Wavelet Analysis in Solving the Cauchy Problem for the Wave Equation in Three-Dimensional Space

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.

“…For example, the decomposition of the solution of the initial value problem for the wave equation in terms of wavelets has been proposed by us in [20].…”

Section: Introductionmentioning

confidence: 99%

New physical wavelet ‘Gaussian wave packet’

Perel¹,

Sidorenko²

2007

“…(60) Here we used the isometry property (25) for w(r) and β(r). Next we consider the time derivative at the moment of time t = 0.…”

Section: Justification Of Resultsmentioning

confidence: 99%

Wavelet-based integral representation for solutions of the wave equation

Perel¹,

Sidorenko²

2009

An integral representation of solutions of the wave equation as a superposition of other solutions of this equation is built. The solutions from a wide class can be used as building blocks for the representation. Considerations are based on mathematical techniques of continuous wavelet analysis. The formulas obtained are justified from the point of view of distribution theory. A comparison of the results with those by G. Kaiser is carried out. Methods of obtaining physical wavelets are discussed.

“…The family of wavelets is constructed by formulas (5). The affine Poincaré wavelet transform F j (µ) of the function f j is defined as follows [1]:…”

Section: Basics Of Affine Poincaré Wavelet Analysismentioning

confidence: 99%

“…The domain of integration is determined by the numerical support of the coefficient in decomposition (25) , which is the wavelet transform of boundary data. The wavelet transform is non-negligible if the supports of a signal (the domain where the signal is non-zero) and of a wavelet from the family (5) have an intersection. The supports of Fourier transforms of a signal and a wavelet should also have an intersection.…”

Section: Numerical Calculations Of Coefficients In the Wavelet Decompmentioning

confidence: 99%

Integral representations of solutions of the wave equation based on relativistic wavelets

Perel¹,

Gorodnitskiy²

2012

A representation of solutions of the wave equation with two spatial coordinates in terms of localized elementary ones is presented. Elementary solutions are constructed from four solutions with the help of transformations of the affine Poincaré group, i.e., with the help of translations, dilations in space and time and Lorentz transformations. The representation can be interpreted in terms of the initial-boundary value problem for the wave equation in a half-plane. It gives the solution as an integral representation of two types of solutions: propagating localized solutions running away from the boundary under different angles and packet-like surface waves running along the boundary and exponentially decreasing away from the boundary. Properties of elementary solutions are discussed. A numerical investigation of coefficients of the decomposition is carried out. An example of the field created by sources moving along a line with different speeds is considered, and the dependence of coefficients on speeds of sources is discussed.