An analytical model for turbulence scattered rays in the shadow zone for outdoor sound propagation calculation

In outdoor sound propagation, an inherent problem of the ray tracing method is its inability to determine the sound pressure level in the shadow zone, where geometrical rays do not penetrate. This is a serious problem in a turbulent atmosphere where significant sound energy will be scattered into the shadow. Empirical corrections that are determined from measurements or numerical simulations are limited to situations within the bounds of the empirical corrections. This paper describes a different approach where the ray tracing model is modified analytically into a scattered ray model. Rays are first diffracted from the shadow boundary, which is determined by the geometrical ray paths. The diffracted rays are then scattered by turbulence in their way to the receiver. The amount of scatter is determined from turbulence statistics that are determined from a Gaussian turbulence model. Most of the statistics are determined analytically except one element, which is determined empirically from numerical simulations. This turbulence scattered ray model is shown to have good accuracy against calculations based on the parabolic equation, and against previously published measurement data. It was found that the agreement is good both with and without turbulence, at distance up to 2 km from the shadow boundary.