The effect of finite duration sources on modes and generalization of the d'Alembert solution

TL;DR

This work addresses how finite-duration sources modify mode content in dispersive linear wave equations with constant coefficients. It develops a comprehensive mode-based framework (the Method of Modes) to relate pre-source and post-source modal amplitudes, deriving explicit formulas in both wavenumber and spatial representations. The paper covers multiple regimes, including single-mode, two-mode symmetric systems, and the standard wave equation, and culminates in a generalization of the d'Alembert solution for arbitrary linear wave equations. The results provide a rigorous, operator-based description of source-induced mode coupling and yield exact-to-practice formulas for predicting wave evolution beyond the source interval, with potential applications across dispersive wave problems in physics and engineering.

Abstract

We investigate the evolution of dispersive waves governed by linear wave equations, where a finite duration source is applied. The resulting wave may be viewed as the superposition of modes before the source is turned on and after it is turned off. We consider the problem of relating the modes after the source term is turned off to the modes before the source term was turned on. We obtain explicit formulas in both the wavenumber and position representations. A number of special cases are considered. Using the methods presented, we obtain a generalization of the d'Alembert solution which applies to linear wave equations with constant coefficients.