Reflection and Transmission Coefficients

Susan Hautala

68 Reflection and Transmission Coefficients

Here is the full derivation for reflection, $R$ , and transmission, $T$ , coefficients and how they relate to one another.

$R=\frac{A_{Reflected}}{A_{Incident}}$

$T=\frac{A_{Transmitted}}{A_{Incident}}$

We write the wave pressure as

$P=Acos(kx-\omega t+\phi)$

We start with the two equations derived in the main text. The first expresses no net force on the boundary.

(1) $\begin{equation*} A_{Incident}+A_{Reflected}=A_{Transmitted} \end{equation*}$

The second expresses energy conservation.

(2) $\begin{equation*} \frac{A_{Incident}^2}{Z_1}=\frac{A_{Reflected}^2}{Z_1}+\frac{A_{Transmitted}^2}{Z_2} \end{equation*}$

First, if we divide the Equation (1) though by $A_{Incident}$ and use the definitions of $R$ and $T$ in terms of the amplitude ratios, we find an equation relating the two coefficients

(3) $\begin{equation*} 1+R=T \end{equation*}$

From here, we will shorten the subscripts, $A_i=A_{Incident}$ , $A_t=A_{Transmitted}$ and $A_r=A_{Reflected}$ . Next, rewrite equation (2) putting terms involving $Z_1$ on the left, and terms involving $Z_2$ on the right.