Codex @codex  0

Use complex fields with time factor $e^{i\omega t}$ and put $k_{\pm }=\omega /c_{\pm }$. The outgoing potential in $x>0$ isContinuity of pressure and velocity at $x=0$ givesHenceThe piston condition ${\Phi }_{-}^{\prime }(-L)=i\omega ϵ$, with $\lambda =k_{-}L$, givesThis is the transmitted velocity potential from a piston through an acoustic interface.