Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 40B / b / i / Solution

Let ${\varphi }_{+}$ and $p_{+}^{\prime }$ denote the fields in $y>0$, and ${\varphi }_{-}$ and $p_{-}^{\prime }$ those in $y<0$. The linearized kinematic boundary condition equates the normal fluid velocity on both faces to the membrane velocity:The linear acoustic pressure relation isPressure below pushes upward and pressure above pushes downward. The transverse force balance for a membrane with mass per area $m$ and tension $T$ is consequentlyor, in terms of the velocity potentials,