Acoustic Wave Produced in a Conical Horn by Stopping a Converging Steady Flow Instantaneously

Abstract

A rigid, infinite, conical horn containing a converging, steady flow of water is considered. The flow velocity is u=−u0 at the horn throat r=r0. The throat is open to empty the water. At time t=0, the opening is suddenly closed. The abrupt stoppage of the water generates a strong compressional wave, which propagates in the direction of the horn's mouth. A description of the wave, with finite-amplitude effects included, is desired. In the linear approximation, the pressure waveform is headed by a sharp jump of magnitude ρ0C0u0r0/r. The head appears at the point r=r0+c0t and is followed by an exponentially decreasing tail. When finite-amplitude effects are taken into account, it is found that linear theory underestimates the propagation speed of the pulse and overestimates its amplitude. The true magnitude of the pressure jump is the small-signal value multiplied by the factor 2/[1+(1+2βε ln r/ro)12], where ε=u0c0 and β ≐ 4. Therefore, if attenuation caused by nonlinear effects is to be kept negligible, ε must be chosen small and r0 large. [Work supported by General Dynamics/Electronics, Rochester, N.Y.]