Transient near-field acoustical holography (NAH) formulation is derived from the Helmholtz equation least squares (HELS) method to reconstruct acoustic radiation from a spherical surface subject to transient excitations in a free field. To facilitate derivations of temporal solutions, we make use of the Laplace transform and expansion in terms of the spherical Hankel functions and spherical harmonics, with their coefficients settled by solving a system of equations obtained by matching an assumed-form solution to the measured acoustic pressure. To derive a general form of solution for a temporal kernel, we replace the spherical Hankel functions and their derivatives by polynomials, recast infinite integrals in the inverse Laplace transform as contour integrals in a complex s-plane, and evaluate it via the residue theorem. The transient acoustic quantities anywhere including the source surface are then obtained by convoluting the temporal kernels with respect to the measured acoustic pressure. Numerical examples of reconstructing transient acoustic fields from explosively expanding, impulsively accelerating, and partially accelerating spheres, and that from a sphere subject to an arbitrarily time-dependent excitation are depicted. To illustrate the effectiveness of HELS-based transient NAH formulations, all input data are collected along an arbitrarily selected line segment and used to reconstruct transient acoustic quantities everywhere.
The HELS method [Wu, J. Acoust. Soc. Am. 107, 2511–2522 (2000)] is extended to reconstruction of transient acoustic radiation from a highly nonspherical structure. The test object is a thin disk subject to an impulsive acceleration in an unbounded fluid medium. Since the HELS method allows piecewise reconstruction of acoustic quantities on the source surface, it is possible to focus on one side of the disk at a time. Also, since the origin of coordinates is arbitrary, one can set the spherical coordinates in such a way that the spherical surface looks almost flat locally. This treatment legitimizes the Rayleigh hypothesis and facilitates reconstruction of the normal surface velocity on the disk front surface. Reconstruction of normal surface velocity on the opposite side of the disk can be done in a similar manner. The input acoustic pressure signals are collected using an array of microphones in front of the disk and reconstructed acoustic quantities are compared with the analytic results [Wu, J. Acoust. Soc. Am. 94, 542–553 (1993)]. Results show that the accuracy of reconstruction depends on that of input signals, and convergence of the reconstructed normal surface velocity improves with an increase in the cutoff frequency of input data. [Work supported by NSF.]
The HELS method (Wu, 2000) is extended to reconstruction of transient acoustic radiation from an impulsively accelerated object. The temporal acoustic pressure field is reconstructed by taking an inverse Fourier transformation of the acoustic pressure in the frequency domain. The infinite integral is replaced by a contour integral and evaluated using the residue theory. The formulations thus derived are valid for a spherical surface with an arbitrary normal velocity distribution. These formulations are used to reconstruct the normal surface velocities and transient acoustic fields generated by explosively expanding sphere, impulsively accelerating sphere, and impulsively accelerating baffled sphere. Results show that satisfactory reconstruction can be obtained with relatively few measurements taken around the object under consideration.
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