Unsteady Isentropic Flow through Ducts with Prescribed Sound Pressure Level Distribution

An analytical method for the determination of the required shape of a duct for a prescribed sound pressure level distribution is presented in this paper. The physical model involves a sound wave propagating in an unsteady flow of compressible fluids through ducts. Two cases are considered. In the first case, the channel shape, F(X), is given as either an exponential function or a linear function of the distance along the axis with an unknown parameter in the expression for F(X). The unknown parameter is determined by the prescribed ratio of the sound pressure level at the exit section of the duct to that at the entrance. In the second case, the sound pressure level is specified at every point along the length of the duct, and the duct shape, F(X), is sought. The governing differential equations of the model are presented. The method of complex superposition is used to separate the real and the imaginary parts of the perturbation quantities. The results show that the cross-sectional area is sensitive to the flow speed and the frequency of the sound source. Furthermore, a convergent/divergent duct has to be used to achieve a linear sound pressure level distribution.