A unified solution for longitudinal wave propagation in an elastic rod

This paper proposes a unified analytical solution of the wave equation governing the propagation of the longitudinal stress wave in an elastic rod. Within a single formula derived by using the Laplace transform and inverse transform, the solution covers the contributions of the external excitations, the nonzero initial conditions and the inhomogeneous boundary conditions altogether, including such boundary conditions that the dependent variables at the ends of the rod are restricted with an equation. The proposed formula, particularly suitable for transient problems, could be regarded as an exact interpolation function in the time domain, provided the rod works as a component in a complex system, etc. Four examples are presented to show the applications of the solution.