Chapter 7 Solution for Wave Interactions Using A PMM

7.1 Waves Advancing on Currents

7.2 Short Waves Modulated by Long Waves

7.3 Wave Modulation Revealed by A MCM Solution

7.4 Phase Modulation Method (PMM)

7.5 Conformal Mapping Approach (old approach)

7.1 Waves Advancing on Currents

Waves on steady and uniform currents: Doppler‘s effect

, (7.1.1)

where -intrinsic frequency,

-absolute (apparent) frequency

-current velocity, -wavenum vector

U k

U k

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����������������������������

Waves on unsteady and non-uniform currents: Modulation

Wave frequency, wavenumber vector and amplitude changes

Peregrine (1976)

Figure 7.1 Wave advance over a converging and diverging velocity field

A converging velocity field: the velocity of currents decreases in the direction of currents. Wavenumber and amplitude increase.

A diverging velocity field: the velocity of currents increases in the direction of currents. Wavenumber and amplitude decrease.

7.2 Short Waves Modulated by Long Waves

Velocity induced by a long-wave train looks like current toA short-wave train. It is unsteady and non-uniform.

Figure 7.2: A short-wave train interacts with a long-wave train.

The velocity field induced by a long-wave train is convergingat its forward face and diverging at its backward face.

When the directions of both wave trains are co-linear, the short-wave train always moves backwards with respect to the long-wave train because the phase velocity of the long-wave train is greater.

The short-wave train experiences compressing and stretching respectively on the forward and backward face of the long-wave train.

Consequently, their characteristics, such as wavenumber and amplitude increase and decrease on the forward and backward face of the long-wave train. The phenomenon of changing short-wave characteristics is known as wave modulation.

7.3 Wave Modulation Revealed by A MCM Solution

Examining the interaction between a long- and short-wave train, following Longut-Higgins anf Stewart (1960).

(1)1 1 2 22

(2) 21 2 2 1 2 1

1

1 2 2 1 2 1

cos cos , (5.2.2)

1 1cos 2 ( )cos( )

2 2

1 ( ) cos( ). (5.2

2

j j jj

a a

a k a a k k

a a k k

1 2 2 1 2 1 1 2 2 1 2 1

1 2 1 2 1 1 2 2 2 1

1 1( )cos( ) ( ) cos( )

2 2cos cos sin s

. )

i

10

n .

a a k k a a k k

a a k a a k

2 2 2 1 2 1 2 1 1 2 2 2 1

2 2

2 2 1 2 1

2 2 1 1

cos cos cos sin sin

cos , (7.3.1)

where

sin , (7.3.2a)

(1 c

a a a k a a k

a

a k

a a a k

1os ), (7.3.2b)

The modulated phase is not a linear function of time and horizontal Coordinates.

For the validity of the approximation, the terms, such as

2 2 2 2 21 2 1 1 2

1 1sin 1, or 1.

2! 2!a k a k

2 12 2 1 2 1

2

22 2 1 1 1

1 cos , (7.3.3)

(1 cos ). (7.3.4)

a kt

k k a kx

Modulated wavenumber and frequency.

The modulated amplitude and wavenumber reach the maximum at the crest and the minimum at the trough of the long-wave train. These results are expected according to the converging and diverging velocity field induced by the long-wave train. (see the applet: long & short wave interaction)

The above example reveals the physics of bound waves that they essentially describe the modulated wave characteristics.

Limitations of revealing wave modulation based on MCM solutions.

1) The approximation in (7.3.1) is not valid if a1k2 is not much smaller than unity. This requirement is also necessary to have a convergent MCM at second order in wave steepness as discussed in Chapter 5.

2) The corresponding solution for the potential of the modulated short-wave train could not be revealed by the above rudiment approach. Hence, an approach that allows for directly deriving the solution for modulated short-wave elevation and potential was developed, which is known as Phase Modulation Method (PMM).

7.4 Phase Modulation Method (PMM)

A PMM is also a perturbation method, but different from conventional perturbation methods in the following key aspects.

1. Formulation of the phase of a modulated free wave elevation

MCM: Always Linear Phase

PMM: Nonlinear Phase

2. Formulation of the potential of a modulated free wave

MCM:

PMM:

2 2 2 2

2 2 1 2 1sin

k x t

a k

2

2 2

2

( ) ( , , ),

e.g. ( ) exp( ), ( , , ) sin

( , , , ) ( , , , ) .

V z H x y t

V z k z H x y t

V z x y t H x y t z

3. The modulation of a free wave in a MCM, is indirectly modeled by bound waves. The modulation of a free wave in a in a PMM, is directly described, e.g. the modulated phase and amplitude, etc.

4. In describing the interaction between a short- and a long-wave train.

MCM solution may not converge, if k1 << k2.

