On the local and global behavior of acceleration waves

Abstract

where t denotes the time. Formula (1.1) is the well-known Bernoulli equation. In general, the coefficients/~ and fl are functions of t. A careful examination of these coefficients reveals that there are two common features. Briefly, (1) the coefficient # depends on the material under consideration and the consequences of the physical assumptions introduced regarding the conditions of the material ahead of the waves, (2) the coefficient fl depends on the elastic response of the material alone. In other words, the coefficient # depends on whether we are considering acceleration waves propagating in, e.g., materials with memory, elastic nonconductors of heat, or inhomogeneous elastic materials, and whether we assume that the material regions ahead of the waves are at rest in homogeneous configurations, in mechanical equilibrium, or in dynamic equilibrium. On the other hand, the coefficient fl depends on the instantaneous elastic response of materials with memory, the isentropic elastic response of elastic non-conductors of heat, or the local elastic response of inhomogeneous elastic materials. In this paper, we shall show that knowing the sign of fl(t) not only allows us to determine the local behavior of amplitudes f rom (1.1) but it also tells us a great deal about their global behavior. Some of our conclusions on the global behavior would not be expected f rom the local behavior alone.