Large amplitude waves in bounded media I. Reflexion and transmission of large amplitude shockless pulses at an interface

Abstract

Techniques which can be used to analyse the interaction of large amplitude elastic waves in a bounded medium are described. Although presented in the context of uniaxial stretching deformations in an elastic string or bar, these techniques can be used to analyse the behaviour of any system whose response is described by the nonlinear one-dimensional wave equation. In this first paper the bounded medium is contained between two parallel planes which separate it from other similar media. These are of semi-infinite extent along the axis of propagation which is normal to the interfaces. The paper is in two parts. In the first part the reflexion and transmission of an incident pulse when it arrives at an interface with a semi-infinite medium is described and the ideas of nonlinear impedance, reflexion coefficient and transmission coefficient are introduced. The results are quite general: no special forms for the stress-strain relations of either elastic materials is assumed. The results for a single interface are used to analyse the decay of a pulse as it moves back and forth between two interfaces. This decay occurs because at each contact with the interface energy is radiated across the interface to the surrounding medium. The algorithm s obtained have simple graphical interpretations. The general theory is used to discuss the decay of a pulse in a layer of saturated soil which is bounded from above by sea water and from below by rock. This pulse is triggered by a seismic disturbance deep inside the rock. The theory is also used to analyse the decay in the oscillation which occurs in a shock tube w hen a diaphragm separating air at high pressure from air at atmospheric pressure is ruptured. The bound gas is contained between the closed end of the tube and the contact discontinuity which is generated w hen the diaphragm bursts. In the second part of the paper a more detailed account is given of what happens when a pulse is partially reflected and partially transmitted at an interface. This is achieved by noting that the responses of m any elastic materials can be correlated, both qualitatively and quantitatively, by a family of stress-strain laws for which the governing nonlinear equations for this problem can be solved exactly. These laws are sufficiently general to locally curve fit any prescribed stress-strain law to an error 0 ([stra in ]4) in some vicinity of the unstrained state. They can also be used to fit the response of a polytropic gas during isentropic flow to within an error of 1 % as the density changes by a factor of ten! The reflexion of a large amplitude pulse from rigid and perfectly free interfaces is given special emphasis as is the reflexion from an interface with a Hookean material.