Simple explicit solutions of the linear wave equation in three dimensions are presented which describe wave packets exponentially localized near a point moving with the wave speed. For large values of a certain free parameter these new solutions are localized in Gaussian manner with respect to longitudinal and transverse variables and time. This agrees with considerations by Babich–Ulin and Ralston who have presented an asymptotic description of solutions exhibiting such local behavior. Global estimates and large-time asymptotics of these solutions are given.

An exact solution of the homogeneous wave equation, which was found previously, is treated from the point of view of continuous wavelet analysis (CWA). If time is a fixed parameter, the solution represents a new multidimensional mother wavelet for the CWA. Both the wavelet and its Fourier transform are given by explicit formulas and are exponentially localized. The wavelet is directional. The widths of the wavelet and the uncertainty relation are investigated numerically. If a certain parameter is large, the wavelet behaves asymptotically as the Morlet wavelet. The solution is a new physical wavelet in the definition of Kaiser, it may be interpreted as a sum of two parts: an advanced and a retarded part, both being fields of a pulsed point source moving at a speed of wave propagation along a straight line in complex space-time.

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