A STUDY ON THE MULTIPLE RECIPROCITY METHOD AND COMPLEX-VALUED FORMULATION FOR THE HELMHOLTZ-EQUATION

The relation between the multiple reciprocity method and the complex-v alued formulation for the Helmholtz equation is re-examined in this pa per. Both the singular and hypersingular integral equations derived fr om the conventional multiple reciprocity method are identical to the r eal parts of the complex-valued singular and hypersingular integral eq uations, provided that the fundamental solution chosen in the multiple reciprocity method is proper. The problem of spurious eigenvalues occ urs when we use either a singular or hypersingular equation only in th e multiple reciprocity method because information contributed by the i maginary part of the complex-valued formulation is lost. To filter out the spurious eigenvalues in the conventional multiple reciprocity met hod, singular and hypersingular equations are combined together to pro vide sufficient constraint equations. Several one-dimensional examples are used to examine the relation between the conventional multiple re ciprocity method and the complex-valued formulation. Also, a new compl ete multiple reciprocity method in one-dimensional cases, which involv es real and imaginary parts, is proposed by introducing the imaginary part in the undetermined coefficient in the zeroth-order fundamental s olution. Based on this complete multiple reciprocity method, it is sho wn that the kernels derived from the multiple reciprocity method are e xactly the same as those obtained in the complex-valued formulation. ( C) 1998 Elsevier Science Ltd.