Dual Boundary Integral Equations for Helmholtz Equation at a Corner Using Contour Approach Around Singularity

Abstract

A dual integral formulation for the Helmholtz equation problem at a corner is derived by means of the contour approach around the singularity. It is discovered that employing the contour approach the jump term comes half and half from the free terms in the L and M kernel integrations, respectively, which differs from the limiting process from an interior point to a boundary point where the jump term is descended from the L kernel only. Thus, the definition of the Hadamard principal value for hypersingular integration at the collocation point of a corner is extended to a generalized sense for both the tangent and normal derivative of double layer potentials as compared to the conventional definition. The free terms of the six kernel functions in the dual integral equations for the Helmholtz equation at a corner have been examined. The kernel functions of the Helmholtz equation are quite different from those of the Laplace equation while the free terms of the Helmholtz equation are the same as those of the Laplace equation. It is worth to point out that the Laplace equation is a special case of the Helmholtz equation when the wave number approaches zero.本文探討經由推到邊界及繞道奇異點的方法導出在角點荷姆茲方程的對偶積分表示式。結果發現，利用環繞邊界法它的跳躍項是由L及M核函數經積分各貢獻一半，這與經由極限過程所得自由項完全由L核函數貢獻有所不同。在超強奇異積分方程中阿達馬主値的觀念在此從雙層勢能的法向微分推廣到切向微分以便於與傳統的定義對照。同時對於荷姆茲方程對偶邊界積分方程式中的六個核函數在角點的自由項也予以檢驗。荷姆茲方程的核函數與拉普拉斯方程的核函數完全不同，但是它們的自由項卻相同。值得一提的是拉普拉斯方程僅爲荷姆茲方程當波數k趨近於零時的一個特例而已。