Mathematical theory of the propagation of multidimensional shock and nonlinear waves

Abstract

This thesis embodies the results of our investigation on a mathematical discussion of hyperbolic nonlinear waves under the high frequency approximation. It is divided into four chapters, of which Chapter I is the introduction to the subject. Here, relevant results of the linear and nonlinear theory are presented in detail. The unacceptable predictions of the linear theory-such as the amplitude becoming infinite on certain surfaces called caustics-motivate, in a natural way, the application of the nonlinear theory. A brief summary of Whitham’s shock dynamics and Maslov’s work on systems of conservation laws is presented here. This chapter also contains a short report of the author’s contribution and the results emerging from the author’s investigations. Chapter II starts with the derivation of the usual Rankine–Hugoniot conditions (which lead to the shock manifold equation and a definition of shock rays) for a multidimensional shock propagating into a polytropic gas medium, using the theory given by Maslov. The highlights of this chapter are the derivation of two exact first order quasilinear equations (along shock rays) for the amplitude of the shock and for the inclination of the normal to the shock in the case of two space dimensions for a shock of arbitrary strength. A small correction in a lemma-which plays a major role in Maslov’s theory-is also discussed. One more compatibility condition along shock rays has been derived for the jump in the second derivative. In Chapter III, an initial value problem for a single conservation law in one space variable is studied, which involves a shock. This example clearly shows that the nonlinear waves behind the shock significantly affect the successive positions of the shock front, and these effects are not properly accounted for in Whitham’s shock dynamics. Highly convinced and motivated by this example, a comparison has been made between the exact equation for the shock and the corresponding equations of Whitham’s shock dynamics. These two equations, which themselves form a hyperbolic system, are analysed further for the appearance of “shock shocks”. Appropriate conservation laws for these equations are derived. By using a classical method, which is much simpler than that of Maslov, exact equations for the amplitude of a shock (in one space dimension) are derived in two cases-one through a stratified layer and the other down a non uniform tube. Finally, we have examined Maslov’s perturbation theory, which gives estimates in a procedure to find the successive positions of a multidimensional shock and the amplitude variation along it. In Chapter IV, the question of approximation of a hyperbolic system of quasilinear first order equations in the neighbourhood of a characteristic manifold is dealt with. A model equation is derived under the assumption that the disturbance is non zero only in a small neighbourhood behind the wavefront, but is of arbitrary amplitude. As an example, the equations representing the motion of an inviscid, compressible gas are taken. This, for a nonlinear wave propagating in two space dimensions, leads to a pair of first order coupled equations. These two equations, completely kinematic as a nonlinear wave, are studied further. A simple wave solution of these equations (which are also reducible) may develop a discontinuity, leading naturally to the concept of wavefront shocks. Again, conservation form of the corresponding equations is derived. Figures representing the propagation of nonlinear wavefronts of various shapes and amplitude distributions are given and discussed. An equation for the amplitude variation along a ray tube is also derived.