Acceleration wave propagation in an inhomogeneous heat conducting elastic rod of slowly varying cross-section

Acceleration wave propagation in an inhomogeneous, heat-conducting elastic rod of slowly varying cross-sectional area is treated as a problem in one-dimensional wave propagation. The wave speed is found to be independent of the varying cross section. The equation governing the growth of the wave amplitude is found to be of the Bernoulli type. The varying cross section affects only the coefficient of the linear term. The term contributed by the varying cross section is exactly the same as that contributed by the varying area of a ray tube due to geometric spreading in the theory of three-dimensional acceleration wave propagation. In a sense to be made precise later, a cross-sectional area that is increasing as the wave propagates renders an acceleration wave less likely to build up into a shock after a finite distance of propagation, while a decreasing cross-sectional area renders an acceleration wave more likely to build up into a shock. The inclusion of thermal effects leads to a damping of the waves independently of the effect of varying cross section.