111EQUATION CHAPTER 1 SECTION 1HO CHI MINH CITY NATIONAL UNIVERSITY

UNIVERSITY OF POLYTECHNIC

THE FACULTY OF MECHANICAL ENGINEERING

THE PFIEV PROGRAM

Graduate thesis

STUDY ON DESIGN OF 2D OCEAN WAVE MAKER

Advisor: Assoc Prof PhD Nguyễn Tấn TiếnDone by: Nguyễn Hồng QuânStudent code: 20402050

Ho Chi Minh City 2009

Acknowledgement

Abstract

Content

I. INTRODUCTION

The wave generation using a wave maker in a test basin has then become an important technology in the field of the coastal and ocean engineering. To date most laboratory testing of floating or bottom-mounted structures and studies of beach profiles and other related phenomena have utilized wave tanks, which are usually characterized as long, narrow enclosures with a wavemaker of some kind at one end; however, circular beaches have been proposed for littoral drift studies and a spiral wavemaker has been used. For all of these tests, the wavemaker is very important. The wave motion that it induces and its power requirements can be determined reasonably well from linear wave theory.

Wavemakers are, in fact, more ubiquitous than one would expect. Earthquake excitation of the seafloor or human-made structures causes waves which can be estimated by wavemaker theory; in fact, the loading on the structures can be determined. Any moving body in a fluid with a free surface will produce waves: ducks, boats, and so on. Wavemakers are also used in experimental wave basins to measure wave effects on various types of structures and vessels, including models of ships, offshore platforms, and other bodies.

The theory of water waves has attracted scientists in fluid mechanics and applied mathematics for at least one and a half centuries and has been a source of intriguing - and often difficult - mathematical problems. Apart from being important in various branches of engineering and applied sciences, many water-wave phenomena happen in everyday experience. Waves generated by ships in rivers and waves generated by wind or earthquakes in oceans are probably the most familiar examples. The mathematical theory of water waves consists of the equations of fluid mechanics, the concepts of wave propagation, and the critically important role of boundary conditions. The results obtained from theory may give some explanation of a natural phenomenon or provide a description that can be tested whenever an expanse of water is at hand: a river or pond, the ocean, or simply the household bath or sink. However, obtaining a thorough understanding of the relevant physical mechanisms presents fluid dynamicists and applied mathematicians with a great challenge

In practical use, two types of wavemakers which use a paddle with two type of moving to produce water wave are the most popular. They are so-called piston-type and flap-type wavemaker.

Figure 1.1. A piston-type wavemaker (HR Wallingford Ltd)

Figure 1.2. HR Wallingford's multi-element wavemakers

Figure 1.3. Edinburgh Designs Ltd’s flap type wavemaker

Figure 1.1. Some DHI Group’s wavemakers

Figure 1.2. An engineering testing tank with the use of wavemaker

II. WAVEMAKER THEORY FOR PLANE WAVES PRODUCED BY A PADDLE [1]

Assumptions:

1) The fluid is incompressible, irrotational. The water displaced by the wavemaker should be equal to the crest volume of the propagating wave form (Figure 2.1).

2) The paddle moves with small amplitude and the wave height is small.3) The water wave propagates in x+¿ direction which tends to infinity. By using a wave

absorber at the other end of the basin, we can consider it equivalent.

Figure 2.1

1. BOUNDARY VALUE PROBLEMS

In formulating the small-amplitude water wave problem, it is useful to review, in very general terms, the structure of boundary value problems, of which the present problem of interest is an example. Numerous classical problems of physics and most analytical problems in engineering may be posed as boundary value problems; however, in some developments, this may not be apparent.

The formulation of a boundary value problem is simply the expression in mathematical terms of the physical situation such that a unique solution exists. This generally consists of first establishing a region of interest and specifying a differential equation that must be satisfied within the region. Often, there are an infinite number of solutions to the differential equation and the remaining task is selecting the one or more solutions that are relevant to the physical problem under investigation. This selection is effected through the boundary conditions, that is, rejecting those solutions that are not compatible with these conditions.