PMM solution always converge, even if k1 << k2.

5. MCM solution: relatively simple

PMM solution: relatively complicated

Relationship between the solutions obtained respectively by MCM & PMM

When both solutions are truncated at the same order and they converge,

1) they are consistent up to the truncated order in general wave steepness.

2) they are exactly identical after the two solutions are converted to the same formulation, say in the formulation used in a MCM.

The above relationship is crucial to a) the development of solution matching approach and b) to the use of both methods in quantifying strong wave interactions in irregular waves.

Techniques to derive PMM solutions

1) Conformal mapping approach. (Zhang et al. 1993)

It was the first approach used to obtain the solution for modulated free waves. The derivation is independent of that of a MCM and more lengthy than the latter.

2) Solution-matching approaches

It was developed based on the fact that the truncated solutions obtained using PMM and MCM are identically matched up to the truncated order when both of them converge and are converted to the same formulation. This approach is only marginally complicated than the MCM in deriving solutions.

7.5 Conformal Mapping Approach

1) Conformal mapping the physical fluid domain of one long wavelength onto a rectangular domain whose coordinates correspond to streamlines and equi-potentials of the long-wave train in the physical domain (Chapter 3).

See handout Zhang et al. (1993)Differences in notations between the handout and the notes

1

1

Handout Notes

' (in - )

C C

K k

x z

1 1 2 2 32 2

1 1 1 1 1

21 1 1 1 1

( , ) ( , ), truncated mapping functions in deep water

/ , / , , , / ,

' sin sin 2 ( ) , (2.6a)

1cos cos 2

2

Kn Kn

Kn Kn

x z s n

s C n C a K a k K k

x x Ct s a e Ks a e Ks O a

z n a a e Ks a e Ks

21 1

21 1 1

21 1 1 1 1

1

( ) , (2.6b)

sin ( ) , (2.7a)

1cos ( ) , (2.7b)

2

For 0, si

Kn

Kn

O a

s x Ct a e O a

n z a a e O a

n s x Ct a

21 1 1 1

1 2 1 21

2 22'

1n sin 2 ( ) , (2.7c)

2/ ', / ', ( , ) / ,

/ / , (2.14)

a O a

H ds ds H dn dn H s n H H U C

H x s z s

Sketch of the Conformal Mapping

Governing Equations in the (s,n) plane:The value of the short-wave potential and elevation are the same as in the (x,z) plane.H(s,n) is a scale factor relating the increments in the two plane.The equations are derived at the (x-z) plane. See Figures 1 &2.

2 2

2 2

2 22 '

0

2

0, (2.9)

1cos , at (2.10)

2

, at . (2.11)

( ', )

ns n

U H H g g C nt s n

H U H H nt s s n

U x z

'

0, / 0, as (2.12) (2.13)

dnC n n

H

Expanding the free surface boundary condition at n=0 (which is the long-wave surface) and subtracting the stead solution for the long free wave.

2 2 2 2 2 20 1 0 0 2 2

2 2 2 2 2 3 '0 0 1,2 2

2 2 2 20 00 0 0 0 0

2 2 3 '0 1,2 2

1 1( ) ( )

2 2

1 1( ) ( ) ( ) , at 0 (4.1)

2 2

( ) ( ) 2

1( ) ( ) ,

2

t s t s n n

t s nn s n n

t s n s s s n

sn s

H C g H C H g

H C H O A n

H HH C H H H C H

n n

H O a

20 00 1 0

0 0

2 22 30 0

2 12 20

at 0 (4.2)

where

cos( ,0), , (4.3)

1 cos( )

2

n

n

H HH gH H s g H C

n n H n

H Hgg C O Kg

n H n

Solution for the short-wave train Using standard perturbation method, we solve for the potential and elevation in the (s,n) plane.

'(1) ' ' 3 '2 1,2 2

(1) ' ' '3 '2 ' 3 '2 2 1,2 2

' ' ' ' '2 2 1

2 ' 2 20 1 0

sin ( ) , (4.4)

3cos cos 2 ( ) , (4.5)

8

/ , / , (4.6) / (4.9)

= , (4.7)

k n

I

I I

A e O A

a a k O a

s k t a A g

H Ck g H k

'

' '2 '22

(2) ' ' ' (2 ) '1 2 2 2

(2) '2 ' '2

'2 ' '1 2 2

(1 ). (4.8)

3 / 2 /sin(2 ), (4.10)

1 /

1cos 2

22 3 / 2

cos(2 ), (4.11)1 /

I k K n

I I

I

I I

a k

a k A e Ks

a k

a k Ks

(3) 3 '1,2 2

(3) '3 '2 ' 3 '2 1,2 2

( ) (4.12)

3cos3 ( ) , (4.13)

8

O A

a k O a

Solution for the long-wave train

2(0) '2 '0

2

22 (0)'2 ' 202 02

2 2 2 21 2

1, (4.14)

2

1, (4.15)

2

(1 2( / ) ).

