Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2015 / Paper 1 / 1 / b / Solution

Let all velocity amplitudes refer to the same positive normal direction, from medium 1 into medium 2. For a right-going linear sound wave, its pressure amplitude is $p^{\prime }=Z{u}^{\prime }$ with acoustic impedance $Z=\rho c_s$; for a left-going wave it is $p^{\prime }=-Z{u}^{\prime }$. Continuity of normal velocity and pressure at the interface givesEliminating $u_t$ yields the normal-incidence acoustic velocity reflection coefficientThe common adiabatic index and mean particle mass $\bar{m}$ give $c_{s,j}=\sqrt{\gamma k_B{T}_j/\bar{m}}$, soA negative coefficient reverses the reflected velocity perturbation's phase in this fixed coordinate system. The reflected pressure coefficient has the opposite sign. Pressure balance also gives ${\rho }_1{T}_1={\rho }_2{T}_2$, so the impedance ratio can equivalently be written $Z_2/Z_1=\sqrt{{\rho }_2/{\rho }_1}$. These are normal-incidence, small-amplitude results without a surface mass or tension.