Figure 2.2

For the geometry depicted in Figure 1.1, the governing equation for the velocity potential is the Laplace equation:

∂2 ϕ∂ x2 +

∂2 ϕ∂ z2 =0 (2.1)

Kinematics boundary conditions

At any boundary, whether it is fixed, such as the bottom, or free, such as the water surface, certain physical conditions must be satisfied by the fluid velocities. These conditions on the water particle kinematics are called kinematic boundary conditions. At any surface or fluid interface, it is clear that there must be no flow across the interface; otherwise, there would be no interface.

The mathematical expression for the kinematic boundary condition may be derived from the equation which describes the surface that constitutes the boundary. Any fixed or moving surface can be expressed in terms of a mathematical expression of the form F ( x , y , z ,t )=0. If the surface vanes with time, as would the water surface, then the total derivative of the surface with respect to time would be zero on the surface. In other words, if we move with the surface, it does not change.

DF (x , y , z , t)Dt

=0=∂ F∂t

+u∂ F∂ x

+v∂ F∂ y

+w∂ F∂ z

(2.2a)

or

−∂ F∂ t

=u ∙∇F=u ∙ n|∇F| (2.2b)

where the unit vector normal to the surface has been introduced as n=∇F /|∇F|

Rearranging the kinematic boundary condition results:

u ∙ n=

−∂ F∂ t

|∇F| on F ( x , y , z , t )=0 (2.3)

where

|∇F|=√( ∂ F∂ x )

2

+( ∂ F∂ y )

2

+( ∂ F∂ z )

2

This condition requires that the component of the fluid velocity normal to the surface be related to the local velocity of the surface. If the surface does not change with time, then u ∙ n=0 ; that is, the velocity component normal to the surface is zero.

The Bottom Boundary Condition (BBC)

The lower boundary of our region of interest is described as z=−h (horizontal bottom) for a two-dimensional case where the origin is located at the still water level and h represents the depth. If the bottom is impermeable, we expect that u ∙ n=0, as the bottom does not move with time.

The surface equation for the bottom is F ( z )=z+h=0. Therefore

w=0 on z=−h (2.4)

Kinematic Free Surface Boundary Condition (KFSBC)

The free surface of a wave can be described aF ( x , z , t )=z−η ( x ,t )=0s, where η(x , t) is the displacement of the free surface about the horizontal plane z=0. The kinematic boundary condition at the free surface is

u ∙ n=

∂ η∂ t

√( ∂ η∂ x )

2

+1

on z=η ( x , t ) (2.5)

where

n=

−∂ η∂ x

i+1 k

√( ∂ η∂ x )

2

+1

Carrying the dot product yields

w= ∂ η∂ t

+u∂ η∂ x|on z=η ( x ,t )

(2.6)

Dynamic Free Surface Boundary Condition

A distinguishing feature of fixed (in space) surfaces is that they can support pressure variations. However, surfaces that are “free”, such as the air-water interface, cannot support variations in pressure (neglecting surface tension) across the interface and hence must respond in order to maintain the pressure as uniform. A second boundary condition, termed a dynamic boundary condition, is thus required on any free surface or interface, to prescribe the pressure distribution pressures on this boundary.

As the dynamic free surface boundary condition is a requirement that the pressure on the free surface be uniform along the wave form, the Bernoulli equation with pη=constant is applied on the free surface z=η (x , t).

−∂ ϕ∂t

+pη

ρ+ 1

2 [( ∂ η∂ x )

2

+( ∂ η∂ x )

2]+gz=C (t ) , z=η(x , t) (2.7)

where pη is a constant and usually taken as gage pressure, pη=0.

