The Imaginary Boundary Equivalent Source Method for Acoustic Computation

This paper establishes the equivalent equation of the Equivalent Source Method (ESM) and, under the theoretical framework where the equivalent sources are distributed on a spherical surface, provides a detailed analysis of the factors influencing the errors of ESM. The study shows that, when the equivalent sources are distributed on a sphere, the zeros of the spherical Bessel functions are the primary cause of the non-uniqueness of solutions at characteristic frequencies. Moreover, during numerical inversion, the alternating behavior of zeros of spherical Bessel functions of different orders amplifies the influence of boundary condition errors and discrete orthogonality errors of the equivalent sources on the sound field calculation. To overcome these problems, the properties of the zeros of integer-order spherical Bessel functions in the complex domain are analyzed, and the Imaginary Boundary Equivalent Source Method (IB-ESM) is proposed. In the imaginary domain, spherical Bessel functions of integer order have no zeros, and their function values decrease monotonically with increasing order. This means that IB-ESM not only eliminates the non-uniqueness problem at characteristic frequencies but also effectively suppresses the discrete orthogonality errors in the high-order spherical wave spectrum. The theoretical analysis under these simplified conditions provides a rational basis for the proposed method. To further verify the applicability and practicality of IB-ESM over a broader scope, various numerical simulations are conducted, including configurations with equivalent sources distributed on both spherical and cubic surfaces. The simulation results demonstrate that IB-ESM is effective and superior in sound field reconstruction in acoustic holography, as well as in solving sound scattering problems involving soft, hard, and impedance boundaries, and sound radiation problems of rigid enclosures. Compared with the conventional ESM, IB-ESM completely eliminates the non-uniqueness problem and achieves higher computational accuracy across a wide frequency range.