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A reaction–diffusion approximation is a method that solutions of multi-component reaction–diffusion systems approximate those of differential equations. We introduce the reaction–diffusion approximations of a semilinear wave equation and a semilinear damped wave equation under some assumptions of a reaction term. These approximation systems consist of a two-component reaction–diffusion system with a small parameter. In this paper, we prove that a first component of a solution for the system converges to a solution for the semilinear damped wave equation as the parameter tends to zero. Moreover, let us show the numerical results of reaction–diffusion approximation for the wave equation and the damped wave equation, respectively. [ABSTRACT FROM AUTHOR]