Lateral Boundary Conditions

Consider a vertical paddle acting as a wavemaker in a wave tank. If the displacement of the paddle may be described as x=S(z , t), the kinematic boundary condition is

u ⋅n=

∂ S ( z , t )∂t

√1+( ∂ S∂ z )

2

where

n=1 i−∂ S

∂ zk

√1+( ∂ S∂ z )

2

or, carrying out the dot product,

u−w∂ S∂ z

=∂ S∂ t |on x=S ( z ,t )

(2.8)

which, of course, require that the fluid particles at the moving wall follow the wall.

2. SOLUTION TO LINEARIZED WATER WAVE BOUNDARY VALUE PROBLEM FOR A HORIZONTAL BOTTOM

Solution of the Laplace equation

A convenient method for solving some linear partial differential equation is called separation of variables. For our case

ϕ ( x , z ,t )=X ( x ) ⋅Z ( z ) . T (t) (2.9)

For waves that are periodic in time, we can specify T (t )=sin σt. The velocity potential now takes the form

ϕ ( x , z ,t )=X ( x ) ⋅Z ( z )⋅sin σt (2.10)

Substituting into the Laplace equation and dividing by ϕ gives us

1X

∂2 X∂ x2 + 1

Z∂2 Z∂ z2 =0 (2.11)

Clearly, the first term of this equation depends on x alone, while the second term depends only on z. If we consider a variation in z in Eq. (2.11) holding x constant, the second term could conceivably vary, whereas the first term could not. This would give a nonzero sum in Eq. (2.11) and thus the equation would not be satisfied. The only way that the equation would hold is if each term is equal to the same constant except for a sign difference, that is,

d2 X ( x )d x2

X ( x )=−k2 (2.12a)

d2 Z ( z )d z2

Z ( z )=+k2 (2.12b)

The fact that we have assigned a minus constant to the x term is not of importance, as we will permit the separation constant k to have an imaginary value in this problem and in general the separation constant can be complex.

Equations (2.12) are now ordinary differential equations and may be solved separately. Three possible cases may now be examined depending on the nature of k ; these are for k real, k=0, and k a pure imaginary number. Table 2.1 lists the separate cases. (Note that if k consisted of both a real and an imaginary part, this could imply a change of wave height with distance, which may be valid for cases of waves propagating with damping or wave growth by wind)

Table 2.1 Possible Solutions to the Laplace Equation, Based on Separation of Variables

Character of k, the Separation Constant

Ordinary Differential Equations

Solutions

Real

k 2>0

d2 Xd x2 +k2 X=0

d2 Zd z2 −k2 Z=0

X ( x )=A cos kx+B sin kx

Z ( z )=C ekx+D e−kx

k=0

d2 Xd x2 =0

d2 Zd z2 =0

X ( x )=Ax+B

Z ( z )=Cz+D

Imaginaryk 2<0 , k=i|k|

|k|=magnitude of k

d2 Xd x2 +k2 X=0

d2 Zd z2 −k2 Z=0

X ( x )=A e|k|x+B E−|k|x

Z ( z )=C cos|k|z+D sin|k|z

Linearization of dynamics free surface boundary condition

The Bernoulli equation must be satisfied on z=η (x , t), which is a priori unknown. A convenient method used to evaluate the condition, then, is to evaluate it on z=η (x , t) by expanding the value of the condition at z=0 (a known location) by the truncated Taylor series.

(gz−∂ ϕ∂ t

+ u2+v2

2 )z=η

=(gz−∂ ϕ∂ t

+ u2+v2

2 )z=0

+η [g− ∂2 ϕ∂ z ∂ t

+

12

∂

∂ z(u2+w2 )]

z=0

+…=C (t )

where p=0 on z=η.

