The Analysis of Plane Discontinuities in Cylindrical Tubes. Part I Available to Purchase

The effect of a plane discontinuity on a plane wave propagated in a cylindrical tube of arbitrary cross section is calculated by considering the higher order modes excited at the discontinuity. 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 Part I, the equations of motion for the propagation of a small disturbance in a cylindrical tube are assumed at the outset and are shown to yield the two‐dimensional wave equation, the solutions to this equation constituting an infinite set of modes, in addition to the plane wave usually treated in the literature. Following a discussion of these solutions, 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. 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 which gives a lower bound to the true answer. In Part II, this method will be applied to a window in a rectangular tube, to a change of cross section in a rectangular tube, to windows and changes of cross section in circular tubes and to the calculation of resonance in certain types of cavities. The ordinary reflection and transmission coefficients will also be correlated with the theory of Part I. To the author'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 (reference 1).