Finite-Amplitude Standing Waves in Rigid-Walled Tubes

Abstract

A perturbation expansion is formulated for the one‐dimensional, nonlinear, acoustic‐wave equation with dissipative term describing the viscous and thermal energy lossesencountered in a rigid‐walled, closed tube with large length‐to‐diameter ratio. The resulting set of iterative, linear equations is solved for a finite‐amplitude standing wave.Solutions lead to a steady‐state distribution of harmonics of the fundamental, the amplitude and phase of each term being strong functions of frequency and the absorptive process. Necessary features of the approach include characterizing all absorptive processes by a bulk absorption coefficient and requiring a boundary‐layer depth much less than the tube diameter. The solution, while strictly limited to the preshock régime, can be used to predict certain features of the onset of shock. Intense longitudinal standing waves were generated within a rigid‐walled tube of 6‐ft length and 212‐in. diam. The tube contained air, at ambient pressure and temperature, which was excited into vibration by a piston at one end. A microphone at the rigid end of the tube was used to observe the pressure as a function of time. For input frequencies around either the fundamental or the first overtone of the tube, the amplitudes of the second and third harmonics of the finite‐amplitude wave were in excellent agreement with the predictions of the theory. Waveforms reconstructed from the predicted amplitudes and phase angles of the solution compared very well with the observed microphone output. An extension of the theory to the shock régime provided qualitative agreement with observations of the frequency dependence of both the intensity needed to produce shock and the phase of the onset of shock.