Sound field of a high speed airborne source in a horizontally stratified fluid medium above an impedance plane

The sound field for a stationary source in a variety of atmospheric conditions has been studied extensively in the past several decades with the advent of modern computing, but relatively little work has been performed for a source in motion. One commonly used model in outdoor sound propagation for a moving source is the heuristic model, which attempts to model a moving source as a collection of stationary sources along the moving source\u27s trajectory. To account for the relative motion between the source and receiver, the Dopplerized frequency is often used in the Weyl–Van der Pol formula along with a source strength correction factor. This heuristic modification gives an approximate sound field for a moving source in a homogeneous atmosphere above an impedance plane. However, the heuristic approach does not take into account fluid convection effects due to the continuous motion of the source so its results are questionable, especially at high source speeds. By considering a specialized geometry of a semi–infinite half–space with horizontal stratification and a moving source of constant velocity moving parallel to a planar impedance surface, a Lorentz–type transformation can be established. The Lorentz transform allows the moving source to be treated as a stationary source in the Lorentz frame, and provides a continuum model for the source\u27s motion. Applying traditional outdoor sound propagation techniques such as the Fast Field Program (FFP) allows the sound field to be computed in the transformed space as if it were stationary. The FFP implementation in this paper is based on a one–dimensional integral transform. Discretization of the horizontally stratified medium (refractive atmosphere) is accomplished via the bounded Green\u27s function in the wavenumber domain. The wavenumber domain boundary conditions are transformed into the Lorentz frame and a global matrix approach was used to compute the complex layer reflection coefficients. Extensions to a refractive atmosphere is accomplished by perturbing the vertically varying sound speed gradients to study its effects at a variety of source speeds. All sound field computations are performed on the analogous stationary source in the Lorentz frame, then an inverse Lorentz transform is performed on the spatial sound field in the Lorentz frame to map the results into a time history (in physical space). This proposed approach offers great computational efficiency, and allows preexisting algorithms for a stationary source to be applied to a moving source. The analysis in this paper is restricted to a scalar sound speed profile—wind effects are ignored in the present study. Due to the algorithmic considerations discussed in this paper, the sound field can only be computed along the plane perpendicular to the source\u27s motion (i.e., overflight conditions). The results show a substantial sound pressure level difference of up to 40 dB in the situation of strong atmospheric refraction and a source Mach number of 0.8. Even in the case of a weak sound speed gradient, the level differences can become quite substantial due to the amplification effects of a high Mach number source at horizontal receiver distances comparable to the source\u27s elevation. These results highlight the importance of incorporating a refractive atmosphere model into the far–field sound level predictions—especially at high source Mach numbers