The limiting amplitude principle is a well-known result connecting the solution of the Helmholtz equation with the large-time behavior of timedependent wave equations with a source term which is periodic in time. Motivated by numerical analysis of time-domain methods for stationary scattering problems in heterogeneous media, we quantify the solution convergence of such time-dependent wave equation towards the stationary solution, under some assumptions on the coefficients and the source term. We also generalise the formulation of the limiting amplitude principle to the one-dimensional setting where the classical statement of the principle is known to be violated.