I

I

C Ha k

n

C Hg ga k gH C

t z t n s

gK a K a kK

(4.16)

Conversion of the PMM solution

1. from the (s,n) plane to the (x,z) plane: focusing on the solution for the short wave train.

2. the modulated phase in the short-wave potential and elevation.

3. the structure of potential

4. from modulation formulation to traditional (MCM) formulation which has linear phase and a potential function consisting of two functions depending on different variables (for the purpose of comparison).

12 2 2 2 32

1 1 1 1

12 2 32

0 1 1 1 1

2 31 1 1

Scale factors

(1 2 cos 4 cos 2 ) ( ) (5.1)

(1 2 cos 4 cos 2 ) ( ) (5.2)

Noticing Figure 2, (5.1), (5.2) and (4.3c),

[1 ( )]

Kn Kn KnH e Ks e e Ks O

H Ks Ks O

g O g

2'301,2' 2

0

' ' 2 20 1 1 1

(5.3)

Based on the phase conservation,

1 1( ) (5.5)

1 2 cos 4 cos 2

dHdkO

k ds H ds

k k Ks

3 '1,2 0

' 20 1

2 2 22

( ) (5.6)

(1 ) (5.7)

(1 ) I

Ks O k

k k

gk a k

(5.8)

' ' 20 1 1 1 1

' 2 20 1 1 1 1

' 21 1 1 1 1

1 2 sin 2 sin 2 (5.9)

1 sin ( ) (5.10)

(2 )sin (2 )sin 2 ( ) , (5.11)

when 1, 2

Kz

Kz Kz

K

k s a Ks a Ks t

k s t ka e O ka

a k e a k e O ka

Kz e

2 31 1 1 1,2

2 31 1 1 1,2

' 21 1 1 1 10

'

0

=

1( sin 2 sin 2 sin ) ( ) (5.12a)

21

(1 cos 2 cos 2 cos ) ( ) (5.12b)2

1sin sin 2 ( ) (5.13)

2

cos

z Kz

n

n

e

x Ks Ks K Ks O

z Ks Ks K Ks O

a k a k O ka

2 2 21 1

1cos (1 sin ) sin ( sin ) (5.14)

2a k a k

2 2 '2 '2 21 1 2 2

2 3 21 1

' 2 2 '2 2 22 1 2 1 1

2 2 21 1

'22 1 3

3 1( , ) 1 cos

8 4

1 3cos 2 cos3 (5.15)

2 8

1 3( , ) 1 cos

4 8

1cos 2 cos

4

321 3 1 2

2 2 4 I

x t a a K a kK a K

a K a K

x t a a k a k

a k

a k

2

2 '21 2 3 2

' 3 '2 1 1 1,2 2

/cos cos 2

(1 / )

32 /1 21 sin sin 2

2 (1 / )

(sin sin 2 )sin ( ) (5.16)

I

I

I

I I

a k a

a a k O a

(1) ' ( )2 1 1 1

2 2 ( 2 ) 3 3 '1 1 1 1 2

(2) (2 )1 2 2 2

11 sin sin( )

2

1sin( 2 ) ( ) (5.17)

2

3/

2sin(2 ) (5.18

(1 / )

kz k K z

k K

k K z

I I

A a k e a ke

a k a k e O a k A

a kA e

1

)

sin (5.19)KzA e

Comparison

The basis for the comparison is that the leading-order solution must be the same.

In this case letting the potential amplitudes in the PMM solution and MCM solution be the same, we obtain the relation between the elevation amplitudes in these two methods.

The results up to third order in wave steepness are identical if both solutions are convergent.

'2 1 1 2

1

' 22 2 1 1 1 1 1

We let

11 (5.20)

2

Allowing for the differences between and ,

and ,

11 (5.21)

2

Then we have

I

A a k A

g g

a a a k a k

i between the

solution for potential and elevation obtained using

the MCM and PMM for the case 2 .k K

dentical matching

Summary

1. Conformal Mapping: Governing equations in the (s,n) plane.

2. Solving for potential and elevation in the (s,n) plane.

3. Converting the solution in terms of (x,z)

4. Comparison: expanding the solution in traditional form, that is, the linear phase and the potential function constructed based on a variable separation method.

5. Identical matching between the two solutions when they are convergent and k>2K, which leads to the solution-matching technique.

