Sound-field reconstruction via the equivalent source method presents an ill-conditioned problem. This paper proposes a method in which traditional monopole equivalent source spherical wave functions are replaced with strong directional wave functions such that an ill-conditioned problem can be improved upon when building the integral equations of a near-field acoustic holography model. Using directional wave functions as an integral kernel function can cause the transfer matrix to be main-diagonally dominant. The accuracy of the proposed method for sound-field reconstruction is verified using numerical simulations, and reconstruction stability in the presence of measurement noise is considered. Furthermore, a method for selecting directional wave functions that are appropriate for various model parameters is introduced. The results show that the proposed method can greatly reduce the condition number of the transfer matrix and improve reconstruction stability.

Based on the fast Fourier wave superposition spectrum method, a new equivalent source method (ESM) with a sparse sampling technique is proposed. First, the equivalent source intensities are expanded on a rectangular virtual surface using a bidirectional Fourier series, resulting in a semi-analytic and half-numerical acoustic pressure expression. The Fourier coefficients result in good sparsity for continuous acoustic pressures from structural vibration sources, and the proposed sparse sampling method can further reduce correlation in the measurement matrix. Better results can be obtained by solving the l₁ norm optimization problem. Finally, the method was verified using several examples. The proposed method offers two main advantages compared with the traditional compressive equivalent source method: (1) the unknown source intensity vector is expanded into a bidirectional Fourier series, thereby transforming an unknown source intensity vector into a sparse Fourier coefficient vector, which has better sparsity; (2) the proposed method constructs a random sampling matrix, which is expanded into a sparse sampling matrix by random distribution, thereby improving the reconstruction accuracy of planar near-field acoustic field compared with the traditional random position sampling method reducing correlation in the transfer matrix.

This paper proposes a complex boundary wave superposition method with a unique solution of full wavenumber based on the similarity between the acoustic wave superposition method (WSM) and the external excitation response of dynamic systems and combined with the idea that a damping system has a unique solution in dynamic theory. By placing the virtual equivalent source on the virtual boundary of complex space, the conventional WSM acquires damping properties comparable to those of the dynamical system. This approach successfully addresses the non-uniqueness of the solution at the eigenfrequency and is more efficient than the conventional combined layer potential method. The paper presents a comprehensive description of the proposed method, with a focus on theory, modeling, and parameter selection. The effectiveness of this method is evaluated by applying it to two types of acoustic problems, namely, radiation and scattering. The numerical results indicate that this method effectively addresses the non-unique problems encountered in conventional WSM. Furthermore, the proposed method is as accurate and efficient as the conventional WSM.