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A compressive nonstationary near-field acoustic holography based on time-domain plane wave superposition method is proposed to precisely reconstruct the instantaneous sound field using the fewer measurement points. In the proposed method, by means of an instantaneous propagation kernel, a time-domain convolution superposition formula is established for relating the time-variant pressure of the hologram plane to the pressure time-wavenumber spectra of the virtual source plane. Next, compressed sensing theory and sparse representation framework are introduced into the time-marching solving inverse process of the established formula, and the smoothed l 0 norm and revised Newton optimization algorithm are employed to solve the serious cumulative ill-posed problem over time for acquiring the pressure time-wavenumber spectra. The key is to approximately replace the discontinuous ℓ 0 norm by using a suitable continuous function, and the revised Newton algorithm is applied to minimize the continuous function for obtaining the optimal solution. Finally, with the aid of a time-domain convolution formula, the solved pressure time-wavenumber spectra are employed to calculate the time-variant pressure on the reconstruction plane. The proposed method not only can evidently suppress the ill-posed problem of time-marching solving inverse process for improving the reconstruction accuracy, but also can greatly reduce the measurement points and microphone number for drastically decreasing the measurement cost on the premise of ensuring the reconstruction accuracy and spatial resolution. A numerical simulation is conducted, and the simulation effects prove that the proposed method can accurately reconstruct the instantaneous sound field with the fewer measurement points in time–space domains. The reconstruction results are also compared to those of time-domain plane wave superposition method with Tikhonov regularization to verify the superiority of the proposed method with fewer measurement points under different parameter configurations. An experiment of an impaction steel plate is implemented for further validating the competence of the proposed method. [ABSTRACT FROM AUTHOR]

• A multistep acoustic method is extended to suppress the reconstruction instability of instantaneous vibration of a structure. • The more serious time-domain instability of the reconstruction between the pressure and acceleration is proved. • The superiority of the extended M-TDPWSM is demonstrated by comparing with the S-TDPWSM. A multistep acoustic method is extended to suppress the more serious instability in the inverse process between the time-evolving pressure on the hologram plane and the time-wavenumber acceleration spectrum on the virtual source plane, and further stabilizes the reconstruction of the instantaneous vibration of an exciting planar structure. In the extended method, the solving equation relating the time-evolving pressure to the time-wavenumber acceleration spectrum at each time step is first established. Then, the established equations at several time steps are combined to constitute a new large solving equation. Finally, after one-time solving inverse calculation of the new equation, the time-wavenumber acceleration spectra at these several time steps are solved simultaneously. The multistep calculation in the expanded method not only can reduce the total number of solving the time-wavenumber acceleration spectra for reducing the accumulation errors in the time domain, but also can expand the dimension of the solving inverse matrix to add the number of the nonzero element for improving the filtering effect. A planar plate excited by a steel ball is designed as the numerical simulation to test the ability of the extended method in suppressing the instability of the inverse process of reconstructing the structural surface acceleration. In the simulation, some important parameters, such as the position of the virtual source plane, the sampling number in the wavenumber domain, the noise and the multistep number, are discussed. The simulation results demonstrate that in the reconstruction of the instantaneous vibration, the single-step time domain plane wave superposition method is more sensitive to these parameters and the divergence problem of its solution is more serious, but the extended method has a stronger adaptation ability in the changes of these parameters and can effectively suppress the reconstruction instability of the instantaneous vibration of an exciting structure for providing a more stable solution. An experiment with a steel plate excited by a steel ball is further employed to verify the superiority of the expanded method in suppressing the serious instability of the inverse process of reconstructing the instantaneous vibration. [ABSTRACT FROM AUTHOR]