Huygens Method of Construction of Weakly Nonlinear Wavefronts and Shockfronts with application to Hyperbolic Caustics

Abstract

This thesis embodies the results of investigations of the effect of nonlinearity on the propagation of weakly nonlinear and arbitrarily curved wavefronts and shockfronts. It is classified into six chapters, of which Chapter I is the introduction. Here, various results pertaining to the application of the linear theory in the high frequency limit to the problem of the propagation of a concave wavefront focusing into a gas at rest are presented. The formation of the caustic in the linear theory explains in a natural way the need for the application of the nonlinear theory for the problem. A brief summary of existing results emerging out of theoretical and experimental investigations on the nonlinear problem is also presented. This chapter also contains a short résumé of the author’s contributions and results based on the author’s investigations. In Chapter II, the kinematics of a weak nonlinear wave in multi dimensions is developed using high frequency approximation. A hyperbolic system of quasilinear partial differential equations describing the motion of a weakly nonlinear wavefront in space time and the distribution of the intensity on it is obtained. The resulting theory stands as an extension to the Huygens’ method of construction of a nonlinear wavefront in multi dimensions. Chapter III deals with the derivation of a nonlinear first order partial differential equation for the calculation of the successive positions of a weak shockfront governed by a general hyperbolic system of n conservation laws in m+1 independent variables. This partial differential equation, called the shock manifold equation, is shown to be useful for problems involving a multi dimensional weak shockfront when the flow ahead is a uniform state or at rest. Finally, a theoretical comparison of the theory developed here with the existing theory of Whitham’s shock dynamics is carried out to estimate the improvements achieved. Chapter IV deals with the application of the theory to the problem of the sudden introduction of a cylinder in a uniform compressible flow. The kinematics of the nonlinear wavefront is used to compute the nonlinear flow created due to the presence of the cylinder, following a single mode of propagation. The kinematics of a weak shockfront developed is then employed to plot the successive positions of the shock emanating from the cylinder. Chapter V gives some exact solutions of the nonlinear wavefront equations giving rise to simple waves on the nonlinear wavefront. A condition for the breakdown of these solutions, and in that case a critical time up to which these solutions are valid, is estimated. In Chapter VI, the question of resolution of a hyperbolic caustic due to nonlinearity is taken up in detail. The effect of various factors such as initial intensity, distribution of the intensity along the initial wavefront, curvature of the initial front, and the nonlinear interactions are studied. The behaviour of the solutions has been delineated for two types of initial parabolic configurations with varying intensity and distribution of intensity along the initial front. The effect due to nonlinearity in a focusing front is thus examined. Interesting interpretations and discussions on the results obtained are the highlights of the concluding pages of the thesis.