Finite part representations of hypersingular integral equation of acoustic scattering and radiation by open smooth surfaces
The use of the Green function method for the solution of the Helmholtz equation for both acoustic scattering by hard surfaces or radiation by vibrating surfaces is shown to lead to hypersingular surface boundary integral equations. For a general open surface a simple proof is given that the integral term should be interpreted as the Hadamard finite part of a divergent integral in one variable. The problem is reformulated as a Cauchy principal value integral equation which also contains the potential at the control point. This is then amenable to numerial treatment by conventional methods. An alternative formulation is given in the better known form containing the tangential derivative of the potential. The simpler two-dimensional problem for an open arc is treated separately.
hypersingular surface boundary integral equations
Hadamard finite part of a divergent integral
Cauchy principal value integral equation
scientific article; zbMATH DE number 955210