Is the Burton–Miller formulation really free of fictitious eigenfrequencies?

This paper is concerned with the fictitious eigenfrequency problem of the boundary integral equation methods when solving exterior acoustic problems. A contour integral method is used to convert the nonlinear eigenproblems caused by the boundary element method into ordinary eigenproblems. Since both real and complex eigenvalues can be extracted by using the contour integral method, it enables us to investigate the fictitious eigenfrequency problem in a new way rather than comparing the accuracy of numerical solutions or the condition numbers of boundary element coefficient matrices. The interior and exterior acoustic fields of a sphere with both Dirichlet and Neumann boundary conditions are taken as numerical examples. The pulsating sphere example is studied and all fictitious eigenfrequencies corresponding to the related interior problem are observed. The reasons are given for the usual absence of many fictitious eigenfrequencies in the literature. Fictitious eigenfrequency phenomena of the Kirchhoff–Helmholtz boundary integral equation, its normal derivative formulation and the Burton–Miller formulation are investigated through the eigenvalue analysis. The actual effect of the Burton–Miller formulation on fictitious eigenfrequencies is revealed and the optimal choice of the coupling parameter is confirmed.