Now for infinitesimally small waves, η is small, and therefore it is assumed that velocities and pressures are small; thus any products of these variables are very small: η≪1, but η2 ≪η

, or uη≪η. If we neglect these small terms, the Bernoulli equation is written as

(−∂ ϕ∂ t

+gη)z=0

=C (t)

The resulting linear dynamic free surface boundary condition relates the instantaneous displacement of the free surface to the time rate of change of the velocity potential,

η= 1g

∂ ϕ∂ t |z=0

+C ( t )

g

Since by our definition η will have a zero spatial and temporal mean, C ( t )=0, thus

η= 1g

∂ ϕ∂ t |z=0

(2.13)

Linearization of kinematics free surface boundary condition

Using the Taylor series expansion to relate the boundary condition at the unknown elevation, z=η (x , t) to z=0, we have

(w−∂ η∂t

−u∂ η∂ x )

z=η

=(w−∂ η∂ t

−u∂ η∂ x )

z=0

+η∂

∂ z (w−∂ η∂ t

−u∂ η∂ x )

z=0

+…=0

Again retaining only the terms that are linear in our small parameters, η, u, and w, and recalling that η is not a function of z, the linearized kinematic free surface boundary condition results:

w= ∂ η∂ t |z=0

(2.14a)

or

−∂ ϕ∂ z |

z=0

=∂ η∂t

(2.14b)

3. APPLICATION TO THE PLANE WAVE PRODUCED BY A PADDLE

Let’s recall boundary conditions:

The linearized form of the dynamic and kinematics free surface boundary conditions:

η=1g

∂ ϕ∂ t

, z=0 (2.15)

−∂ ϕ∂ z

=∂η∂ t

, z=0 (2.16)

The bottom boundary condition is the usual no-flow condition

−∂ ϕ∂ z

=0 , z=−h (2.17)

To the lateral boundary condition, in the positive x direction, as x becomes large, we require that the waves be outwardly propagating, imposing the radiation boundary condition (Sommerfield, 1964). At x=0, a kinematics condition must be satisfied on the wavemaker. If S(z ) is the stroke of the wavemaker, its horizontal displacement is described as

x=S (z )

2sin σt (2.18)

where σ is the wavemaker frequency.

The function that describes the surface of the wavemaker is

F ( x , z , t )=x−S ( z )

2sin σt=0 (2.19)

The general kinematics boundary condition is

u ∙n=

−∂ F ( x , z , t )∂ t

|∇F| on F ( x , z , t )=0(2.20)

where u=u i+w k and n=∇F /|∇F|. Substituting for F (x , z ,t ) yields

u−w2

dS ( z )dz

sin σt=S ( z )2

σ cos σt on F ( x , z , t )=0 (2.21)

For small displacement S(z) and small velocities, we can linearize this equation by neglecting the second term on the left-hand side.

As at the free surface, it is convenient to express the condition at the moving lateral boundary in terms of its mean position, x=0. To do this we expand the condition in a truncated Taylor series

[u−S ( z )2

σ cosσt ]x=[ S ( z )2 ]sinσt

=[u− S ( z )2

σ cos σt ]x=0

+S (z )2

sin σt∂

∂ x [u−S ( z)2

σ cosσt ]x=0

+…

(2.22)

Clearly, only the first term in the expansion is linear in u and S(z ), the others are dropped, as they are assumed to be very small. Therefore, the final lateral boundary condition is

u (0 , z , t )=S ( z )2

σ cosσt (2.23)

Now that the boundary value problem is specified, all the possible solutions to the Laplace equation are examined as possible solutions to the determine those that satisfy the boundary conditions. Referring back to Table 2.1, the following general velocity potential, which satisfies the bottom boundary conditions, is presented.

ϕ ( x , z ,t )=A pcosh k p(h+z)sin(k p x−σt )+( Ax+B )+C e−k s x cosk s(h+z )cosσt (2.24)

When using wavemaker to generate water wave, in addition to the desired progressive wave, there exist several evanescent standing waves. The subscripts on k indicate that that portion of ϕ is associated with a progressive ( p) or a standing wave (s). For the wavemaker problem, A must be zero, as there is no uniform flow possible through the wavemaker and B can be set to zero without affecting the velocity field. The remaining terms must satisfy the two linearized free surface boundary condition, made up of both conditions. This condition is

∂ ϕ∂ z

−σ2 ϕg

=0 , z=0 (2.25)

which can be obtained by eliminating the free surface η from Eqs. (2.15) and (2.16). Substituting our assumed solution into this condition yields

