Chapter 5 Solutions for Interacting

Waves Using A MCM

5.1 Governing Equations and Hierarchy Eq.s

5.2 An Example of Applying A Mode Coupling Method (MCM)

5.3 A General Third-Order Solution for Uni- Directional Irregular Waves

5.4 Validity and Convergence of Solutions of A MCM

5.1 Governing Eq.s and Hierarchy Eq.s Assumption: incompressible and irrotational flow, no wind and h=constan, the governing equations for the surface gravity waves are:

2

2

0 - (5.1.1)

0 (5.1.2)

1( ) at (5.1.3)

2

0 at h h

h z

at z hz

g C t zt

zz t

(5.1.4)

22

2

2

Applying ( ) on Eq. (5.13) and noticing (5.1.4), we eliminate

in (5.13) to have

10, at . (5.1.5)

2

t

g zt z t

Mode Coupling Methods (MCM), also known as Stokes Expansion, are applied to a wave filed in water depth considering as deep or intermediate with respect to its typical wavelength. They were widely used for the studies of 1) resonant quartet wave interaction, 2) modulation of a short wave by a long wave, 3) nonlinear correction on linear frequencies and 4) high-order wave loads on structures .

Using a MCM to solve Equations (5.1.1), (5.1.2) and (5.1.5) for the potential and then substitute it into (5.1.3) for the solution of wave elevation.

Perturbation Series:

1 2 3

1 2 3

1 2 3

(5.1.6)

(5.1.7)

(5.1.8)

j

j

jC C C C C

an idea small parameter in a wave field consisting of more than one free wave cannot be uniquely defined. It is vaguely refers to general wave steepnesses,

, and may or may not .i ja k i j

How to identify the orders of perturbed solutions related and forcing terms ? Noticing there is no power index attached to each term in perturbed series

In Chapter 4, the magnitude of wave-wave interactions was identified based on the power index of the free-wave amplitudes involved in the corresponding forcing terms.

Accordingly, the order of terms in perturbed series or forcing terms in the free-surface conditions is determined by counting the number of free-wave amplitudes involved in the corresponding solutions.

( ) ( )

, and invlove mulptiplication of number of

free-wave amplitudes.

Hence, a forcing term is of order ( ).

j j j

j ixz x

C j

i j

Convergence of Perturbed SeriesAll general wave steepnesses must be much smaller than unity. Under weakly nonlinear assumption

This requirement may be satisfied in a wave field of a relatively narrow banded spectrum. However, in the case of a wave field of a broad-banded spectrum one or several wave steepnesses may be close to or even greater than unity. Thus, certain second- and high-order solutions related to these wave steepnesses may not be much smaller then lower-order solutions. Under this circumstance, truncated MCM solutions converge slowly or even diverge.

1 for 1,.... , and 1,.... .i ja k i N j N

(1), if , even if 1.i j j i i ia k O k k a k

•Expanding the free-surface boundary conditions (5.1.5) and (5.13) at the still water level. •The perturbed series given in (5.1.6)-(5.1.8) are substituted into the expanded free-surface boundary conditions, the Laplace equation and bottom boundary condition. •The equations are sorted and grouped according to the order in general wave steepnesses.

2 ( )

21 1

2

0, 0 (5.1.9)

0, at (5.1.10)

, ,

j

j

j jj j j

h z

z hz

g Pzt

1 1 ( )

at 0, (5.1.11)

1, , at 0. (5.1.12)

jj j j j j

z

Q C zg t

1 1 (1)

21

12 (1)2 1

2

12 22 1 1

12 (1)3 2 1

2

0, 0, and 0,

, (5.13a)

1, (5.13b)

2

P Q C

P gz z tt

Qz t

P gz z zt

22 (2)

2

1 212 2 (1)2

12 2

212

21 (1) (1)

223 1 2 1 1

2

1 2

2

1 , (5.14a)

2

1

2

gzt

gz tz t

t z

Qz t

1 12 32

12

1. (5.14b)

2z t z t

5.2 An Example of Applying A MCM

•Applying a MCM to the above nonlinear wave equations, we give details of deriving the solutions truncated at second order for two interacting free waves in constant water depth h.

•Solutions are derived from low to high order.

•At leading order, the solutions for the potential and elevation of each free wave are independent and same as those linear solution. They are simply the superposition of the corresponding linear solutions for individual free waves.

1 2(1) 1 21 2

1 1 2 2(1)

1 1 2 2

cosh ( cosh (sin sin , (5.2.1)

cosh cosh

cos cos , (5.2.2)

k z h k z ha g a g

k h k h

a a

The dispersion relation between the wavenumber and frequency is also identical to linear solution.

