Three dimensional analyses of scattering by pressure-release sinusoidal surfaces

Abstract

Scattering by pressure-release sinusoidal surfaces in three dimensions is analyzed using the Fresnel phase approximation and realistic source and receiver directivity approximations. Geometrical shadowing and second-order scattering are explicitly included to explore the validity of the Kirchhoff approximation. No restrictions on the surface heights and slopes are made. The "goodness" of the resulting expressions is verified by requiring the pressure scattered by a sinusoidal surface to reduce to the image solution as the surface amplitude goes to zero. The first-order scattered pressure achieves a very good approximation to the image solution, and the second-order scattered pressure goes to zero, as expected, under this requirement. The theory is compared with available experimental scattering measurements, and the agreement is good. Because the slopes on the experimental surface are very steep, shadowing corrections are indispensible to achieving accurate first and second order scattering results. Shadowing has a greater impact on the scattering prediction than the second-order scattering contribution. This suggests that the Kirchhoff approximation may be more robust when incorporated into a theory including a detailed shadowing treatment as well as the Fresnel phase approximation and a good directivity approximation.

Cross Correlation of Omnidirectional, Broadband Signals Scattered by a Random Pressure-Release Surface

“…The resulting Fresnel expansions are given by (23) and (24) Substituting (23) and (24) into (7), and making the coordinate transformations given in (15) yields (25) Performing the and integrals gives a pair of Dirac delta functions (26) that allow the and integrals to be trivially evaluated to yield (27) In the narrowband case, the magnitude of the complex amplitude is given by , and when the time delay between hydrophones is steered to the image source the horizontal coherence in (27) reduces to (28) where . When the surface roughness goes to zero, both (20) and (28) reduce to the image solution [34], [35] for the acoustic intensity scattered by an omnidirectional CW source in the specular direction (29) This demonstrates the validity of the Fresnel phase approximation for an omnidirectional CW source; this is significant since the simplicity of the coherence expressions achieved in this paper relied on the use of the Fresnel phase approximation with the origin at the specular point. Finally, it also confirms that for an omnidirectional CW source the angles everywhere on the surface can be represented by their value at the origin without any loss in accuracy.…”

“…Here it will be assumed that the Kirchhoff approximation is valid when the surface slopes are not too steep. A recent study [35] showed that even for very steep slopes multiple scattering is insignificant at low to moderate grazing angles (below 60 ), and contributes to the scattered field only at very high grazing angles. At normal incidence the field due to multiple scattering was still 5-10 dB below the field due to single scattering.…”

“…The Kirchhoff approximation [40]- [52] has been widely studied, and was used in all of the coherence studies [1]- [20] and most of the scattering studies [21], [23]- [31], [35], [37], [38] cited in the Introduction. Here it will be assumed that the Kirchhoff approximation is valid when the surface slopes are not too steep.…”

“…This paper focuses on the spatial variation of the coherence of the acoustic field scattered Manuscript by pressure-release random surface that is assumed to be motionless. The Fresnel phase approximation was used in a number of the papers [4]- [6], [10], [18]- [35] cited above and is required to achieve the image solution [25], [34]- [39] when the roughness of the scattering surface goes to zero. In a series of papers, McDonald and Spindel [4], McDonald and Schutheiss [5], and Tuteur [7] used the Fresnel approximation centered on the origin defined by the specular point to obtain expressions for the temporal coherence of omnidirectional acoustic waves scattered by random surfaces.…”

Time-domain Helmholtz-Kirchhoff integral for surface scattering in a refractive medium

“…Our approach will be based on ray theory incorporated into the H-K integral. The benefit of this ray-based formulation is that it enables to correct geometric shadowing effects, 12 which are normally neglected in the Kirchhoff approximation along with multiple scattering effects.…”