Decompositions of solutions of the wave equation in terms of localized solutions from the wide class of the same equation are constructed by means of the affine Poincaré continuous wavelet analysis. Two particular decompositions are considered. One of them is represented in terms of nonstationary particle-like solutions, the Gaussian wave packets. The another one is a decomposition of a monochromatic field in terms of monochromatic Gaussian beams. A comparison with known results is given.