Using the example of the plane motion of a membrane, an analytical method is developed for finding the approximate solution of the second‐order wave equation whose right‐hand side describes a pressure wave moving with decreasing velocity. It is assumed that the variation of the velocity includes the characteristic speed of wave propagation in the membrane. The method is elaborated for obtaining a solution which is valid in the early‐time region. It exploits the Laplace integral transform technique with appropriate approximate methods of inversion. The latter are improved here in comparison with those which have been previously developed in the analyses of similar problems, and can be easily generalized to approximate analyses of the early‐time stage of elastic waves which are caused in thin shells by a pressure wave.