Transfer matrix analysis of a duct with gradually varying arbitrary cross-sectional area

A method is presented to determine the low-frequency sound field in a duct with a cross-section of arbitrary shape and area that varies gradually without discontinuities. The duct is modeled as several fictitious segments, each of which has a cross-sectional area that is a quadratic function of position along its axis. A closed-form solution does not exist for arbitrary quadratic variation. Therefore, a series solution with two coefficients that depend on the boundary conditions and the radius of convergence (ROC) of the series is presented. Losses due to absorption are included. The solution corresponds to standing waves and is recast in terms of traveling waves. A transfer matrix is developed and used to express the pressure and velocity at the throat of a segment in terms of the pressure and velocity at its mouth. The series solution converges very rapidly when the length of the segment is less than half the ROC. The transfer matrix for the entire duct is the product of the transfer matrices of the segments. Numerical results are presented for open-closed ducts with the same length but different variations in area to illustrate the effect of the latter on the lowest resonance frequency and the pressure amplification.