some nonlinear surface wave propagation problems

Abstract

This thesis contains the results of investigations of some nonlinear surface wave propagation problems. It is divided into six chapters, of which Chapter I is the general introduction and the concluding remarks constitute Chapter VI. In Chapter I, a general introduction to nonlinear wave propagation and the different methods used to tackle problems in this field are mentioned briefly. The significance of the methods of solution employed by the author has been brought out. The chapter concludes with a plan of the thesis. In Chapter II, the details of the singular perturbation method used to study the asymptotic behaviour of the nonlinear surface acoustic waves on an isotropic elastic solid are given. A uniformly valid solution in the interior of the medium is derived. Also, numerical solutions are obtained to study the growth-decay cycles of various harmonics. Perspective drawings to show the shock formation have also been done. Appendix B contains the study of two model equations with harmonic initial conditions. This is to simulate the velocity components and the displacement components in the nonlinear propagation of surface acoustic waves on an elastic solid. Chapter III deals with the systematic derivation of the infinite set of coupled amplitude equations governing the propagation of surface waves generated at the interface of an elastic membrane and a fluid medium. Here a modified Fourier series method is adopted to study the problem analytically. Numerical solutions for the membrane displacement and velocity components of the fluid medium have been obtained. Chapter IV contains the study of nonlinear Kelvin waves using the method adopted in Chapter II. The infinite set of coupled amplitude equations governing the propagation of nonlinear Kelvin waves with geostrophic approximation in an ocean of constant depth is obtained. Appendix D contains the results obtained for the propagation of nonlinear Kelvin waves without geostrophic approximation in a channel of finite width. Chapter V deals with the application of the modified Fourier series method to the problem of propagation of nonlinear Poincaré waves in a channel. The ultimate aim of obtaining the infinite set of coupled amplitude equations governing this type of surface wave propagation is achieved here also. The problems investigated show clearly that the methods employed in this thesis can be exploited fruitfully to study nonlinear surface wave propagation problems with cross-space boundary conditions. A computer program has been suitably developed to simplify the numerical study of these problems. The methods used give a good insight into the nonlinear propagation characteristics such as growth-decay cycles of amplitudes, asymptotic variation of velocity components and displacement components, and shock formation.