Two definitions of wave reflection coefficient

[namphcar22]

The waves book by A.P.French gives two characterizations of the reflection coefficient for a 1-D traveling wave encountering an interface between two media. On one hand, he writes
$$R = \frac{v_2 - v_1}{v_2 + v_1}$$
where $v_i$ are the wave speeds in the two media. Later on, he writes the reflection coefficient in terms of mechanical impedance:
$$R = \frac{Z_1 - Z_2}{Z_1 + Z_2}$$
where $Z_i$ denote the mechanical impedances for the media. My question is: are these two definitions always consistent? I two media have the same wave speed, do they automatically have the same impedance?

 

[Valerie Park]

No, the two definitions are not always consistent. While the wave speed may be the same in both media, the mechanical impedance can still be different. The mechanical impedance is determined by the medium's properties, such as its density and elasticity, which can be different even if the wave speed is the same.