6. The effects of SW on LW are of 3rd order while those of LW on SW are of 2nd order.

Shortcomings of Conformal Mapping Approach

1. The derivation and conversion is relatively lengthy.

2. It is not easy to use the conformal map for a long-wave train involving several free waves because it is not steady in the moving coordinates.

3. It is very awkward to apply to the interaction between directional short and long wave interaction. The conformal mapping is mainly a 2-D approach. (Hong 1993)

It is deseriable to develop a simpler approach to derive thePMM solution

Chapter 7 Solution for Wave Interactions Using A PMM

7.6 PMM Solution Obtained Using Solution-Matching Approach

7.7 Solution for A Short Wave Modulated by A Long Wave

7.8 Solution for A Short Wave Modulated by Two Long Waves

7.9* Solution for Two Short Waves Modulated by One Long Wave 7.10* Phase-Modulation Solution for Directional Wave Interaction

7.6 PMM Solution Obtained Using Solution-Matching Approach

The derivation of the solution using the conformal mapping approach is lengthy and complicated because of the conformal mapping between the two solution domains.

To simplifying the derivation, a new approach, known as the solution-matching approach, is developed. It focuses on how to derive these modulation functions depending upon the characteristics of the underneath long-wave train without invoking conformal mapping.

An early version of the solution-matching approach was employed to derive the solution for a short free wave modulated by a long free wave advancing in different directions (Zhang et al. 1999).

The modulation functions were assumed to be periodic in the horizontal plane in terms of the long-wave phase but perturbed in the vertical direction by a power series in terms of kl z. It is understood that the change in kl z is relatively slow in the region where the short-wave potential remains significant. Using a multiple-scale perturbation scheme, the coefficients of the expansion power series were determined by the Laplace equation and free-surface boundary conditions. Although the above procedure is simplified in comparison with the derivation using the conformal mapping approach (Hong 1993), it is still much more complicated than the derivation using a MCM.

1) Taking the advantages of the fact that the solutions given by a PMM are identical to those given by a MCM up to the truncated order, the new approach determines the modulation functions in the PMM solution by matching with the corresponding MCM solution.

2) After the modulation functions are obtained through the matching, the remaining tasks are to prove the solution satisfying the Laplace equation, bottom and free-surface boundary conditions. Of course, it also needs to show that the PMM solution is convergent when the corresponding MCM solution may diverge.

3) This approach greatly simplifies the derivation. It is only marginally more time-consuming than the derivation using a MCM.

7.6.1 Governing Equations for Long- and Short-Wave Trains

The total potential and elevation are divided into the long-wave and short-wave parts,

, (7.6.1a) . (7.6.1b)l s l s

In general, either short- or long-wave train may consist of one or multiple free waves.

The difference in frequencies of the free waves within a wave train is assumed to be relatively small. However, the typical wave frequencies of the two wave trains are quite different.

The directions of the short-wave and long-wave train may be different. However, all free waves in either short-wave or long-wave train are virtually uni-directional.

1) The F-S boundary conditions are expanded at the undisturbed surface of the long-wave train using the Taylor expansion.

2) Truncating the expansion series at the third order in wave steepnesses, we split all equations into two sets with respect to the short- and long-wave trains.

2

2 32

2

2 2

0, - , (7.6.2a) 0, (7.6.2b)

1

21

( ), at (7.6.2c)2

s l s

s l ss s s s l

l s s ls

s sh s h l h l h s h

z z

gt t z t z

MND C t zz

t z z

2 3

22 3

1, at . (7.6.2d)

2

l h l h s h s s

l ss s ls

MNK zz z

2

2

0, (7.6.3a) 0, (7.6.3b)

1( ), at (7.6.3c)

2

, at . (7.6.3d)

l l l

ll l l ll

l lh l h l ll

z z

g MND C t zt

MNK zz t

1) Splitting of linear terms is straightforward, following whether a term in is of long- or short-wave phase (wavenumber & freq.).

2) Splitting of the nonlinear terms is not obvious because the phases of a nonlinear term (the interacting between short- and long-wave trains) are not exactly the same phase of either wave trains or their high harmonics. The rule for assigning a nonlinear term to either corresponding short wave’s or long wave’s boundary condition is to examine whether the phase of this term is close to that of the short- or long-wave train or their high harmonics.

For example, a nonlinear term,

, has phases of at second order in wave

steepness and 2 or 2 at third order. Thus,

the wavenumber and freq. of this term are respectively

and at second order, and

l s l s

s l s l

s l s l sk k k

2 and

2 or 2 and 2 at third order in wave

steepness. Since the PMM is applied to interactions

between two wave trains of disparate wavelength and

frequency scales, and , th

l

s l s l s l

s l s l

k

k k

k k

e wavenumber

and frequency of the above nonlinear term are close to

those of the short-wave train or its second harmonic.

