Here is the full derivation for reflection, , and transmission, , coefficients and how they relate to one another.
We write the wave pressure as
We start with the two equations derived in the main text. The first expresses no net force on the boundary.
The second expresses energy conservation.
First, if we divide the Equation (1) though by and use the definitions of and in terms of the amplitude ratios, we find an equation relating the two coefficients
From here, we will shorten the subscripts, , and . Next, rewrite equation (2) putting terms involving on the left, and terms involving on the right.
Now divide by
Now, multiply by , and use the definitions of and in terms of the amplitude ratios
Now we will factor the quadratic term, and also use the relationship between and Equation (3) to substitute for
Divide through by
And solve for R
We can find by using Equation (3)