A study on the multiple reciprocity method and complex-valued formulation for the Helmholtz equation

Abstract

The relation between the multiple reciprocity method and the complex-valued formulation for the Helmholtz equation is re-examined in this paper. Both the singular and hypersingular integral equations derived from the conventional multiple reciprocity method are identical to the real parts of the complex-valued singular and hypersingular integral equations, provided that the fundamental solution chosen in the multiple reciprocity method is proper. The problem of spurious eigenvalues occurs when we use either a singular or hypersingular equation only in the multiple reciprocity method because information contributed by the imaginary part of the complex-valued formulation is lost. To filter out the spurious eigenvalues in the conventional multiple reciprocity method, singular and hypersingular equations are combined together to provide sufficient constraint equations. Several one-dimensional examples are used to examine the relation between the conventional multiple reciprocity method and the complex-valued formulation. Also, a new complete multiple reciprocity method in one-dimensional cases, which involves real and imaginary parts, is proposed by introducing the imaginary part in the undetermined coefficient in the zeroth-order fundamental solution. Based on this complete multiple reciprocity method, it is shown that the kernels derived from the multiple reciprocity method are exactly the same as those obtained in the complex-valued formulation.