφ∗x|x=−x0

dz =

(|I|2 − |R|2

)Re [λ]

λ′2 sinh2 λ′H

(H

2+

sinh 2λ′H

4λ′

). (C.3)

137

The second integration is considered in x < 0 and x > 0, first we have using the plate

equation in x < 0

Im

∫ 0

−x0

φφ∗z|z=0 dy = Im

∫ 0

−x0

D1

ρω2

(∂2

∂x2− k2

)2

φzφ∗zdx.

Note that we used Imφzφ∗z = 0. After succesesive use of the integral by parts, we have

Im

∫ 0

−x0

D1

ρω2

(∂2

∂x2− k2

)2

φzφ∗zdx = Im

D1

ρω2φ∗

z

[φzxxx − (2 − ν) k2φzx

]0−x0

− ImD1

ρω2

[νk2φ∗

zφzx + φ∗zxφzxx

]0−x0

. (C.4)

Similarily, we have for x > 0

Im

∫ x0

0

D2

ρω2

(∂2

∂x2− k2

)2

φzφ∗zdx = Im

D2

ρω2φ∗

z

[φzxxx − (2 − ν) k2φzx

]x0

0

− ImD1

ρω2

[νk2φ∗

zφzx + φ∗zxφzxx

]x0

0. (C.5)

We find that for any combination of the transition conditions given in chapter 2, the first

terms of the right hand side of Eqn. (C.4) and Eqn. (C.5) at x = 0 vanish, and fiurthermore

the second terms of the right hand side of Eqn. (C.4) and Eqn. (C.5) at x = 0 either vanish

or are real. Hence, evaluating the terms at x = ±x0, we have

2D1 Re [λ]λ′2

ρω2

(|I|2 − |R|2

)− 2D2 Re [µ]µ′2

ρω2|T |2 (C.6)

Adding Eqns. (C.2, C.3, C.6), we have

− |T |2

2D2 Re[µ]µ′2

ρω2 + Re[µ]

µ′2 sinh2 µ′H

(H2

+ sinh 2µ′H4µ′

)

+(|I|2 − |R|2

)2D1 Re[λ]λ′2

ρω2 + Re[λ]

λ′2 sinh2 λ′H

(H2

+ sinh 2λ′H4λ′

)= 0

⇔ λ′2 sinh2 λ′H

µ′2 sinh2 µ′H

2D2 Re [µ]µ′4 sinh2 µ′H + Re [µ] ρω2

(H2

+ sinh 2µ′H4µ′

)

2D1 Re [λ]λ′4 sinh2 λ′H + Re [λ] ρω2

(H2

+ sinh 2λ′H4λ′

) T 2 + R2 = 1

⇔ Re [µ]λ′2 sinh 2λ′H

Re [λ]µ′2 sinh 2µ′H

2µ′H (D2µ′4 + b2) + (5D2µ

′4 + b2) sinh 2µ′H

2λ′H (D1λ′4 + b1) + (5D1λ′4 + b1) sinh 2λ′HT 2 + R2 = 1.

138

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