Hence, it is assigned to the boundary condition for

the short-wave train.

Quite a few nonlinear terms in the free-surface B. Cs need to be further divided into sub-terms of long- and short-wave phases, respectively. They are denoted by

2 32 2

2

2 32

2 3

1 1+ , (7.6.4a)

2 2

1

2

. (7.6.4b)

s ls s s l s s

s lh s h s s s

h s h l h l h s s

MNDt z t z z

MNKz z

z

Further division of them into sub-terms involving long-wave or short-wave phases requires the knowledge of the solution for the first harmonic of short-wave train. The related sub-terms are vaguely denoted respectively by subscript ‘s’ or ‘l’ in Equations (7.6.2c), (7.6.2d), (7.6.3c) and (7.6.3d).

Although the details of these sub-terms have not been given yet, based on the order analysis it can be shown that the sub-terms of the SW phases are of 2nd order in wave steepness and those of the LW phases are of 3rd order at most.

An important conclusion: the effects of the LW train on the SW train are of 2nd order while the effects of the SW train on the LW train are of 3rd order at most.

Based on this conclusion, we may greatly simplify the computation in examining whether or not the solutions satisfy nonlinear B. Cs. For example, in computing the solutions truncated at 3rd order in wave steepness, the LW related variables or their derivatives in a nonlinear terms remains the same as the ones in the absence of the SW train. In other words, we only need 2nd-order solution for the LW train in computing the modulation of the SW train.

7.6.2 General Formulations for A Modulated Short Wave

The solution for the SW train obtained using conformal-mapping approach indicates a general solution for the 1st harmonic (free) SW potential and elevation,

(1)3

(1)33

3

( , , , ) ( , , , ), (7.6.5a)

(1 ( , , )) cos , (7.6.5b)

where

( , , , ) exp ( ( , , , ) , (7.6.6

s

s a

k

A V z x y t H x y t z

a f x y t

V z x y t k z f x y z t

3

3 3 3

3 3 3 3 3

a)

( , , , ) sin , (7.6.6b)

( , , , ), (7.6.6c)

, x y

H x y t z

k f x y z t

k x k y t

3 3 3

(7.6.6d)

( , , ) , (7.6.6e)lz

k f x y t

Examining under which conditions the 1st harmonic SW potential satisfies the Laplace equation.

1) For uni-directional long- and short-wave trains, the modulated potential satisfies the Laplace equation exactly, if

2 2 2 2

2 2 2 20 and 0, (7.6.7a)

and . (7.6.7b)

k k

k k

f f f f

x z x zf f f f

x z z x

2) For directional LW and SW trains, the modulated potential does not satisfy the Laplace equation exactly but satisfies the Laplace equation up to the truncated order at which it matches the corresponding MCM solution.

Examining whether or not the first-harmonic short-wave potential satisfies the bottom boundary condition.

fk varies slowly with respect to z, then

3exp ( ) 0, when .kk z f z

Hence, the short-wave potential satisfies the bottom boundary condition.

7.7 Solution for A Short Wave Modulated by A Long Wave

1 1

1 1

21 1 1 1 1

21 1 1 1 1

2 2 23 3 1 3 1 1 1 1 1

cos cos 2 , (7.7.1a)

sin sin 2 , (7.7.1b)

3 1( ) cos cos 2 ,

8 2

k z k zk

k z k z

a

f a e a e

f a e a e

f a k a k

1

32 2 2 2

3 3 3 3 1 1 3

(7.7.1c)

0, (7.71d) , (7.7.1e)

[1 2 ] (7.7.1f)

Average pot

f

gk a k a k k

33

3ential and elevation amplitude arerelated by

a gA

Exercises for the matching between the PMM & MCM solutions

and kf f explicitly describe the modulation effects on the SW potential in vertical and horizontal directions, respectively.

fa mainly depicts the modulation of the amplitude of the SW elevation.

The 1st term in (7.7.1c) results from the nonlinear effects of the SW train itself, same as in the case of a single Stokes wave train (i.e., in the absence of the LW train). The 2nd term is due to the nonlinear effects of the LW train. Both of them are independent of LW phase and of 3rd order in general wave steepness.

The 3rd and 4th terms respectively represent the modulation by 1st and 2nd harmonics of the LW and are of 2nd and 3rd order, respectively.

