A fast Green’s function method for one-way sound propagation in the atmosphere

Abstract

A Green’s function method is used to derive a fast, general algorithm for one-way wave propagation. The algorithm is applied to outdoor sound propagation. The general method is not limited to atmospheric sound propagation, however, and can be applied to other problems, such as sound propagation in the ocean and electromagnetic wave propagation. The new algorithm, called ‘‘GF-PE’’ (Green’s function method for the parabolic equation), reduces to the well-known Fourier split-step algorithm for the parabolic equation (PE) when no boundary conditions are imposed (e.g., at a ground surface). With the GF-PE, range steps many wavelengths long are possible, while with a PE algorithm based on a finite-difference range step, such as the Crank–Nicolson method, the range steps are typically limited to a fraction of a wavelength. Because of its longer range step, the new algorithm is 40–450 times faster than PE algorithms that use the Crank–Nicolson method. For outdoor sound propagation over a locally reacting ground surface, the computed GF-PE field is the sum of three terms: a direct wave, a specularly reflected wave, and a surface wave. With the new method, the air–ground impedance condition is treated exactly and results in an analytic expression for the surface wave contribution. Numerical results from the GF-PE model are presented and compared to exact calculations, fast-field program (FFP) calculations, and PE results computed with the Crank–Nicolson method. The GF-PE algorithm is shown to be accurate and approximately two orders of magnitude faster than a PE based on the Crank–Nicolson method. Hence, the new algorithm opens the door to some useful new computational capabilities such as real-time predictions on desktop computers, fast pulse calculations, and practical three-dimensional calculations.