Summary The paper presents some efficient algorithms based on the spectral Laguerre approximations of temporal derivatives. The key to the efficiency of these algorithms is to construct new trial functions, which lead to the systems with sparse matrices. Such trial functions represent the integrals of a combination of the Laguerre functions with the damped exponential. For the sake of exposition we consider propagation of waves in one spatial dimension for the first and the second order equations with respect to time. The resulting linear system with the right-hand side has a sparse matrix independent of number n the degree of the Laguerre polynomial, its right-hand side having the recurrent dependence on the parameter n. The preceding allows us to use fast met hods of solving the linear system with a great number of the right-hand sides. In this case the matrix is only once transformed. It is evident that the method is readily extended to 2D and 3D elastic wave propagation.