Finite and Small Amplitude Underwater Gas Bubble Oscillations Compared

Abstract

Bubble oscillation is well described by the nonlinear Keller-Kolodner equation. Although compressibility effects, which strongly influence the motion, are considered, analysis starts with incompressible approximation. The key parameter is p1/p0, the ambient/bubble pressure ratio. This ranges from zero to unity; near the lower limit finite amplitude pulsations occur; near the upper, simple harmonic motion ensues. The equation determining the maximum and minimum bubble radii, with p1/p0 as parameter, is graphed; these endpoints coalesce at a level where p1/p0 = 1. Further, taking compressibility into account, at each depth the end-points also converge to an equilibrium radius. T̄ has a simple power law dependence on p1/p0; this is generally not true for the finite amplitude period T1. The curves for T̄ and T1, as calculated from the Willis formula, cross; hence the actual T1 vs depth may be inferred. The shape of the bubble radius vs time curve changes from pulsation to simple oscillation with depth; this is asymptotically correct at any level, since T1 → T̄.