On Slowly Varying Stokes Waves

ON SLOWLY VARYING STOKES WAVES by VINCENT HWA-HING CHU In 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. It is believed that basic studies of this kind may ultimately be of importance to ocean engineering and oceanography as they permit a closer look at the significance of the classical Stokes waves which have led to an oversimplified picture of reality. Thesis Supervisor: Chiang C. Mei Title: Associate Professor of Civil Engineering