Source directivity in the parabolic equation method using an inverse Fourier transform technique

The parabolic equation (PE) method is accurate for prediction of low-frequency noise in the situation that the starting field does not emit significant energy at large elevation angles. The typical PE sound source is assumed to be omnidirectional and represented with a Gaussian starting field, defined at all heights of the grid. A method of representing a source with arbitrary directivity has been developed for the unique form of the PE starting field. By extending the defined vertical wave number spectrum of the arbitrary far-field directional function of the source to include evanescent wave numbers, the approach uses array theory and Fourier transform techniques to define the starting field. The approach recognizes the elevation angle limitation of the PE method and adheres to the accepted wavelength-dependent vertical grid point spacing, satisfying the fundamental requirements of the PE method. Results for horizontal and vertical dipole sources in the free field and above a ground surface are presented and compared with analytical solutions within the valid PE elevation angle range. [This research was funded by the Federal Aviation Administration (FAA) Western-Pacific Region through U.S. Department of Transportation Volpe National Transportation Systems Center.]