The Non-Linear Interaction of a Plane Progressive Wave with a Small Sphere

Abstract

Under certain circumstances an acoustic wave exerts an average force (on particles) several orders of magnitude greater than the force arising from radiation pressure. Preliminary measurements of the force generated in a wave containing second harmonic distortion were reported previously. The force was 11 orders of magnitude greater than that caused by radiation pressure. The dependence of this force has been evaluated from Oseen's relation for the drag on a sphere. If the sound particle velocity is u = u0[sinωt + f sin(2ωt + φ)], the average force is FAv = − 3r2ρ0u02f sin φ to within 10 percent provided f < 0.4; but FAv = − 2.05r2ρ0u02 sin φ if f = 1; and FAv→ − 2.25r2ρ0u02 sin φ as f→ ∞. Here r is the radius of the sphere, small compared with wavelength; ρ0 is the density of the medium; and f is the fraction of second harmonic. The sphere is supposed not to follow the first order motion of the medium. The previously reported experiment, for which f = 1 and φ = ±π/2 is found to agree well with the theory. Another interesting case arises when a d.c. velocity is superposed on a sinusoidal wave. If u = u0[1 + f sinωt], then FAv = (7.07 + 3.52f2)r2ρ0u02 provided f < 1: If u = u0[sinωt + f], then FAv = 2.25[(1 + 2f2) sin−1f + 3f(1 − f2)12]r2ρ0u02 provided f < 1. In addition to the usual viscous and thermal boundary losses, other losses at the sphere will occur either at large acoustic amplitudes (non-linear absorption) or at small amplitudes provided the sphere drifts through the medium (differential absorption). The non-linear cross section based on the Oseen force, σNL = 9r2u0/c, is directly proportional to the ratio of the sound particle velocity to the velocity of sound c. The differential cross section, σD = 14.1r2udcc−1 | cos(k, udc) |, is proportional to the ratio of drift velocity udc to sonic velocity and to the magnitude of the cosine of the angle between the drift velocity and the propagation vector k. Rough computations indicate that the absorption cross section in the direction of fall of a 1-mm radius rain drop will be two orders of magnitude greater than the viscous cross section of the drop.