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 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.

Wavelet-based integral representation for solutions of the wave equation

“…It has infinitely many zero moments with respect to spatial coordinates. As is shown in [22], its asymptotics coincides with the Morlet wavelet [9,10] as p → ∞ and the time t is fixed:…”

“…We also consider exponentially localized physical wavelets found and generalized in [19,20,21], wavelet properties of which has been studied in [22].…”

“…We also restrict the class of admissible proxy wavelet to the class L 1 (R) L 2 (R). A special case of solutions of the class (81) named the Gaussian wave packet was found in [19,20] and studied in [22]:…”

The Poincar´e wavelet transform: Implementation and interpretation

“…The The construction of the wavelets Ψ j is not so obvious, and we briefly discuss the procedure. The starting point for the wavelet construction is the 'Gaussian packet' [8,9]. We construct wavelets in the Fourier domain, because all calculations are performed in the Fourier domain by convolution.…”

“…It is easy to see that the function (18) does not satisfy the admissibility condition (8), but the second derivative of it does. Just this derivative with a small modification is taken as a mother wavelet in the domains D 1,2 :…”