2 tanh , for =1 and 2 (5.2.3) j j jgk k h j

•Solving for 2nd-order solution

Substituting (5.2.1) and (5.2.2) into (5.1.13a) and using (5.2.3),the forcing term for the second-order potential is equal to,

2 22

1

2 3 21 2 2 21 2 2 1 2 1

2

2 3 21 2 2 21 2 2 1 2 1

2

1

3sin 2

sinh 2

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

2 (1 )( 1) 1 ( 1) sin( ),2

where coth and /

j j jj

jj

j j

ga kP

k h

ga a k

ga a k

k h

2.

is of second order because all of its terms have the product of two free-wave amplitudes, consistent with our definition on second-order forcing terms. •Based on the phases involved in the forcing term, 2nd –order potential should have the following form.

(2)P

22 1(2) (2) (2)

2 11 2 1

2 1(2)2 1

2 1

cosh ( )( )cosh 2 ( )sin 2 sin( )

cosh ( )

cosh ( )( ) sin( ) (5.2.4)

cosh ( )

j j jj

k k h zA k h z A

k k h

k k h zA

k k h

Each term in the potential is proposed such that it satisfies the Laplace equation and the bottom boundary condition.

•Substituting (5.2.4) into (5.1.11), the amplitude of each term is obtained by matching the coefficients of the same phases on both side of the equation.

2(2)

4

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

2 22 1 2 1

3, for 1 and 2, (5.2.5a)

8 sinh

[2 (1 )( 1) ( 1) 1]. (5.2.5b)

2 (1 ) ( ) tanh ( )

j jj

j

aA j

k h

a aA

k k h

•Similarly, we substitute linear solution into (5.1.13b) to compute (2)Q

2 22 2(2) 1

1 1

21 2 21 2 2 1

2

21 2 21 2 2 1

2

1cos 2

2 sinh 2 2sinh 2

(1 ) ( 1) cos( )2

(1 ) ( 1) cos( )2

i i i ii i

j ji j

a k a kQ

k h k h

a a k

a a k

•Substituting the forcing term and (5.2.4) into (5.1.12) for j = 2, we obtain the solution for second-order elevation.

22(2)

31

(2) (2)2 1 2 1

22(2)

1

cosh2 cosh 2 cos 2

4sinh

cos( ) cos( ), (5.2.6)

, 2sinh 2

j j jj j

j j

j j

j j

a k k hk h

k h

a a

a kC

k h

(2)(2) 21 2 2 2

1 22

(2)(2) 21 2 2 2

1 22

(5.2.7)

where

(1 ) ( 1) (1 ),2

(1 ) ( 1) (1 ).2

a a k Aa

g

a a k Aa

g

Convergence

Because the solutions for intermediate-depth water are lengthy, to simplify the comparison we examine the corresponding solution in deep water.

2 1 1( ) and k k h k h

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

2(2) 2

1 2 2 1 2 11

1 2 2 1 2 1

sin( ), (5.2.9)

1 1cos 2 ( )cos( )

2 2

1 ( ) cos( ), (5.

2

k k z

j j jj

a a e

a k a a k k

a a k k

(2)

2.10)

0.C

The corresponding 2nd-order solution in deep water.

Figure 5.1 Amplitude spectrum of two interacting periodic waves of relatively close frequencies (truncated at 2nd order)

0 0.30

1

2

3

4

5

6

f ( Hz )

Am

plit

ud

e (

m )

f1 f

2

2f1

2f2

f1+ f

2

f2- f

1

Figure 5.2: Amplitude spectrum of two interacting periodic waves of quite different frequencies (truncated at 2nd order)

0 0.60

1

2

3

4

5

6

f ( Hz )

Am

plit

ud

e (

m )

f2- f

1

f1

f2

2f1

f1+ f

2

2f2

The amplitude ratio of the second-order (difference-frequency) potential to the free wave of shorter wavelength (short wave) at the still water level is .1 2a k

1 2

1 2

1 1

1 2 1 2

When 1, that is, per the dispersion relation, the

above ratio possibly (1) even if 1 for 1,2.

For example, if 0.12 and the wavelength ratio

/ 1/ 9 ( 1/ 3), 1.08 1.

j j

k k

a k O a k j

a k

k k a k

This example indicates slow convergence or divergence of the solution truncated at second order in the case of two free waves of drastically different wavelengths (frequencies).

• Similar conclusion can be drawn by comparing the elevation amplitudes of sum- and difference-frequency bound waves with that of the short free wave.

Reason for comparing the bound-wave amplitudes to that of short free wave instead of the long free wave:the wavenumber and frequency of those bound waves are close to those of the short wave.

2 1 2 2 1 2 2 1( ) and ( ) , when .k k k k k

In the frequency domain, these bound waves hence are located closer to the short free wave. In studying an irregular wave fieldof a broad-banded spectrum, those bound waves are in the same frequency range as the short free wave not the long free wave. Consequently, the above comparison with the short wave is justified.

5.3  A General Third-Order Solution for

Uni-Directional Irregular Waves Concept of general solutionsThe total potential and elevation of an irregular wave train involving multiple free waves can be obtained by appropriately superposing these general solutions over these free waves.

