本文主要是利用數值方法求解二維薄物體的聲學散射問題，並考慮到Dirichlet和Neumann的條件。適用於薄物體的積分方程式中含有弱奇異點和超奇異點的問題。因此本文主要是處理這些奇異點的問題，並採用基本的減加技巧。
合併邊界表面積分方程式之參數的效果是雙重的。
首先是幫助數值方法求解任意形狀的物體，其次是將未知函數展開成CHebyshev多項式級數。有些合成積分是將級數的係數提出到積分符號外面後再使用高斯-Chebyshev積分法（Gauss-Chebyshev integration rules）來求值，其他的則可求得正確的值；包括去奇異核後的積分式。
本文在數值上的成就包含了兩個部分：
1. 求得常積分式（ordinary integrals）的解
2. 求解一系列的代數方程式
因為上述兩種求解的過程是簡單並明確的，因此本文的方法具有很高的效率和準確性。
最後數值計算的結果是符合平板和曲板的聲學散射問題。並成功地比較平板的解析解。

This paper presents a numerical method for predicting the acoustic scattering from 2-D thin bodies. Both the Dirichlet and Neumann problems are considered. Applying the thin-body formulation leads to the boundary integral equations involving weakly singular and hypersingular kernels. Completely regularizing these kinds of singular kernels is thus the main concern of this paper. The basic subtraction-addition technique is adopted. The purpose of incorporating a parametric representation of the boundary surface with the integral equations is twofold. The first is to facilitate the numerical implementation for arbitrarily shaped bodies. The second one is to facilitate the expansion of the unknown function into a series of Chebyshev polynomials. Some of the resultant integrals are evaluated by using the Gauss-Chebyshev integration rules after moving the series coefficients to the outside of the integral sign；others are evaluated exactly,including the modified hypersingular integral. The numerical implementation basically includes only two parts, one for evaluating the ordinary integrals and the other for solving a system of algebraic equations. Thus, the current method is highly efficient and accurate because these two solution procedures are easy and straightforward. Numerical calculations consist of the acoustic scattering by flat and curved plates. Comparisons with analytical solutions for flat plates are made.

1. Martin P.A.“End-point behaviour of hypersingular integral equations.”Proceedings of the Royal Society London, Series A 1991; 432: 301-320.
2. Seybert A.F., Cheng C.Y.R. and Wu T.W.“The solution of coupled interior/exterior acoustic problems using the boundary element method.”Journal of the Acoustical Society of America 1990; 88: 1612-1618.
3. Wu T.W. and Wan G.C.“Numerical modeling of acoustic radiation and scattering from thin bodies using a Cauchy principal integral equation.”Journal of the Acoustical Society of America 1992; 92: 2900-2906.
4. Burton A.J. and Miller G.F.“The application of integral equation methods to the numerical solution of some exterior boundary value problems.”Proceedings of the Royal Society London,Series A 1971; 323: 201-210.
5. Schenck H.A.“Improved integral formulation for acoustic radiation problems.”Journal of the Acoustical Society of America 1968; 44: 41-58.
6. Maue A.W.“Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integraleichung.”Zeitschrift für Physik 1949; A 126: 601-618.
7. Meyer W.L., Bell W.A., Zinn B.T. and Stallybrass M.P.“Boundary integral solutions of three-dimensional acoustic radiation problems.”Journal of Sound and Vibration 1978; 59: 245-262.
8. Chien C.C., Rajiyah H. and Atluri S.N.“An effective method for solving the hypersingular integral equations in 3-D acoustics.”Journal of the Acoustical Society of America 1990; 88: 918-937.
9. Yang S.A.“ A boundary integral equation method for two-dimensional acoustic scattering problems.”Journal of the Acoustical Society of America 1999; 105: 93-105.
10. Ih K.-D. and Lee D.-J.“ Development of the direct boundary element method for thin bodies with general boundary conditions.”Journal of Sound and Vibration 1997; 202: 361-373.
11. Wu T.W.“ A direct boundary element method for acoustic radiation and scattering from mixed regular and thin bodies.” Journal of the Acoustical Society of America 1995; 97: 84-91.
12. Fox L. and Parker I.B. Chebyshev Polynomial in Numerical Analysis. London: Oxford University Press. 1968.
13. Elliott D.“The numerical solution of integral equations using Chebyshev polynomials.”Journal of the Australian Mathematical Society 1959-60; 1: 344-356.
14. Scraton R.E.“ The solution of integral equations in Chebyshev Series.”Mathematics of Computation 1969; 23: 837-844.
15. Piessens R. and branders M.“ Numerical solution of integral equations mathematical physics,using Chebyshev polynomials.”Journal of Computation Physics 1976; 21: 178-196.
16. Hadamard J. Lectures on Cauchys’Problems in Linear Partial Differential Equations. New Haven:Yale University Press.1923.
17. Kaya A.C. and Erdogan F.“ On the solution of integral equations with strongly singular kernels.”Quarterly of Applied Mathematics 1987; ⅩLⅤ: 105-122.
18. Morse P.M. and Ingard K.U. Theoretical Acoustics. New York: McGraw-Hill Book Co., 1968; Chapter 8.
19. Abramowitz M. and Stegun I.A.（editors） Handbook of Mathematical Functions. New York: Dover, 1965.
20. Beyer W.H.（editor） CRC Standard Mathematical Tables and Formulae. Boca Raton: CRC Press, 1991.
21. Smirnov V.I. A Course of Higher Mathematics. Oxford: Pergamon, 1964; Ⅳ.
22. Martin P.A. and Rizzo F.J.“ On the boundary integral equations for crack problems.”Proceedings of the Royal Society London 1989; Series A 421:341-355.
23. Bowman J.J., Senior T.B.A. and Uslenghi P.L.E.（editors） Electromagnetic and Acousic Scattering by Simple Shapes. New York: Hemisphere, 1987; Chapter 4.
24. Stratton J.A. Electromagnetic Theory. New York: McGraw-Hill Book Co.,1941; Chapter Ⅵ.