σ 2=g k p tanh k p h (2.26)and

σ 2=−gks tan ks h (2.27)

The first equation is the dispersion relationship for progressive waves, while the second relationship, which relates k s to the frequency of the wavemaker, determines the wave numbers for standing waves with amplitudes that decrease exponentially with distance from the wavemaker. Rewriting the two equations as

σ 2hg k p h

=−tanh k p h and σ2 h

g ks h=− tan k sh (2.28)

The solutions to these equations can be shown in graphical form (see Figure 2.3 and 2.4).

The first equation has only one solution or equivalently one value of k p for given values of σ and h whilst there are clearly an infinite number of solutions to the second equation and all are possible. It means that there exist one progressive wave and countless standing wave. Each k s solution will be denoted as k s(n), where n is an integer. The final form for the boundary value problem is proposed as

ϕ=A p coshk p (h+ z )sin(k p x−σt )+∑n=1

∞

Cn e−k s ( n) x cos [k s (n ) ( h+z ) ]cos σt

(2.29)

The first term ( Ap cosh k p (h+z ) sin(k p x−σt )) represents a progressive wave, made by the

wavemaker, while the second series of waves (∑n=1

∞

Cn e−k s ( n) x cos [k s (n ) (h+z ) ]cos σt) are standing

waves which decay away from the wavemaker.

Figure 2.3

Figure 2.4

To determine how rapidly the exponential standing waves decrease in the x direction, let us examine the first term in the series, which decays the least rapidly. The quantity k s (1 )h, from

Figure 2.3, must be greater than π /2, but for conservative reason, say k s (1 )h=π /2, therefore,

the decay of standing wave height is greater than e−π

2xh . For x=2 h, e

−π2

xh=0.04, for x=3 h, it is

equal to 0.009. Therefore, the first term in the series is virtually negligible two to three water depths away from the wavemaker.

For a complete solution, Ap and Cn need to be determined. These are evaluated by the lateral boundary condition at the wavemaker.

u (0 , z , t )=S ( z )2

σ cosσt=−∂ ϕ∂ x

(0 , z , t )=−A pk pcosh k p(h+z)cos σt+∑n=1

∞

Cn ks (n ) cos [ks (n ) (h+z )]cosσt

orS (z )

2σ=−A p k p coshk p (h+ z )+∑

n=1

∞

Cn k s (n )cos [ks (n ) (h+z ) ] (2.30)Now we have a function of z equal to a series of trigonometric functions of z on the right-hand side, similar to the situation for the Fourier series. In fact, the set of functions,

{cosh k p ( h+ z ) , cos [ ks (n ) (h+z ) ] , n=1 , ∞ } form a complete harmonic series of orthogonal

functions and thus any continuous function can be expanded in terms of them (Sturm-Liouville theory). Therefore, to find Ap, the equation above is multiplied by cosh k p(h+z) and integrated from −h to 0. Due to the orthogonality property of these functions there is no contribution from the series terms and therefore

Ap=−∫

−h

0S ( z )

2σ cosh k p (h+z ) dz

k p∫−h

0

cosh2k p (h+ z )dz

(2.31)

Multiply Eq. (2.17) by cos [ ks (m ) ( h+ z ) ] and integrating over depth yields

Cm=∫−h

0S (z )

2σ cos [k s (m) (h+z ) ] dz

ks (m )∫−h

0

cos2¿¿¿¿(2.32)

Depending on the functional form of S(z ), the coefficients are readily obtained. For the simple cases of piston and flap wavemaker, the S(z ) are specified as

S ( z )={ S ,piston motion

S (1+ zh )flap motion (2.33)

Thus

Ap={ −2 Sσk p

sinh k p h2k p h+sinh k p h

piston motion

−2 Sσ

k p2 h

k p h sinh k p h−cosh k p h+1

2k p h+sinh 2k phflap motion

(2.34)

Cm={ 2 Sσks (m )

sin k s ( m) h2ks (m )h+sinh 2ks (m )h

,∧piston motion

2 Sσks (m )h

ks (m ) hsin k (m) h+cos ks (m ) h−1

2 ks (m ) h+sinh 2 ks (m ) h,∧flap motion

(2.35)

The wave height for the progressive wave is determined by evaluating η far from the wavemaker.

