Wave equation subject to a non-local integral condition: derivation of exact solutions

Purpose This study aims to derive explicit analytical solutions for a wave equation subject to a non-local integral conservation condition and provide benchmark solutions for the validation of computational methods. Design/methodology/approach Two analytical approaches are developed to obtain exact solutions. In the first approach, the governing equation is transformed into its characteristic form, where characteristic variables are determined along characteristic lines and integrated piecewise across subregions. In the second approach, the solution is constructed based on the exact solution in an infinite domain, with boundary and integral conservation conditions incorporated to supply the missing information outside the domain of interest. Findings Explicit analytical solutions are obtained for the wave equation under non-local integral constraints. The results reveal key structural features of the solution, including wave propagation characteristics and periodic behavior. These solutions can serve as benchmark references for verifying numerical simulations. Originality/value This study provides explicit and practically implementable analytical solutions for wave equations with non-local integral conditions. The results offer valuable benchmark solutions for computational modeling and numerical method verification in engineering applications.