Consider a longitudinal elastic wave incident normally on the boundary between two media, labeled 1 and 2. By matching the displacement and the normal component of stress at the boundary, show that the ratio of the transmitted wave amplitude to the incident amplitude is given by

where Z_{1, 2} = [ρ_{1, 2}(K_{1, 2} + 4μ_{1, 2}/3)]^1/2 is known as the acoustic impedance. (The impedance is independent of frequency and is just a characteristic of the material.) Likewise, evaluate the amplitude reflection coefficient, and verify that wave energy flux is conserved.

t= 2Z Z + Z