Uniform asymptotic expansions of an inverse-Laplace-transform integral with applications to problems of wave propagation

In this paper, we study an inverse-Laplace-transform integral whose phase function depends on an auxiliary parameter. When this parameter approaches a critical value, the saddle point of the phase function tends to infinity. As a consequence, the classical saddle-point method cannot be used to derive the asymptotic expansions of the integral. Here, we use some recently developed techniques of uniform asymptotic expansions to tackle this problem. Two asymptotic expansions are obtained, which are uniformly valid for the parameter in two different intervals, and rigorous mathematical proofs are provided. These mathematical results are then applied to two physical problems. The first problem concerns impact waves in a semi-infinite viscoelastic rod. Asymptotic expansions for the axial stress are constructed, which hold uniformly in two intervals behind the wavefront. In the first interval (immediately behind the wavefront), we find that the value of the axial stress is amplified from the wavefront value by a modified Bessel function. In the second interval (behind the first interval), the value of the axial stress is mainly amplified by an exponentially large factor. Our results extend those of Achenbach and Reddy obtained by the singular surface method. The second problem deals with an integral occurring in combustion. The uniform asymptotic expansions which we have obtained give a justification to a heuristic result in the literature. Furthermore, the detailed asymptotic structure of the wavefront is determined.