Weakly nonlinear waves and acoustic streaming produced by an oscillating rigid cylinder

Abstract

The weakly nonlinear propagation of waves radiated from a rigid cylinder executing harmonic translatory oscillations in a direction perpendicular to its axis into an ideal gas is studied theoretically, under the condition that an acoustic Reynolds number is sufficiently large. The entire propagation process, from generation of sound to the stage where the wave amplitude is saturated, is analyzed in the leading order approximation. It is shown that in the near-field acoustic streaming occurs around the cylinder in the next order. The compressibility in fluid together with the nonlinearity plays an essential role in this acoustic streaming. For the problem up to the shock formation a far-field equation is exactly solved and the subsequent evolution into the sawtooth wave is examined using the equal-areas rule. These multidimensional waves have the directional quality proportional to cos θ (θ is the angle measured from the direction of oscillation of the cylinder). At a great distance it appears that the saturation of the wave and the amplitude simultaneously becomes independent of cos θ there.