New physical wavelet ‘Gaussian wave packet’

Abstract

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 formulae 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 spacetime.