The Analysis of Discontinuities in Cylindrical Tubes Propagating Sound Waves

Citation

Miles, John Wilder (1944) The Analysis of Discontinuities in Cylindrical Tubes Propagating Sound Waves. Engineer's thesis, California Institute of Technology. doi:10.7907/58H4-E268. https://resolver.caltech.edu/CaltechETD:etd-06152004-114239

Abstract

The effect of a plane discontinuity on a plane wave propagated in a cylindrical tube is calculated by variational methods. In carrying out the calculations a transmission line analogy is used and the effect of the discontinuity at a distance is represented by a capacitance placed at the discontinuity. In section 1 the equations of motion for the propagation of a small disturbance in a cylindrical tube are established, culminating in the tyro-dimensional wave equation, while the solutions to the equations are discussed in section 2. These solutions constitute an infinite set of modes, in addition to the plane wave usually treated in the literature. In section 3 the analogy between propagation of sound and an electrical transmission line is established, and it is shown that each mode requires a separate transmission line. In section 4 it is shown that the effect of the higher modes excited by a plane discontinuity may be represented by a lumped capacitance, and this capacitance is given by a variational expression which gives a systematic method of calculation yielding an upper bound to the true answer. For the case of a window, a variational principle is produced inihich gives a lower bound to the true answer. In section B this method is applied to a window in a rectangular tube in some detail, in section 7 it is applied to a change of cross section in a rectangular tube, and in section 8 it is applied to windows and changes of cross section in circular tubes. In section 9 the reflection and transmission factors are calculated, and in section 10 the resonant frequency of certain types of resonators is calculated. To the authors's knowledge this is the first rigorous treatment of the above problems with the exception of an earlier paper calculating the reflection due to a change of cross section in a circular tube.

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