We extended the pseudo-Laplacian to staggered grids based on the concept of normalized pseudo-Laplacian and applied it to constructing the pseudoanalytical formulations for the variable-density acoustic wave equation and the elastic wave equation. Acoustic wavefields only contain P-waves and therefore only P-wave pseudo-Laplacians are required for acoustic wave propagation. In comparison, two sets of staggered grid pseudo-Laplacians are needed in the elastic case in order to properly compensate for time stepping errors for both P-waves and S-waves. We gave a thorough derivation of the pseudoanalytical method for the elastic wave equation, based on normalized pseudo-Laplacians implemented on staggered grids, and presented the resulting complete discretized formulas. We proved that the staggered grid pseudo-Laplacian reduces to the pseudo-Laplacian for the scalar wave equation on standard grids. When using zero compensation velocities for normalized pseudo-Laplacians, the pseudoanalytical formulas simply reduce to the pseudospectral equations. We demonstrated with numerical examples that staggered grid pseudo-Laplacians effectively compensate for second-order time stepping errors and help generate highly accurate acoustic and elastic wave solutions in variable-density media.