(2) 3 1 333 1 12

(2) 23 33 3 2 1 1

22 1 1

(3 2 )exp (2 )( ) sin(2 ), (7.7.2a)

2(1 )

1(1 )cos 2 (1+2 )cos(2 - ) , (7.7.2b)

2where

3(2 )1 2(2 )cos , (7.7.2c)

2 (1

s k

s a

a

Ak k z f

a k f

f

2. (7.7.2d)

)

2nd Harmonic short-wave potential and elevation

Modulated short-wave wavenumber and frequency

3 23 3 1 1 1 1

3 1 2 13 3 1 1 1 1

= (1 cos cos 2 ) (7.7.3a)

= (1 cos cos 2 ) (7.7.3b)

k kx

t

When k2 >2k1 and a1k3<<1, it can be proved by invoking the Taylor expansion that they are identical to the corresponding solutions derived using a MCM up to 3rd order.

Substituting the above solutions for the SW potentials and elevations into (7.6.2c) and (7.6.2d), it can be shown that the solutions satisfy the free-surface B. Cs for the SW.

We now examine the convergence of the PMM solution for potential and elevation.

1) the perturbed solutions decay with the increase in orders. E.g., the magnitude of 2nd-harmonic solution is at least one order high the 1st harmonic solution and so is 3rd-harmonic solution than the 2nd-harmonic solution.

2) the modulation of the amplitude of SW elevation is of 2nd order at most, which is hence much smaller than the SW amplitude.

3) Although the modulation in the phases of the potential and elevation is of order when , the order of the SW amplitudes remains unchanged regardless the large changes in their phases.

Hence, the PMM solutions are well behaved.

Long-wave solution

1 3 (1),a k O 3 1k k

(1) 2 2 21 1 3 3 1 1

2 2 2 21 1 1 1 3 1 3

3 1[1 ( )]cos , (7.7.4a)

8 4

[1 2 ] . (7.7.4b)

l a a k k

gk a k a k k

Matching Procedures

• Expanding the modulation function in perturbation series

• Converting the general formulation (modulation solution) in the form of the solutions consisting of linear vertical and horizontal functions (MCM solution) using Taylor expansion

• Matching with the corresponding conventional solution (MCM solution) from the low order to high order

• Examining if the PMM solution satisfies the Laplace, and B.Cs

• Examining if it is convergent when the corresponding MCM solution does not converge.

7.8 Solution for A Short Wave Modulated by Two Long Waves 

For the solution truncated at third order, the general solution for wave-wave interactions involves at most three distinct free waves. Hence, to study a short wave modulated by multiple long free waves, a general solution can be derived in the case of a short wave (denoted by subscript 3) interacting with a long-wave train consisting of two distinct long free waves (denoted by subscripts 1 and 2, respectively).

The general formulation for the first harmonic of the short wave modulated by the long-wave train remains the same as shown in Equations (7.6.5a) and (7.6.5b). For simplification, we assume all three free waves are in the same direction and the water depth is deep.

1 2

2 1 2 1

1 2

22 ( )

(1 2) 1 2

1( ) ( )

(2 1) 2 1 (2 1) 2 1

( )(1 2) 1 2

[ cos cos2 ] cos( + )

cos( ) cos( )

cos( + )

i ik z k z k k zk i i i i i

ik k z k k z

k k z

f a e a e C e

C e C e

C e

1 2

2 1 2 1

1 2

22 ( )

(1 2) 1 2

1( ) ( )

(2 1) 2 1 (2 1) 2 1

( )(1 2) 1 2

(7.8.1a)

[ sin sin 2 ] sin( + )

sin( ) sin( )

sin( + ) ,

i ik z k z k k zi i i i i

ik k z k k z

k k z

f a e a e C e

C e C e

C e

(7.8.1b)

22 2 2

3 3 31

(2 1) 2 1 (2 1) 2 1

(2 1) 2 1 (2 1) 2 1

3

3 1[ ( ) cos cos 2 ]

8 2

cos( + ) + cos( ) (7.8.1c)

sin( ) sin( ) , (7.8.1d)

, 1, 2.

a i i i i i i ii

ii

f a k a k

D D

f H H

i

22 2 2 2

3 3 3 3 31

33

3

(7.8.1e)

[1 2 ] (7.8.1f)

i i ii

gk a k k a k

a gA

As expected. the modulation by the LWs includes the superposition of the modulation by individual LWs, which is accounted by the summation notations. (cf. Sec. 7.7)

The modulation functions also involve the contribution from the BWs resulting from the interaction between two LWs, which is absent in the corresponding functions Sec. 7.7. The magnitude of the modulation by BWs is denoted by C, D and H. The subscripts are designed to reflect which types (sum- or difference-phase) of BWs result in these modulation terms.

The coefficients (C, D and H) were determined based on matching the PMM solution with the corresponding MCM solution up to 3rd order in general wave steepness.

