1. 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)

2. 7.1 Waves Advancing on Currents Waves on steady and uniform currents: Doppler‘s effect Waves on unsteady and non-uniform currents: Modulation Wave frequency, wavenumber vector and amplitude changes Peregrine (1976)

3. 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.

4. 7.2 Short Waves Modulated by Long Waves Velocity induced by a long-wave train looks like current to A short-wave train. It is unsteady and non-uniform. Figure 7.2: A short-wave train interacts with a long-wave train.

5. The velocity field induced by a long-wave train is converging at 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.

6. 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).

7. The modulated phase is not a linear function of time and horizontal Coordinates. For the validity of the approximation, the terms, such as

8. 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.

9. Limitations of revealing wave modulation based on MCM solutions. 1)The approximation in (7.3.1) is not valid if a 1 k 2 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).

10. 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:

11. 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 k 1 << k 2. PMM solution always converge, even if k 1 << k 2. 5. MCM solution: relatively simple PMM solution: relatively complicated

12. 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.

13. 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.

14. 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

16. Sketch of the Conformal Mapping

17. 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.

18. 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.

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

20. Solution for the long-wave train

21. 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).

26. 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.

28. 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 3 rd order while those of LW on SW are of 2 nd order.

29. 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 the PMM solution

30. 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

31. 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.

32. 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 k l z. It is understood that the change in k l 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.

33. 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.

34. 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, 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.

35. 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.

36. 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.

37. For example, a nonlinear term,

38. 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 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).

39. 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 2 nd order in wave steepness and those of the LW phases are of 3 rd order at most. An important conclusion: the effects of the LW train on the SW train are of 2 nd order while the effects of the SW train on the LW train are of 3 rd 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 3 rd 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 2 nd -order solution for the LW train in computing the modulation of the SW train.

40. 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 1 st harmonic (free) SW potential and elevation,

41. Examining under which conditions the 1 st 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) 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.

42. Examining whether or not the first-harmonic short-wave potential satisfies the bottom boundary condition. f k varies slowly with respect to z, then Hence, the short-wave potential satisfies the bottom boundary condition.

43. 7.7 Solution for A Short Wave Modulated by A Long Wave Exercises for the matching between the PMM & MCM solutions

44. explicitly describe the modulation effects on the SW potential in vertical and horizontal directions, respectively. f a mainly depicts the modulation of the amplitude of the SW elevation. The 1 st 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 2 nd term is due to the nonlinear effects of the LW train. Both of them are independent of LW phase and of 3 rd order in general wave steepness. The 3 rd and 4 th terms respectively represent the modulation by 1 st and 2 nd harmonics of the LW and are of 2 nd and 3 rd order, respectively.

45. 2 nd Harmonic short-wave potential and elevation Modulated short-wave wavenumber and frequency

46. When k 2 >2k 1 and a 1 k 3 <<1, it can be proved by invoking the Taylor expansion that they are identical to the corresponding solutions derived using a MCM up to 3 rd 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 2 nd -harmonic solution is at least one order high the 1 st harmonic solution and so is 3 rd -harmonic solution than the 2 nd -harmonic solution.

47. 2) the modulation of the amplitude of SW elevation is of 2 nd 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

48. 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.

49. 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.

52. 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 3 rd order in general wave steepness.

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

56. 3) As we discussed in Sect. 7.7, to have the accuracy of the solution for the 1 st -harmonic SW up to 3 rd order, the LW elevation and potential occurred in (7.8.3a) and (7.8.3b) need to be accurate up to 2 nd order. 4) The solution satisfies the B.Cs.

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

58. Convergence of the solution of SW In Chap. 5, it was shown that MCM solution for a SW interacting with two LW encounters two different types of convergence difficulties. The 1 st 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 The 2 nd type occurs at 3 rd 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.

59. Physical interpretation of the 2 nd 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 2 nd 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,

61. The approximations show that Small denominators in f k is cancelled and hence f k 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 1 st - harmonic SW. Since the modulated short-wave frequency and wavenumber are related to, we examine if they are convergent.

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

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

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