Wave propagation in dissipative materials

Abstract

The laws of wave propagation cast light upon the nature of material response by analysis. They show how the material reacts, locally and instantaneously, to a small change or impulse in a tiny region. To compose the small effects into a motion of the body as a whole is a much harder problem even in the simplest of theories, a problem generally too hard to solve, yet we gain insight and assurance by taking the preliminary step even though we rarely follow through. The term “small” has two distinct meanings: that the disturbance itself is small, and that it is confined to a small region. There are several different approaches to wave motion, resting upon different concepts of smallness. In the commonest of these, the differential equations of motion are shorn of their non-linear terms so as to yield a linear system which may be visualized as an assembly of harmonic oscillators, whose motions may be described in the terms hallowed by centuries of contemplation of the pendulum: frequency, amplitude, wave length, phase shift. It was to this method that HUGONIOT referred when, in 1885, he wrote: “… hypotheses have been imposed upon the equations of hydrodynamics which are disguised, it is true, by the word of approximation, but which singularly alter the value of such results as can be deduced from them.” Hugoniot himself developed in fairly general terms a different concept of wave propagation in which the disturbance is limited, rigorously, to a region of no volume at all, namely, a surface, but the disturbance itself may be of any amount.