A perturbation expansion is formulated for the one-dimensional, nonlinear, acoustic-wave equation with dissipative term describing the viscous and thermal energy losses encountered 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.
A three-dimensional mathematical model for acoustical standing waves in lossy fluid-filled cavities has been obtained which requires empirical values for the resonance frequencies fn and quality factors Qn (all measured in the linear acoustic r•gime) of the pertinent standing waves which the cavity can support. The nonlinear distortion of the observed pressure waveform depends strongly on the f's and Q's of those standing waves excited by harmonics of the driving frequency. The model is applicable to non-ideal cavities if the deviations from idealized geometry and boundary conditions are small. It is restricted to small values of M (1 q-1/2 B/A)Q•, where M is the peak Mach number and Q• the quality factor of the fundamental component of the driven standing wave, and B/A is the parameter of nonlinearity of the fluid. Comparisons between the model and experiment are made for a rectangular cavity driven in one-and twodimensional modes. Agreement is excellent except when there are degeneracies involving the predicted nonlinearly excited standing waves and other standing waves of the cavity. Small discrepancies appear to result from the coupling of energy from the nonlinearly excited standing wave into its degenerate neighbor. Subject Classification: 25.25; 55.20. liquid shear viscosity instantaneous and equilibrium densities of the flfiid velocity potential (angular) frequency at which the cavity is driven (angular) frequency of a resonance + 2 ß 1133
It is experimentally well established that the addition of minute amounts of long-chain macromolecules to water can significantly reduce the drag experienced by a submerged body. Although the mechanism responsible is at present unknown, it is reasonable to expect that this drag reduction might be accompanied by some degree of noise reduction. To check this hypothesis, spheres were dropped in a reverberant tank containing first water and then a 100-weight ppm solution of Polyox WSR-301 and the radiated sound measured. In water, the Reynolds numbers (based on the terminal speed) covered the region of either side of the critical Reynolds number, but only those spheres with Reynolds numbers at or above the critical value radiated sufficient energy to be detected above the background. In the polymer solution, where all spheres displayed considerable increase in speed, the sound radiated by all spheres was reduced to below background. This reduction amounted to as much as 15 dB. [This research was partially supported by Naval Ship Systems Command.]
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