Closed form solution for acoustic wave equation between two parallel plates using Euler–Maclaurin sum formula

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

“…The use of the Euler-Maclaurin sum formula to provide an approximate, accurate closed form solution for the 2 Mathematical Problems in Engineering infinite series Green's function that describes partially or fully bounded acoustic and structural systems is developed and verified in [8][9][10] by comparing the closed form solution to the infinite series solution for several types of inputs. In [8], the main part and associated variations of the EulerMaclaurin closed form with the infinite series solutions in a partially bounded three-dimensional acoustic space between two parallel plates are presented for an exponential input.…”

“…In [8], the main part and associated variations of the EulerMaclaurin closed form with the infinite series solutions in a partially bounded three-dimensional acoustic space between two parallel plates are presented for an exponential input. In [9], the Euler-Maclaurin closed form solution is compared with the infinite series solutions in a partially bounded two-dimensional space acoustic space for a narrow duct.…”

“…For all of the Euler-Maclaurin based studies in [8][9][10], state space representations were not considered. In view of the Euler-Maclaurin closed form results given in [8][9][10], in both time domain response and the transfer function based control for frequencies up to 1000 Hz, and the successes in [1][2][3][4][5][6][7] for the truncated state space models for various types of low frequency control in acoustic and structural systems, the work in [11] focuses on the development of a low order but full finite state space model that would naturally include a broadband frequency range and essentially all of the terms required for the convergent infinite series in the Green's function representation of bounded structures.…”

“…The distinguishing difference between this paper and the development in [11] is that, in [11], the Euler-Maclaurin sum formula is applied to an infinite series of string modes. In this paper, the Euler-Maclaurin sum formula is applied directly to the Laplace transform form of the Green's function infinite series for acoustic reverberation in [8] providing all polynomial type functions in the Laplace transform domain which are directly combined into a system function. This system transfer function in [11] leads directly to an input/output ordinary differential equation which serves as the basis for defining the state variables and determining the state equations.…”

“…This system transfer function in [11] leads directly to an input/output ordinary differential equation which serves as the basis for defining the state variables and determining the state equations. For the acoustic space of the current paper, the Euler-Maclaurin sum formula must be directly applied to the time domain form of the Green's function infinite series as is done in [8]. As previously indicated, that direct application of the Euler-Maclaurin sum formula to the time domain form of the Green's function infinite series [8] leads to an integral of a radical function that has no exact analytical solution for providing polynomial type functions in the Laplace transform domain.…”