Application of Euler–Maclaurin sum formula to obtain an approximate closed-form Green's function for a two-dimensional acoustical space

The Green's function solution for the acoustic wave equation in a two-dimensional rectangular space is expressed as an infinite series of terms based on the cross modes of the duct. An approximate closed-form solution is obtained by applying the Euler–Maclaurin sum formula. The procedure provides a closed-form expression in both the space–time and Laplace domains along with an upper bound for a remainder. Plane wave and higher-order waves components are identified. A numerical example for an exponential input gives comparisons of the transient response for the approximate closed form and series Green's function solutions. The time response and analytical transfer function frequency spectrum of the series and Euler–Maclaurin closed-form Green's function are computed. Lastly, the approximate closed-form Green's transfer function expression is used in model-based designed feedforward and feedback control schemes to reduce peaks in the frequency response and provide system damping.