Reflection and Transmission at a Boundary
The wave equation is a second order differential equation, and this implies that the the wave function and its first derivative are both continuous across a boundary.
If we assume the wavefunction is described bythen in region I below the wave is described bysince the wave undergoes partial reflection at the boundary atand in region II it is described by
The first of these implies thatatwhich implies thathence(1)
The second implies(2)
givesso the transmission coefficient is
gives so the reflection coefficient is