Fast Prediction Of Scattered Sound Field Based On Revered Use Of Fourier Diffraction Theorem

When an incident sound wave impinges an object under water, it is reflected andscattered. Scattered sound pressure at different directions is related to geometry andmaterial of the object. The positive acoustic problem is to calculate directiondistribution of scattered field based on shape of the object and acoustic parameters ofthe surrounding medium. Scattering problem can be solved by rigorous analyticalmethod or numerical method. In fact, only several objects with regular geometry canobtain exact solutions of field by using analytical method. While scattering field fromobjects with irregular boundary and complex material can be calculated by finitedifference time domain method (FDTD), finite element method (FEM), boundaryelement method (BEM) and some other numerical methods. These methods divide thefield into grids, which mean heavy computation cost and time consuming. Reversalusage of Fourier diffraction theorem is a fast method to solve the forward acousticproblem without mesh division and iterative computation.Plane wave is incident perpendicular to the infinitely long cylinder submerged inwater. Two-dimensional sound field is described as a binary image based on shape ofthe object and acoustic parameters of the surrounding medium.Scattered field can bepredicted by sampling on spectral domain of image. In this dissertation, we study amethod of fast prediction of scattered sound field based on Fourier diffractiontheorem, and focus on the improvement of prediction accuracy and computationalefficiency. The contributions of this paper are summarized as follows:1. Rotation of the object on polar coordinate system by translation in therectangular grid-represented.Generalized projections in arbitrary directions are calculated by rotating theobject while maintaining the incident wave unchanged. Sufficient fine scattering directional pattern can be obtained by increasing sampling rate of the image orreducing the rotation angle interval. While computation cost and time consuming isincreasing. A fast rotation method is proposed in polar coordinate system, in this way,rotation of the object is achieved by converting multiplication of trigonometric intotranslation in the rectangular grid-represented. Using this method rapid rotation isfinished with improved accuracy and reduced computation cost. The proposed methodcan provide favorable conditions for fast prediction of scattered sound field.2. Numerical calculation of high precision frequency spectrumHigh-precision frequency spectrum is the first and the most important issues forcalculation of scattered field in reversal usage of Fourier diffraction theorem. Coarsesampling in space domain makes jagged edges of objects. There are great errors in thefrequency domain that obtained by direct2D-FFT, especially in high-frequency areathe samples are unavailable. It is difficult to get the correct prediction results byextracting samples along circular arc on2D-DFT transform spectrum directly. Fourierslice theorem says:1D-DFT transform of Radon projection in the spatial domainimage is in corresponding to samples along a slice on two-dimensional frequencydomain. Based on this principle, spectrum correction method is proposed. Radonprojections of the binary image at different viewing angles are extended byinterpolation, then smoothed and decimated. Several repeated operations are used toobtain smoothed projections. The corrected2D spectrum can be obtained byinterpolating1D-DFT of projects form polar coordinate to two-dimensional Cartesiancoordinate system. The improved spectrum is helpful for prediction of scattered soundfield with accurate and reliable results.3. Modified Born approximation to improve directional distribution of scatteredfield.When an incident sound wave impinges an object under water, directionaldistribution of scattered field is calculated based on diffraction CT. From0°to360°,in all direction around the objects, scattered pressure is proportional to samples alonga circular arc, the radius of the circle is equal to incident wave-number k0. First-orderBorn approximation is applicable when density and sound of the object is very close to that of the surrounding medium, namely weak scattering. But if impedance ratio isdepart away from1, there are substantial errors in sound scattering prediction basedon the first-order Born approximation. With the modified Born approximation, thedifference between wave number of the scattering object and that of the surroundingmedium is considered. Improvement of accuracy is expressed as modifying of theradius size and changing of center position. The far-field directional pattern of thescattered sound from a circular cylinder is in good agreement with the rigoroussolution, even when impendace ratio ranged from0.76~1.32. Numerical calculationsfor several objects with different shapes are used to show applicability andeffectiveness of the proposed method.4. Second-order Born approximation is introduced to expand the scope ofapplication of Fourier Diffraction TheoremIn order to improve accuracy and expand the scope of application of the method,the second-order correction term is introduced to the solution. The first-orderscattered wave is calculated by substituting incident wave for the whole pressure intointegration function. Then scattered wave is then used as the source of incrementalquantity or correction term into scattering integral equation. The correction term pluswith the scattered field calculate by first-order Born approximation to getsecond-order scattering results. Following the framework of reversal usage ofFourier diffraction theorem for fast prediction, the incremental values of scatteredfield based on are calculated by sampling in the frequency domain two times.Numerical calculations show that prediction accuracy is improved significantly, evenwhen impedances ratio of the object and the surrounding are in the range of0.72to1.44, the method is still applicable.