Abstract
The acoustic pressure of an unbounded acoustic field with multiple prolate spheroids with the Robin boundary conditions subjected to a time-harmonic point source located at an arbitrary location is solved semi-analytically in this work. This resultant solution is the so-called dynamic Green’s function, which is important for acoustic problems such as sound scattering and noise control. It can be obtained by combining the fundamental solution with a homogenous solution, which is determined by using the collocation multipole procedure to satisfy the required Robin boundary conditions. To consider the geometries as described herein, the regular solution is expanded with angular and radial prolate spheroidal wave functions. As an alternate to the complex addition theorem applied to problems in multiply connected domains, by the directional derivative, the multipole expansion is computed in a straightforward manner among different local prolate spheroidal coordinate systems. By taking the finite terms of the multipole expansion at all collocating points, an algebraic system is acquired, and then the unknown coefficients are determined to complete the proposed dynamic Green’s function by the Robin boundary conditions. The present results of one spheroid agree with the available analytical solutions. For the case of more than one spheroid, the proposed results are verified by comparison with the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. In the end, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheroids, and the aspect ratio of spheroid on the dynamic Green’s functions. The proposed results can be applied to solve the time-harmonic problems for an unbounded acoustic field containing multiple spheroids. In the form of numerical Green's functions, they can improve the computational efficiency and increase the application of the boundary integral equation method.
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13 October 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00707-022-03364-7
References
Kythe, P.K.: Green’s functions and linear differential equations: theory, applications, and computation. Chapman and Hall/CRC (2011)
Greenberg, M.D.: Application of Green’s functions in science and engineering. Prentice-Hall, Englewood Cliffs, N.J. (1971)
Melnikov, Y.A.: Some application of the Green’s function method in mechanics. Int. J. Solids Struct. 13, 1045–1058 (1977)
Melnikov, Y.A., Melnikov, M.Y.: Modified potential as a tool for computing Green’s functions in continuum mechanics. CMES-Comp. Model. Eng. Sci. 2, 291–305 (2001)
Seybert, A.F., Soenarko, B.: Radiation and scattering of acoustic waves from bodies of arbitrary shape in a three-dimensional half space. J. Vib. Acoust. Trans. ASME. 110(1), 112–117 (1988)
Park, J.M.: A boundary element method for propagation over absorbing boundaries. J. Sound Vib. 175(2), 197–218 (1994)
Chandler-Wilde, S.N., Heinemeyer, E., Potthast, R.: Acoustic scattering by mildly rough unbounded surfaces in three dimensions. SIAM J. Appl. Math. 66(3), 1002–1026 (2006)
Allen, J.B., Berkley, D.A.: Image method for efficiently simulating small-room acoustics. J. Acoust. Soc. Am. 65(4), 943–950 (1979)
Newman, J.N.: The Green function for potential flow in a rectangular channel. J. Eng. Math. 26(1), 51–59 (1992)
Tadeu, A., Antonio, J., Godinho, L.: Applications of the Green functions in the study of acoustic problems in open and closed spaces. J. Sound Vib. 247(1), 117–130 (2001)
Wang, X., Sudak, L.J.: Antiplane time harmonic Green’s functions for a circular inhomogeneity with an imperfect interface. Mech. Res. Commun. 34(4), 352–358 (2007)
Lee, W.M., Chen, J.T., Young, W.M.: Dynamic Green’s functions for multiple circular inclusions with imperfect interfaces using the collocation multipole method. Eng. Anal. Bound. Elem. 94, 113–121 (2018)
Lee, W.M., Chen, J.T.: Dynamic Green’s functions for multiple elliptical inclusions with imperfect interfaces. Mech. Res. Commun. 108, 103567 (2020)
Lee, J.W., Chen, J.T., Leu, S.Y., Kao, S.K.: Null-field BIEM for solving a scattering problem from a point source to a two-layer prolate spheroid. Acta Mech. 225, 873–891 (2014)
Chen, J.T., Lee, J.W., Kao, Y.C., Leu, S.Y.: Eigenanalysis for a confocal prolate spheroidal resonator using the null-field BIEM in conjunction with degenerate kernels. Acta Mech. 226, 475–490 (2015)
Okoyenta, A.R., Wu, H., Liu, X., Jiang, W.: A short survey on Green’s function for acoustic problems. J. Theor. Comput. Acoust. 28(02), 1950025 (2020)
Duffy, D.G.: Green's Functions with Applications. Chapman and Hall/CRC (2001)
Telles, J.C.F., Castor, G.S., Guimaraes, S.: Numerical Green’s function approach for boundary elements applied to fracture mechanics. Int. J. Numer. Methods Eng. 38(19), 3259–3274 (1995)
Harwood, A., Dupere, I.: Numerical evaluation of the compact acoustic Green’s function for scattering problems. Appl. Math. Model. 40, 795–814 (2016)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)
Flammer, C.: Spheroidal wave functions. Stanford University Press, Stanford, Calif. (1957)
Stratton, J.A.: Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients. Technology Press of M.I.T. and Wiley, New York (1956)
Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill, New York (1953)
Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. (Vol. 3) McGraw-Hill, New York (1955)
Zhang, S., Jin, J.: Computation of Special Functions. Wiley, New York (1996)
Kleshchev, A.A.: Sound scattering by spheroidal bodies near an interface. Sov. Phys. Acoust. 23(3), 225–228 (1977)
Wu, T.W.: Boundary Element Acoustics: Fundamentals and Computer Codes. WIT Press, Southampton UK (2000)
Acknowledgements
Financial support from the National Science and Technology Council of Taiwan (ROC), under Grant No. MOST 108-2221-E-157-002-MY2, to the China University of Science and Technology is gratefully acknowledged. The author thanks the reviewers for their very constructive comments and suggestions.
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Lee, W.M., Chen, J.T. Construction of dynamic Green’s function for an infinite acoustic field with multiple prolate spheroids. Acta Mech 233, 5021–5041 (2022). https://doi.org/10.1007/s00707-022-03301-8
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DOI: https://doi.org/10.1007/s00707-022-03301-8