A discussion of modal uncoupling and an approximate closed-form solution for weakly coupled systems with application to acoustics
Modal analysis is often used to solve problems in acoustics, leading to a system of coupled equations for the modal amplitudes. A common practice in analytical work utilizing modal analysis has been to assume that weak modal coupling is negligible, thereby enabling the modal coefficients to be solved independently in closed form. The validity of this assumption, as well as the order of the error from neglecting modal coupling, is discussed. It is possible to incorporate the principal effects of weak modal coupling in a very simple way without solving the fully coupled system. An approximate closed-form solution for weakly coupled systems of equations is developed. The procedure gives insight into the errors incurred when coupling is neglected, and shows that these errors may be unacceptably large in systems of practical interest. A model problem involving a pipe with an impedance boundary condition is solved when the one-dimensional sound field is harmonically driven, and when it undergoes reverberant decay from initial conditions. The approximate solution derived in this paper is compared with results for the fully coupled and fully uncoupled equivalent problems. The approximation works well even for systems where the coupling is fairly strong. The results show that modal coupling must be included, at least approximately, if certain salient features of the sound field, such as intensity flow and detailed reverberant structure, are to be predicted correctly.
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