Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 16C / b / iv / Solution

Per unit depth, the bubble volume is its area $V=\pi a^2$. At equilibrium its pressure equals $p_{\infty }$, so $p_{0}V=p_{\infty }\pi a_0^2$. HenceContinuity of fluid pressure at the interface gives $p(a,t)=p_0(t)$. Consequently