η= 1g

∂ ϕ∂ t |z=0

=−A p

gσ cos k p h cos ( k p x−σt )=H

2cos(k p x−σt ) x≫h (2.36)

Substituting for Ap, we can find the ratio of wave height to stroke as

HS

=4sinh k p h

k p h.k p h sinh k ph−cosh k p h+1

sinh 2 k p h+2k p hflap type (2.37)

HS

=2 ( cosh2 k p h−1 )sinh 2 k p h+2k ph

piston type (2.38)

The power required to generate these water can be obtained by determining the energy fluid flux away from the wavemaker

P=ECn (2.39)

where E is the total average energy per unit surface area

E=18

ρg H 2(2.40)

C=σ /k is the phase velocity, and with Cg=Δσ / Δk is the group velocity:

n=Cg

C=1

2 (1+ 2 khsinh2 kh ) (2.41)

Figure 2.4

III. MATHEMATICAL MODELING

This modeling problem consists of:

- With the given wave height, period of the desired water wave, calculating the wave numbers, the wave length and the wavemaker’s stroke to generate that wave.

- Finding the distance beyond which the evanescent standing wave can be neglected.- Calculating the elevation of the water wave at several positions, counting the

standing waves at positions where they are significant.

1. Computing the wave numbers:

The given parameters: the desired wave height: H , the period T (hence, the angular frequency σ=2π /T ) and the basin’s depth h.

The wave numbers of the progressive wave and standing waves are computed from the dispersion equations (2.26) and (2.27)

σ 2=g k p tanh k p h

σ 2=−gks tan ks h

where the first equation has a unique root k p while the second has numerous roots k s but we just need to find some smallest roots, say 3, because the standing wave’s amplitudes decrease exponentially with k s.

These equations is non-linear, and the solutions can be achieved with a programming language’s function, e.g. the Python’s function scipy.optimize.fsolve() which is a wrapper around MINPACK’s hybrd and hybrj algorithms.

Such a numerical root finding subroutine need a starting estimate. For k p we can make use of the approximation suggested by Fenton and McKee (1989) for the wave length:

L=L∞( tanh( σ2hg )

34 )

23

where L∞=gT 2

2π and k=2 π

L.

For k s, three starting estimates of three roots k s(1), k s(2), k s(3) can be deduced from Figure 2.3. They are, respectively, π /h, 2 π /h, 3 π /h.

The wavemaker’s stroke can be calculated using (2.37) and (2.38).

2. Finding the range in which the standing waves is considerable

In order to choose a relevant length for the basin, we must determine the effective range of

the standing waves. We knew that the standing wave amplitudes Cn e−k s (n ) x cos [k s (n )h ]σ /g

decreasing exponentially with k s and the distance x. So we can determine the effective range relied on the first standing wave component, say C1 and k s(1). This range is defined so that the ratio of the standing wave’s amplitude to that of the progressive is less than a threshold, usually 0,001.

Denote D(x )=C1e−k s (1) x cos [ ks (1 ) h ] σ / g. Following is the binary algorithm to search the

nearest distance x p beyond which the standing wave is negligible within the increasing sequence of discrete positions x=[x0 , x1 , x2 , … xn].

1. Begin with pl=0, p=n/2, pr=n

2. At x p, calculate diff =D(x p)/ (H /2)−R

3. If diff <0, let pr=p, then p=pl+(pr – pl)/2; if diff >0, let pl=p, then p=pl+(pr – pl)/2; and if diff =0, break the iteration loop.

