A new method for predicting the far-field scattered pressure due to a plane wave incident upon an infinitely long cylinder of noncircular cross section is presented. The method, referred to herein as the Fourier matching method (FMM), involves conformally mapping the exterior and interior of a closed surface to a semi-infinite strip. This method is new in that the boundary conditions are satisfied using constraints described in the new angular variable. The resultant formulation is numerically efficient (much more so than the T-matrix method under certain conditions, for example) and works well for both small and large deviations from a circular cross section, as well as penetrable and impenetrable materials. Furthermore, the basis functions generated in the calculation can also improve the efficiency of other methods such as the T-matrix method. Example calculations are presented for elliptic, square, and three-leaf clover cross sections for several types of boundary conditions. In all cases, the results compare extremely well with exact or high-frequency asymptotic results.
In the analysis of a fluid loaded line-driven plate, the fields in the structure and the fluid are often expressed in terms of a Fourier transform. Once the boundary conditions are matched, the structural displacement can be expressed as an inverse transform, which can be evaluated using contour integration. The result is then a sum of propagating or decaying waves, each arising from poles in the complex plane, plus a branch cut integral. The branch cut is due to a square root in the transform of the acoustic impedance. The complex layer analysis (CLA) used here eliminates the branch cut singularity by approximating the square root with a rational function, causing the characteristic equation to become a polynomial in the transform variable. An approximate analytic solution to the characteristic equation is then found using a perturbation method. The result is four poles corresponding to the roots of the in vacuo plate, modified by the presence of the fluid, plus an infinity of poles located along the branch cut of the acoustic impedance. The solution is then found analytically using contour integration, with the integrand containing only simple poles.
The backscattering behavior of straight cylinders is examined whose lengths range from much less than the diameter of the first Fresnel zone of the source/receiver pair to much greater than the first Fresnel zone, with special emphasis on the complicated ‘‘transitional region,’’ where the cylinders occupy a finite number of Fresnel zones (≊1–5). In general, the scattering characteristics of cylinders in this region can only be described numerically. The scattering is described by first adapting the deformed cylinder formulation [T. K. Stanton, J. Acoust. Soc. Am. 86, 691–705 (1989)] to the point-source/point-receiver combination. Numerically evaluating this expression showed the scattering characteristics to be dominated by Fresnel zone effects—oscillations in the backscatter versus length curve caused by constructive and destructive wave interferences due to phase shifts from contributions along the cylinder axis. An experiment was performed that involved measurement of backscatter versus cylinder length in the transitional region, and there is reasonable agreement between the results and the trend as predicted by the approximate theory.
In the classic treatment of the line-driven, fluid-loaded, thin elastic plate, a branch cut integral typically needs to be evaluated. This branch cut arises due to a square root operator in the spectral form of the acoustic impedance. In a previous paper [J. Acoust. Soc. Am. 110, 3018 (2001)], DiPerna and Feit developed a methodology, complex layer analysis (CLA), to approximate this impedance. The resulting approximation was in the form of a rational function, although this was not explicitly stated. In this paper, a rational function approximation (RFA) to the acoustic impedance is derived. The advantage of the RFA as compared to the CLA approach is that a smaller number of terms are required. The accuracy of the RFA is examined both in the Fourier transform domain and the spatial domain. The RFA is then used to obtain a differential relationship between the pressure and velocity on the surface of the plate. Finally, using the RFA in conjunction with the equation of motion of the plate, an approximate expression for the Green's function for a line-driven plate is obtained in terms of a sum of propagating and evanescent waves. Comparisons of these results with the numerical inversion of the exact integral show reasonable agreement.
A novel technique is presented for obtaining approximate analytic expressions for an inhomogeneous line-driven plate. The equation of motion for the inhomogeneous plate is transformed, and the transform of the total displacement is written as a sum of the solution for a homogeneous line-driven plate plus a term due to the inhomogeneity. The result is an integral equation for the transform of the inhomogeneous contribution. This expression may in general be solved numerically. However, by introducing a small parameter into the problem, it may be solved approximately using perturbation techniques. This series may not be convergent, but its convergence may be improved using Pade approximation. Results are presented for the case of a single mass discontinuity, and a distribution of mass discontinuities.
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