Redo the computation of radiation reaction for a radially oscillating ball immersed in a fluid without imposing the slow-motion assumption and approximation. Thereby obtain the following coupled equations for the radial displacement ξ(t) of the ball’s surface and the function Φ(t) ≡ a⁻²f (t − ∈a/C), whereψ = r⁻¹f (t − ∈r/C) is the sound-wave field:

Show that in the slow-motion regime, this equation of motion has two weakly damped solutions of the same form as we derived using the slow-motion approximation [Eq. (16.83)], and one rapidly damped solution: ξ ∝ exp(−∈κt/τ). Burke (1970) shows that the runaway solution (16.84) obtained using the slow-motion approximation is caused by that approximation’s futile attempt to reproduce this genuine, rapidly damped solution.

Equation 16.83 and 16.84.

Transcribed Image Text:

²+² = K₂ = − − € (a/C). (16.86)