On slowly-varying Stokes waves

Abstract

ON SLOWLY VARYING STOKES WAVES by VINCENT HWA-HING CHUIn this thesis investigations are made on the theory of a train of slowly modulated gravity waves propagating over uneven bottom topography. The primary object is to study the interplay of amplitude dispersion the frequency dispersion in waves on the surface of water where the depth is not too shallow compared to a typical wave length.The solution of the wave train is expressed in expansions of the WKB type with a small parameter which is proportional to the wave steepness and the rate of modulation. A systematic perturbation scheme that may be carried out for all orders is presented. It is found that new terms directly representing modulation rate must be included to extend the scope of the Whitham's theory based on an averaged variational principle. Several specific examples are discussed and the following main conclusions are reached:1. The Stokes waves of constant amplitude are unstable for all depths under three dimensional disturbances. 2. For quasi-steady waves normally incident on a mild beach, the local rate of depth variation is found to affect the wave phase which in turn gives additional dispersion effect as compared to Stokes waves of constant amplitude. 3. In deep water, there exist solutions of permanent wave envelopes which represent exact balance between amplitude dispersion and frequency dispersion. 4. Numerical computation on the transient development of the wave envelopes reveals that any wave group deviated from permanent form will eventually disintegrate into a sequence of peaks separated by nodal points (where amplitude is zero) and locally the wave group tends to approach the dynamically stable permanent form.

Nonlinear Interaction of a Narrow Surface Spectrum with a Prescribed Surface Current

“…The observed properties of these wave systems seem to be consistent with theoretical expectations [e.g. see, Chu andMei (1970, (1971) soliton solutions for an initially narrow spectrum as discussed by Kadomtsev and Karpman (1971) and Zakharov and Shabat (1972).…”

“…The analysis of Chu and Mei, (1970) however, compares the evolution of the pulse modeled as a soliton with the experimental results. As they pointed out the dominant effect is the attenuation in amplitude due to the short time of evolution; i.e., short compared with the e-folding time which is given in Table I as 11 sec.…”

Extended time-dependent mild-slope and wave-action equations for wave–bottom and wave–current interactions

“…Moreover, application of the results to vortical currents might also prove to be plausible (see discussion in §6). In order to write separate equations for the oscillatory flow and the mean flow, the free-surface elevation and the velocity potential are divided into two parts (see Chu & Mei 1970)-a current and a wave part. The mean current is assumed to be varying slowly on wavelength scale, denoted by mx and my, and the oscillatory part is restricted to small wave steepnesses (3).…”

Note on wavefront dislocation in surface water waves

“…A combination 1 Some authors call this quotient the 'Fornberg-Whitham term' [27], referring to [28]. However, throughout this Letter we call it the 'Chu-Mei quotient', since they introduced it for the first time [29,30] when they derived the modulation equations of Whitham's theory [31] for slowly varying Stokes waves. However, the quotient already appeared earlier in [32,33] when they consider modulated waves in nonlinear media.…”