(1 2) 1 2 1 2

2 2 1(2 1) 1 2 1 2

2 2 1 3 3

1(2 1) 1 2 1 1

1 2

( ), (7.8.2a)

1 , (7.8.2b)(1 )( )

(1 )(

C a a k k

k kC a a k

k k

C a a k

2 1

1 3 3

(1 2)

2(2 1) 1 2 1 2

1 , (7.8.2c))

0 , (7.8.2d)

1( ) ,

2

k k

k k

C

D a a k k

2 21 2 1/ 2 1/ 2 1/ 2

(2 1) 1 2 1 21 2

21 2 2 1

(7.8.2e)

( ) 1 1

(1 )(1 )

1 ( ) ,

2

D a a k k

a a k k

(7.8.2f)

1 2(2 1) 1 2 1 2 1 1/ 2

1 2

1(2 1) 1/ 2

2

(1 )= (1 ) , (7.8.2g)

(1 )(1 )

0, . (7.8.2h)

and satisfy the Laplace Eq. and Cauchy-Riemann

condition. Hence, thk

H a a k k

H

f f

e potential satisfies the Laplace Eq.

Although increases with , because of

is much faster than , the first-harmonic potential

still satisfies the bottom boundary condition

k

k

f z z

f

The F-S B. Cs for the first-harmonic short wave are:

(1) 3 (1) (1)(1) (1) 2 (1) (1)

2

(1) (1)(1) (1)

(1) 3 (1)(1) (1)

1( ) , at (7.8.3a)

2

1

2

s s se s s s s l

s s lh l h s h l h l s

h s sh s s

dg z

dt t z z

d

dt z z z

z

(1) 23

( ) , at (7.8.3b)

, at (7.8.3c) , at (7.8.3d)

where ----- the vertical velocity induced by the long waves,

----- the resultant v

s l

ll l e l

l

e

zz

dwdz g g z

dt t dt

w

g

ertical acceleration, known as the effective

gravitational acceleration at the long-wave surface.

nd

1) and are in the above F-S B Cs. It

is because they contribute only to the 2 -harmonic SW.

2) The terms at the right-hand side of the dynamic B. C. and the

last two terms in

s sMND MNK absent

rd

rd st

rd

the kinematic B.C. are of 3 order in wave

steepness. They result from the interaction of SW with itself.

Hence, they make contribution only to the 3 -order 1 harmonic

and/or 3 harmonic, and the nonlinear dispersion of SW.

3) As we discussed in Sect. 7.7, to have the accuracy of the solution for the 1st-harmonic SW up to 3rd order, the LW elevation and potential occurred in (7.8.3a) and (7.8.3b) need to be accurate up to 2nd order.

4) The solution satisfies the B.Cs.

• Modulation of the 2nd-harmonic SW by the BWs resulting from the interaction between two LWs is of 4th order. •Hence, the solution for the modulated 2nd-harmonic potential and elevation simply includes the superposition of the modulation by the 1st -harmonics of individual LWs and thus similar to that given in Section 7.7.

2(2)

33 3 321

2 22(2) 3 3

3 32 i1 1

22

(3 2 )exp (2 )( ) sin(2 ), (7.8.8a)

2(1 )

(1 )cos 2 (1+2 )cos(2 - ) , (7.8.8b)2

3(21 2(2 )cos , (7.8.8c)

2

i i is i k i

ii

s ai i ii i

i iai i i i i

A k k z f

a kf

f

2

). (7.8.8d)

(1 )i

Convergence of the solution of SWIn Chap. 5, it was shown that MCM solution for a SW interacting with two LW encounters two different types of convergence difficulties.

• The 1st type results from drastic difference in wavelengths of LW & SW, which is the same as described in Sect. 7.7. We find that the solution given by Eq.(7.8.1) remains convergent even if

3 2 3 and (1) for 1& 2.ik k a k O i

• The 2nd type occurs at 3rd order and does not present in the case of one SW interacting with one LW. It occurs when two LWs are of close wave frequencies/wavelengths, but not necessarily much longer than the SW in wavelength.

2 1 1 1( )O

Physical interpretation of the 2nd type of convergence difficulty.

•When the two LWs are close in frequency, the BW resulting from the difference-phase interaction between the two LWs has much longer wavelength than that of the short wave, even if individual LWs are not much longer in wavelength than the SW.

•The drastic difference in wavelength of the SW and difference-phase BW essentially causes the 2nd type of convergence difficulty in the MCM solution.

