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 \(Z_1\) and \(Z_2\). (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 \(\dfrac{p_1^+}{Z_1}\) and \(-\dfrac{p_1^-}{Z_1}\). (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: \[\begin{equation} p_1^+ + p_1^- = p_2^+ + p_2^- \end{equation} \label{1} \] and \[\begin{equation} \dfrac{p_1^+ - p_1^-}{Z_1} = \dfrac{p_2^+ - p_2^-}{Z_2} \end{equation} \label{2}\]

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 \(Z_2\). Instead, it relates to the load impedance \(Z_{load}\), e.g., the input impedance of the system to the right of the discontinuity point. So we will replace the eq. \(\eqref{1}\) and eq. \(\eqref{2}\) by \[\begin{equation} p_1^+ + p_1^- = p_2 \end{equation} \label{3}\] and \[\begin{equation} \dfrac{p_1^+ - p_1^-}{Z_1} = \dfrac{p_2}{Z_{load}} \end{equation} \label{4}\]. This way, we get the reflection coefficient \[\begin{equation} R = \dfrac{Z_{load}-Z_1}{Z_{load}+Z_1} \end{equation} \label{5}\]

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 \(Z_2\). Then we get \[\begin{equation} R = \dfrac{Z_2-Z_1}{Z_2+Z_1} \end{equation} \label{6}\]