1. Chapter 6 Wave Instability and Resonant Wave Interactions in A Narrow-Banded Wave Train

2. Narrow –Banded Wave Train Definition: The frequencies of all free waves in a wave train are close to its spectral peak frequency and almost travel in the same direction. Sketch of a narrow-banded spectrum

3. Order Analysis of Narrow-Band Waves

4. The superposed elevation & potentialin terms of a carrier wave train with slowly varying amplitude

5. 2 nd –order solution expressed in terms of a carrier wave train with slowly varying amplitude The derivatives of ‘zero’ harmonic potential & elevation w.r.t. time, space coordinates are at most of third order.

6. Constraints on A T Laplace Equation Linear free-surface dynamic B.C Based on the definition

7. Derivation of Nonlinear Schrodinger Equation for A Narrow-Banded Wave Train Many different methods (see notes) MCM is used here.

8. Limited to the first harmonic terms (L.H.S) The 1 st & 3 rd terms are of 2 nd order & 2 nd & 4 th Terms are of 3 rd order

9. Limited to the first harmonic terms (R.H.S) 1) is of 3 rd -order but only contribute to zero-harmonic. It has no contribution to the first harmonic up to 3 rd. 2) The 1 st & 3 rd terms in are of 4 th -order at most. 3) The derivatives of 2 nd -order potential with respect to the space coordinates & time are of 3 rd -order the 2 nd & 4 th terms in are of 4 th -order at most. 4) The 5 th term in is calculated below.

10. Nonlinear Schrodinger Equation

11. Steady Solution Of NSE A Periodic Wave Train Nonlinear dispersion relation

12. Solution for Envelope Soliton (uni-directional wave)

13. Snapshot of the elevation of an envelope soliton in deep water at t = 0. The carrier wave’s period and amplitude are 2s and 0.1m, respectively

14. Elevation of the same envelope soliton at x = 410 m as a function of time

15. Solution for Conoidal Envelope Envelope soliton is a special case of Conoidal Envelope See hand-written notes

16. A snapshot of a wave train with a Cnoidal envelope in deep water (Emax=1.0 m 2, Emin=0.1 m 2, Tp=10sec).

17. Side-Band Instability Initial Instability Superposing infinitesimal disturbances on a steady periodic wave train.

18. To have a non-trivia solution for the system, the determinant of the matrix must be zero, leading to the solution for

19. Side-band (Benjamin-Fier) Instability Growth Rate (Yuen & Lake 1982) Ky = 0 K = Kx a o = a p, k o = k p

20. Long-Term evolution ZEM MCM (see notes for the details) Numerical simulation

21. Measured wave spectrum of a wave train experienced the side-band instability.

22. a)at x = 5 ft b)at x = 10 ft c)at x = 15 ft d)at x = 20 ft e)at x = 25 ft f)at x = 30 ft Time aeries at different locations

23. Coupling Eq.s derived using MCM Identify the forcing terms which may force or nonlinear interact related ‘free’ waves. (Resonance conditions) Quartet Resonance Interaction - Coupled Equations - Phase governing equations - Long-term evolutions (energy conservation)

24. MCM Then we consider four- (quartet)-wave interaction among free waves ‘0’, ‘1’,’2’ and ‘3’. The P (3) for 4 free waves can be extended from the general solution of above P (3). How?

25. The related forcing terms are now applied to free wave ‘0’.

26. The free-surface B.C. for free wave ‘0’ are split to

28. Five Coupling Eq.

29. Special case: Side-Band Instability

30. Resonant Wave Interactions in Ocean Waves Quartet wave interaction WAM----- Wave Energy Budget