I got suddenly confused about how the reflection coefficient is computed, and it took me a whole night to figure out my mistake.

Consider such a case, when a wave is propagating, there is an abrupt discontinuity, which separates two different medium M1 and M2 to the left and right of the discontinuity point, respectively. The two medium has two different characteristic impedance Z1 and Z2. (It could also be the same media with different cross-section areas in a tube).

The pressure in M1 is decomposed into a right-going wave p+1 and a left-going wave p−1, which correspond to incident and reflection, respectively. So the right- and left-going volume flow rate are p+1Z1 and −p−1Z1. (the "−" sign means the left-going traveling direction)

Assuming the continuity of the pressure and the conservation of the volume flow at the boundary, we have: p+1+p−1=p+2+p−2(1) and p+1−p−1Z1=p+2−p−2Z2(2)

From here the problem comes. I try to compute the reflection coefficient from the above two equations, but of course, I can't get anything. This is because the reflection coefficient is not related to Z2. Instead, it relates to the load impedance Zload, e.g., the input impedance of the system to the right of the discontinuity point. So we will replace the eq. (1) and eq. (2) by p+1+p−1=p2(3) and p+1−p−1Z1=p2Zload(4). This way, we get the reflection coefficient R=Zload−Z1Zload+Z1(5)

Wait! But why I always see the reflection coefficient with two different characteristic impedances? The answer is, to calculate the reflection coefficient based on characteristic impedance, it is assumed that p−2=0, indicating an infinite length of M2 with no reflection or right-going wave. This way the input impedance of M2 is simply its characteristic impedance Z2. Then we get R=Z2−Z1Z2+Z1(6)