In reference [1], the authors presented the so-called non-dimensional dynamic in#uence function method for membrane vibration. A regular formulation and singularity-free method were obtained. Also, a symmetry and meshless formulation can be achieved. The auxiliary system is a complementary solution instead of fundamental function. It is source free in the in#uence function. Many successful examples of the Dirichlet types were demonstrated. It seems that this method is very attractive. However, this method can be treated as one kind of the Tre!tz method [2}4]. Based on the dual formulation developed by Chen and Hong [5,6], the in#uence function is nothing but the imaginary part of the fundamental solution (;(s, x)"iH (kr)) [7]. The method by Kang et al.
[1] can be treated as a special case of the imaginary-part dual BEM. Also, the real-part dual BEM was developed and many references can be referred [8,9]. MRM formulation can also be viewed as a real-part formulation and its occurrence of spurious eigenvalues have been found in references [10}15]. It is well known that the real-part, imaginary-part formulations and multiple reciprocity method all result in spurious eigensolutions. Particularly, the imaginary-part formulation also results in an ill-posed problem since the condition number for the in#uence matrix is very large. Many approaches have been employed to "lter out the spurious solutions and extract the true solution, for example, residue method [11,12], singular-value decomposition (SVD) technique [9, 13}15], generalized singular-value decomposition (GSVD) technique [16] and domain partition technique [17]. It is expected that the Kang's method has the problems of spurious solutions and ill-conditioned behavior since it is an imaginary-part formulation. However, no such information was addressed. Some points will be discussed as follows.
(1) Spurious eigensolution: It is interesting to "nd that all the examples in reference [1] are of the Dirichlet type. According to the theoretical derivation, the Neumann problem has the problems of spurious solutions using the imaginary-part formulation. We will prove that in the following.
As mentioned earlier, spurious eigenvalues occur in the real-part of MRM formulation [9,11,12]. Also, the imaginary-part BEM results in spurious solutions [7]. Here, we will derive the true and spurious solutions in the discrete system analytically for a circular domain by using the non-dimensional in#uence function method [1] and the imaginary-part dual BEM [7] in a uni"ed way. The degenerate kernels and circulants are employed to study the discrete system in an exact form. The relation between the non-dimensional in#uence function method and the imaginary-part dual BEM is also addressed and is summarized in Table 1. The symbols in Table 1 follow the dual model of Chen and Hong [6].