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.
Chen, I-Lin; Liang, Ming-Te; Kuo, Shyh-Rong; and Chen, Jeng-Tzong
"Dual Boundary Integral Equations for Helmholtz Equation at a Corner Using Contour Approach Around Singularity,"
Journal of Marine Science and Technology: Vol. 9:
1, Article 7.
Available at: https://jmstt.ntou.edu.tw/journal/vol9/iss1/7