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Mathematical Problems in Engineering
Volume 2018, Article ID 2631718, 16 pages
https://doi.org/10.1155/2018/2631718
Research Article

A Finite State Space Model for Representing the Broadband Infinite Series for Acoustic Reverberation between Parallel Reflecting Planes

Gannon University, 109 University Square, Erie, PA 16541, USA

Correspondence should be addressed to Michael J. Panza; ude.nonnag@aznap

Received 27 November 2017; Revised 31 March 2018; Accepted 6 May 2018; Published 4 June 2018

Academic Editor: Salvatore Alfonzetti

Copyright © 2018 Michael J. Panza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The acoustic reverberation between two parallel reflecting planes can be represented by an infinite series of the images caused by the planes. To provide a more useful model for analysis and control, the infinite series version of the Green’s function is converted into a finite state space model that retains the high frequency character that enables broadband noise inputs to be examined. The infinite series is first summed into a very accurate, approximate closed form expression in the time domain in terms of a radical function. The radical is then transformed into an expression containing exponentials which have exact Laplace transforms that lead to an overall closed form transfer function for the system. The system transfer function is transformed into a third-order state space model that theoretically contains all of the frequency characteristics of the infinite series representation. The accuracy of the state space model is examined by comparing it to the infinite series solution for three typical types of acoustical inputs: exponential for impulse noise, single frequency sine for harmonic noise, and a multifrequency Schroeder phased harmonic sequence for random noise.