Fictitious frequency revisited

The nonexistence and nonuniqueness problems associated with integral equation methods for exterior acoustics are revisited. The Fredholm alternative theorem in conjunction with the singular value decomposition updating technique is used to simultaneously determine the fictitious frequencies and corresponding modes in exterior acoustics. After selecting the combined Helmholtz interior integral equation formulation (CHIEF) points, the influence row vectors are obtained. A criterion in selecting the minimum number of CHIEF points and their positions is proposed by testing the orthogonal condition between the influence row vector and the right unitary vector. It is proved in the discrete system for arbitrary-shape problems that the source of numerical instability of irregular frequency originates from the zero divided by zero using the generalized coordinates of unitary vectors. The mathematical structures of the four influence matrices in the dual boundary element method (BEM) are examined by using the left and right unitary matrices. Extracting the true eigenvalue and filtering out the fictitious frequency can be unified by using the updating term and updating document, respectively. Radiation problems of a cylinder and a square rod are demonstrated to see the validity of the present formulation.