The NPE (nonlinear progressive wave equation) and associated computer program is a time-domain representation of acoustic propagation in waveguides that includes the effect of nonlinearity. In that approach a spatial window is initialized with the original waveform and then convected in the primary propagation direction as time evolves. Previous work modified NPE by using cylindrical coordinates to describe a paraxial approximation suitable for axisymmetric sound beams. The present development further modifies NPE by using spherical coordinates. The matter of interfacing it with the cylindrical coordinate version, in order to describe the far field of a sound beam, is described. This simulation technique is used to evaluate the long range propagation of the signal radiating from a piston in an infinite baffle that is subjected to harmonic excitation. It is also applied to a focussed sound beam generated by transient excitation of a concave projector. Comparison of the results with experimental data shows good overall agreement, with the main source of error apparently being due to dissipation, which is not addressed in the present models.

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