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.

    \[ \frac{1}{Z_1}\left(A_i^2-A_r^2\right)=\frac{1}{Z_2}A_t^2 \]

Now divide by A_i^2

    \[ \frac{1}{Z_1}\left(1-\frac{A_r^2}{A_i^2}\right)=\frac{1}{Z_2}\frac{A_t^2}{A_i^2} \]

Now, multiply by Z_1 Z_2, and use the definitions of R and T in terms of the amplitude ratios

    \[ Z_2\left( 1-R^2 \right)=Z_1 T^2 \]

Now we will factor the quadratic term, and also use the relationship between R and Equation (3) to substitute for T

    \[ Z_2 (1-R) (1+R)=Z_1 (1+R)^2 \]

Divide through by 1+R

    \[ Z_2 (1-R) =Z_1 (1+R) \]

And solve for R

    \[ -(Z_2+Z_1)R=Z_1-Z_2 \]

    \[ R=\frac{Z_2-Z_1}{Z_1+Z_2} \]

We can find T by using Equation (3)

    \[ T=1+R=1+\frac{Z_2-Z_1}{Z_1+Z_2}=\frac{Z_1+Z_2}{Z_1+Z_2}+\frac{Z_2-Z_1}{Z_1+Z_2} \]

    \[ T=\frac{2 Z_2}{Z_1+Z_2} \]