Spherical One-Way Wave Equation

Abstract

The coordinate-free one-way wave equation is transferred in spherical coordinates. Therefore it is necessary to achieve consistency between gradient, divergence and Laplace operators and to establish, beside the conventional radial Nabla operator ӘΦ/Әr, a new variant ӘrΦ/rӘr. The two Nabla operator variants differ in the near field term Φ/r whereas in the far field r > 0 there is asymptotic approximation. Surprisingly, the more complicated gradient ӘrΦ/rӘr results in unexpected simplifications for - and only for - spherical waves with the 1/r amplitude decrease. Thus the calculation always remains elementary without the wattless imaginary near fields, and the spherical Bessel functions are obsolete.