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 ∂Φ/∂r, a new variant ∂rΦ/r∂r. 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 ∂rΦ/r∂r 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.

Keywords

• spherical wave
• one-way wave equation
• two-way wave equation
• spherical coordinates
• Nabla operators: gradient, divergence, Laplace
• impulse flow equilibrium