Application of the Euler-Maclaurin Sum Formula to Obtain a Closed form Analytical Solution for Acoustical and Heat Diffusion in a Partially Bounded Region

Michael Panza^1,

¹Mechanical Engineering, Gannon University, 109 University Square, Erie, PA 16541

Cite this paper:
Michael Panza. Application of the Euler-Maclaurin Sum Formula to Obtain a Closed form Analytical Solution for Acoustical and Heat Diffusion in a Partially Bounded Region. American Journal of Mechanical Engineering. 2020; 8(3):111-121. doi: 10.12691/ajme-8-3-3

Abstract

Green’s function in the Laplace transform domain for the sound wave field in an infinite area between two parallel reflecting planes may be represented as an infinite series of images caused by the planes. Based on a comparison of the wave equation and the diffusion equation, the Green’s function representation as an infinite series for diffusion in the same region is directly obtained. A closed form expression in space time for the diffusion problem is obtained by applying the Euler-Maclaurin sum formula to a modified diffusion form of the series in the Laplace transform domain and then inverting to the time domain. Using several sets of numerical values for system parameters applicable to acoustic diffusion in the region, numerical comparisons of the infinite series vs Euler-Maclaurin closed form representation of the Green function is presented. Comparison of the infinite series vs Euler-Maclaurin transient response to an exponential and constant input are presented for cases of acoustical noise diffusion and heat diffusion respectively. Transfer function comparisons are given for the diffusion models along with the use of the closed form representation in a model-based control scheme.

Keywords:
Euler-Maclaurin sum formula acoustic diffusion heat diffusion method of images Green’s function

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References:

[1]	Panza M. J.,“Closed form solution for acoustic wave equation between two parallel plates using Euler-Maclaurin sum formula”, Journal of Sound and Vibration, 277, 123-132, 2004.
[2]	Panza M. J., “Application of Euler–Maclaurin sum formula to obtain an approximate closed-form Green’s function for a two-dimensional acoustical space”, Journal of Sound and Vibration, 311, 269-279, 2008.
[3]	Panza M. J., “Euler-Maclaurin closed form finite state space model for a string applied to broadband plate vibrations”, Journal of Mathematical Problems in Engineering, Vol. 2010, 2010.
[4]	Panza M. J., “A finite state space model for representing the broadband infinite series for acoustic reverberation between parallel reflecting planes”, Journal of Mathematical Problems in Engineering, Vol.2018, 2018.
[5]	Navarroa J. M. and Escolano J. “Simulation of building indoor acoustics using an acoustic diffusion equation model”, Journal of Building Performance Simulation, 8, 1-2, 2015.
[6]	Hernández M., Imbernón, B., Navarro J. M., García, J. M. and Cebrian J. M., “Evaluation of the 3-D finite difference implementation of the acoustic diffusion equation model on massively parallel architectures”, Computers and Electrical Engineering, 46, 190-201, 2015.
[7]	Gül Z. S., Xiang N. and Çalışkan M., “Sound field analysis of monumental structures by the application of diffusion equation model”, in Proceedings 2014 COMSOL, Cambridge Publishers.
[8]	Jing Y. and Xianga N., “A modified diffusion equation for room-acoustic predication”, Journal of the Acoustical Society of America, 121, 3284-3287, 2007.
[9]	Valeau V, Pacuat J. and Hodgson M., “On the use of a diffusion equation for room-acoustic prediction”, Journal of the Acoustical Society of America, 119, 1504-1513, 2006.
[10]	Constanda C., Solution Techniques for Elementary Partial Differential Equations, Chapman & Hall/CRC Publishers, Boca Raton, FL, 207-217, 2010.
[11]	Hayek S. I., Advanced Mathematical Methods in Science and Engineering, Marcel Dekker Publishers, New York, 2001.
[12]	Oberhettinger F. and Badii L, Tables of Laplace Transforms, Springer-Verlag Publishers, New York, 1973, 41.
[13]	Billon A., Picaut J., Valeau V. and Sakout A., “Acoustic Predictions in Industrial Spaces Using a Diffusion Model”, Advances in Acoustics and Vibration, Vol. 2012, 2012
[14]	Panza, M., “A Mathematical Images Group Model to Estimate the Sound Level in a Close-Fitting Enclosure”, Journal of Advances in Acoustics and Vibration, Vol. 2014, 2014.
[15]	Wu Z. C. and Cheng D. L, ”Temperature distribution of an infinite slab under point heat source”, Journal of Thermal Science, 18, 1597-1601, 2014.