4. Return to step 2 and so on, until p=pl, the iteration stops with xp is found

3. Computing the elevation of water wave

The computation of progressive wave’s elevation at numerous positions x=[ x1 , x2 , …xn ] at

several continuous instants t=[ t 0 ,t 0+Δt ,t 0+2 Δt ,…t 0+m Δt ] is massive because it needs

evaluation of many trigonometric functions. It can be lightened with following iterative computational method [2].

The elevation at i-th time step:

ηi=H2

cos (k p x−σ (t 0+i Δt ))

Introduce ξ i in companion with ηi as:

(η i

ξi)= H

2 ( cosk p x sin k p x−sin k p x cos k p x)(cos (t 0+ i Δt ) σ

sin (t0+i Δt ) σ )Denote c=cosk p x, s=sin k p x

The elevation at (i+1)-th time step:

(η i+1

ξi+1)=H

2 ( c s−s c )(cos ( (t0+i Δt ) σ+σ Δt )

sin (( t0+i Δt ) σ+σ Δ t ))¿ H

2 ( c s−s c )(cos σ Δt −sin σ Δt

sin σ Δt cosσ Δt )(cos (t0+i Δt ) σsin ( t0+i Δt ) σ )

Denote α=cos (−σΔt ), β=sin (−σ Δt )

(η i+1

ξi+1)=H

2 ( c s−s c )( α β

−β α )(cos (t 0+i Δt ) σ

sin (t 0+i Δ t ) σ )

¿ H2 ( α β

−β α)( c s−s c )(cos ( t0+i Δt ) σ

sin (t 0+i Δt ) σ )Thus

(η i+1

ξi+1)=( α β

−β α)(ηi

ξ i)

(η0

ξ0)=H

2 ( c s−s c)(cosσ t 0

sin σ t 0)

The elevation component caused by standing waves

ηs=−σg

∑n=1

3

Cn e−k s ( n) x cos [ks (n ) h]sin σt=σg∑n=1

3

Cn e−k s ( n) x cos [ks (n ) h]cos (σt+ π2 )

is only considerable in the range which we determined earlier and can be computed with the method above.

(η0

ξ0)n=1,2,3

=σg

Cn ks (n ) e−k s ( n) x cos [ ks (n ) h ](cosσ t 0

sin σ t0)

(η i+1

ξi+1)=( α β

−β α)(ηi

ξ i)

The final elevation is the sum of those of those waves.

IV. MODELING’S RESULT

Following is the result for the modeling of water wave with wave height H=5 cm, period T=1 s, generated by piston type and flap type wavemaker in a basin with the water level of h=15 cm height.

Piston type Flap type

Wave number (m−1) 5.765 5.765

Wave length (m) 1.09 1.09

Wavemaker’s stroke (cm) 5.84 11.05

Standing wave’s wave numbers (m−1)19.6

41.2462.402

19.641.24

62.402

Standing wave effective range (cm) 17 39

Following is the plots of the wave elevations. We can see in the case of flap type wavemaker, the affection of standing waves is more remarkable than of piston type wavemaker.

Figure 4.1. The total elevation of waves generated by piston type wavemaker

Figure 4.2. The elevation of standing wave in case of piston type wavemaker

Figure 4.3. The total elevation of waves generated by flap type wavemaker

Figure 4.4. The elevation of standing wave in case of flap type wavemaker

Conclusion: In the case of shallow water, the piston type wavemaker is more effective than the flap type.

In addition, we have a comparison between piston type and flap type wavemaker:

Piston type:

- Advantage: Shorter wavemaker stroke, less affection of evanescent standing wave.

- Disadvantage: In deep water, this type wastes more energy to move the lower water layers.

Flap type:

- Advantage: In deep water, this type doesn’t waste energy to move the lower water layers as piston type.

- Disadvantage: Longer wavemaker stroke, more affection of evanescent standing wave.

Reference

[1] Dean & Dalrymble, Water wave mechanics for engineers and scientists, Word Scientific, 1991

[2] Ben T. Nohara, A Survey of the Generation of Ocean Waves in a Test Basin