Complexified Spherical Waves And Their Sources In The Physical Space

We address spherical waves complexified by a complex shift in a coordinate of the point source. These waves have been studied since the early 1970s in both time-harmonic and non-time-harmonic cases as exact localized solutions of the wave equation. We deal with the fundamental mode described by u = f(theta(*))/R-* where R-* = root x(2) + y(2) (z - ia)(2), a > 0 is a free. positive constant, theta(*) = R-* - ct is a complex phase and f(theta(*)) is an arbitrary function describing the waveform. Such a function satisfies the inhomogeneous wave equation u(xx)+u(yy)+u(zz) - c(-2)u(tt) = F with a certain source function F = F(x, y, z, t), which is a generalized function supported by a 2D surface in the real 3D physical space. Here, c > 0 is the constant wave speed. The function F is dependent on the waveform f as well as on the definition of the branch of the square root in the "complex distance" R-*. Unlike several earlier studies, in which sources in the complex space were discussed, we focus on explicitely finding the source function F in the real physical space for a rather arbitrary waveform f. For different definitions of the branch of the square root, we present an example of the waveform describing a Gaussian-localized wave packet.