•In the PMM solution, the modulation by the difference-phase BW on the SW is explicitly considered and hence the solution is convergent,

(1)3

2 1

(2 1) (2 1)

((2 1)

For being significant, the range of is limited to (1),

then ( ) 1. Hence, the magnitude of and related

to and can be respectively approximated by,

:

s

k

kk

z k z O

k k z f f

C C

f C e

2 1 2 1

2 1 2 1

) ( )2 1 (2 1) 2 1

(2 1) (2 1) 2 1

( ) ( )(2 1) 2 1 (2 1) 2 1

(2 1)

cos( ) cos( )

[ ]cos( ),

: sin( ) sin( )

[

k z k k z

k k z k k z

C e

C C

f C e C e

C C

(2 1) 2 1

1 2 1 2(2 1) (2 1) 1 2 1 2 1

1 2

1 2 1 2(2 1) (2 1) 1 2 1 2 1

1 2 2 1

2 22 1

1 2

]sin( )

1 (1 )( )( ), (7.8.4a)

(1 )(1 )

( ) 2( )

(1 )(1 )( )

( )

(1 )(1 )

C C a a k O a

C C a a k

2 12 1

1( ). (7.8.4b)

( )O a

The approximations show that

•Small denominators in fk is cancelled and hence fk is convergent.

•Small denominators in is of second order if and can be even greater, if . Since is contained in the phase of the SW, the large magnitude resulting from the small denominators does not affect the amplitude of the 1st-harmonic SW.

•Since the modulated short-wave frequency and wavenumber are related to , we examine if they are convergent.

f 2 1 1( ) ( )O 2 1 1( ) f

f

23 2

3 31

21 2 1 2 1 2 2 1 (2 1) (2 1)

(2 1) 1 2 2 1 2 1

23

3 3 31

1

= {1 ( cos cos 2 )

1( ) cos( ) ( )

21

( ) cos( )}, (7.8.5a)2

= {1 ( cos cos 2 )

1

2

i i i ii

i i i i i i ii

k kx

a a k k k k C C

H a a k k

k a at

a a

2 3 1 2 1 2 1 2 3 2 1

2 1 (2 1) (2 1) 1 2 2 1 2 1

2 1 3 2 1 2 1

( )( ) cos( ) ( )

1[ ( )]cos( )}. (7.8.5b)

2Notice ( ) ( )( ),

the modulated wavenumber and frequency are convergent.

k k k k

C C H a a k k

k k k

1 2 2 1 2 1 2 1

2 23 3 1 1 2 1 1 1 2 1

Considering a case that the two distinct LWs reduce to a

single LW of amplitude, and , & ,

Eq.s (7.8.5a) and (7.8.5b) reduce to

= [1 ( )cos ( ) cos 2

a a k k

k k k a a k a a

special

21 2 1

12

3 3 3 1 2 1 1 1 1 2 1 1

11 2 1

1

1 4 ], (7.8.6a)

(1 )

= {1 [( ) cos ( ) cos 2

2 ]} . (7.8.(1 )

a a k

k a a k a a

a a k

6b)

Comparing the above eq.s with the corresponding wavenumber

and freq. given in Sect. 7.7 ((7.7.3a) & (7.7.3b)), they are

consistent except for a constant. The additional terms indicate

constant increases in average wavenumber and freq. of the SW.

•Increases in the average freq. and wavenumber should be considered in the nonlinear dispersion relation.

• The corresponding nonlinear dispersion relations shown in Eqs. (7.7.1f) & (7.8.1f) are hence compared.

•When the two LWs reduce to a single LW of amplitude, , to be consistent with (7.7.1f) the last term in (7.8.1f) should be

1 2a a

2 2 23 1 1 1 2 3 1 1 1 22 ( ) instead of 2 ( ).k k a a k k a a

The inconsistency can be reconciled when the two constants in Eqs (7.8.6a) & (7.8.6b) are included in the nonlinear dispersion relation.

Now including the constant increase in Freq. in the nonlinear dispersion relation. We redefine the average wavenumber

23 3 1 2 1

12

2 2 13 3 1 2 1

1

22 2 2 2 1

3 3 3 3 1 1 1 2 1 2 11

2 2 2 23 3 3 3 1 1 1 2

2 411 2 1 3 1 2 1 1

1 1

23 3

1[1 4 ]

(1 )

1 2(1 )

1 2 ( ) 1 2(1 )

1 2 ( )

1 4 4 ( )

(1 ) (1 )

k k a a k

a a k

gk a k k k a a a a k

gk a k k k a a

a a k k a a k O

gk

22 23 33 1 2 1 1[1 2( ) ] . a k a a k k

2 1 2 1 2 1

22 2 23 3 33 3 1 2 1 1

3 31 2 1 1 2 1 1 2 2 1

When 1 & 1, ( ) changes very slowly,

the quasi-average freq. of the SW, it is approximated by

1[1 ( )

2

2 cos( )] ( ( )) . (7.8.7)

k k

gk a k a a k k

a a k k O

Anti-Node

Node a

1+a

2

| a2 - a

1 |

Figure 7.5: A SW riding on a LW train consisting of two LWs with close freq.