•1st-order general solution= linear solution, involves one freewave ---- single summation

•2nd -order general solution involving at most two distinct free waves----- double summation.

•3rd -order general solution involving at most three distinct free waves----- triple summation.

The derivation involves lengthy but straightforward algebraic manipulation and is similar to that of a second-order solution shown in Section 5.2.

For the simplicity our study is limited to three uni-directional free waves in deep water.

The total first-order solution is the superposition of linear solutions.

The wavenumber and frequency of a free wave are always positive.Those of second- and third-order bound waves are expressed in terms of their summation or difference, or both.

The total second-order solution for three interacting free waves are the superposition of the second-order general solution summing up all possible pairs among three free waves.

Four major steps in deriving the third-order solution.

1) Substituting the first- and second-order solutions given in (5.3.1) and (5.3.2) into (5.1.14a) and (5.1.14b) to calculate the forcing terms P(3) and Q(3).

2) Examining how many distinctive phases occurring in P(3) ;• formulating the same number of individual terms in the total third-order potential so that their phases match the corresponding phases in P(3);• each term in the third-order potential is constructed based on a separation variable method. That is, the vertical function (depending on the vertical coordinate, z) of each individual term is determined according to its phase function so that it satisfies the Laplace equation and bottom boundary condition.

3) Substituting the third-order potential and into (5.1.11) and matching the coefficients of individual terms in the third-order potential with those of the same phases in P(3) ,the solution for third-order potential is derived.

4) Substituting the solution for third-order potential and Q(3)

into (5.1.12) to derive the solution for the third-order elevation.•Similar to the derivation of the potential, formulating individual terms in the third-order solution for elevation to match the phases in Q(3) and the third-order potential;• matching coefficients of these individual terms with those of the same phases at both sides of the equation.

The third-order solution is given in handout.

Terms of Different Phases in P(3)

• Their phases have the phase of 1 free wave, same as the free wave three times of the free wave

• Involving the phases of 2 distinct free waves

• Involving the phases of 3 distinct free waves

sin for 1, 2,3j j

sin 3 for 1, 2,3j j

sin(2 ) if

sin(2 ) if 2 or 0,

j i

j i j i

j i

j i k k

3 2 1 3 2 1 3 2 1

3 2 1 3 2 1

sin ,sin ,sin

sin if or 0.k k k

5.4 Validity and Convergence of Solutions of A MCM

•Frequency Range for Convergence

•Convergence Difficulty Due To Drastically Different Frequencies

•Singularity Related To Resonance Interaction

• New Type of Convergence Difficulty At Third Order

•Summary

a. Frequency Range for Convergence

3 2 2 1 1 and (1)O

(1), regardless whether or not, .i ja k O i j

When their frequencies are not very close, the denominators in the above solutions (for example, see (5.3.6c) and (5.3.6d)) are not too small. The magnitude of the 2nd - and 3rd-order solutions decreases with the increase in perturbation orders, indicating rapid convergence of the solutions.

b. Convergence Difficulty Due To Drastically Different Frequencies

When 2 of 3 interacting waves are quite different in frequency (wavelengths are quite different), we already showed that 2nd-order solutions may not converge. At 3rd-order, the solutions may still encounter the same type of convergence difficulties resulting from the disparity in wavelengths of interacting free waves.

1 2 3 1 2 3

3 2 1 3

Let 0.08, 0.05, 0.81, 0.0625 ,

The amplitude ratio / 1.54

as shown in (5.3.6f)

k k k

a a

c. Singularity Related To Resonance Interaction

When the frequencies of all three free waves are very close

3 2 2 1 1 1 and ( )O

the denominators in the solution for certain bound waves, (e.g. see( 5.3.6d) and (5.3.6e)) become extremely small. The related bound wave is a free wave, because its freq. & wavenumber almost satisfy the dispersion relation, e.g.

3 2 1 3 2 1 and A A

4 3 2 1 4 3 2 1

2 2 2 2 24 4 2 1 3 1

, and . (5.4.1)

2( )( ) ( ) ( )i i i i

k k k k

gk O O

d. New Type of Convergence Difficulty At Third Order

2 of the 3 free waves are close in frequency and the 3rd free wave is neither close to and nor much greater than them in frequency.

These 3 waves can not form a quartet resonant interaction although free waves “1'' and “2'' can form a special quartet resonant interaction with other free waves but not with free wave “3''. Also the above scenario about the frequencies of 3 free wavesdoes not belong to that of disparate frequencies.

Examining the solution for (5.3.6g) & (5.3.6h),their amplitudes can be very large due to the small factor in the denominator. For example,

2 1 1 1 3 2 1( ) , (1)O O

3 2 1 3 2 1 and a a

2 1 1 1 3 2 1 3 2 1 3 2 3( ) , and ( )O a a O k a a