D’Alembert sums for a vibrating bar with viscous ends

We describe a new method for finding analytic solutions to some initial-boundary problems for partial differential equations with constant coefficients. The method is based on expanding the denominator of the Laplace-transformed Green’s function of the problem into a convergent geometric series. If the denominator is a linear combination of exponents with real powers, then one obtains a closed-form solution as a sum with a finite but time-dependent number of terms. We call it a d’Alembert sum. This representation is computationally most effective for small evolution times, but it remains valid even when the system of eigenmodes is incomplete and the eigenmode expansion is unavailable. Moreover, it simplifies in such cases. In vibration problems d’Alembert sums represent superpositions of original and partially reflected traveling waves. They generalize the d’Alembert type formulas for the wave equation and reduce to them when original waves can undergo only finitely many reflections in the entire course of evolution. The method is applied to vibrations of a bar with viscous ends and an internal damper. The results are illustrated by computer simulations and comparisons to modal and finite-element method (FEM) solutions.