i think that's not related to the seismic reflection.
I want to know if there is any proofs of
"how do the wave speeds compare for the two strings?
From high speed to low speed (low density to high density)"
located at
http://www.kettering.edu/~drussell/Demos/reflect/reflect.html [Broken]
You write expressions for the waves at the interface (incident, reflected and transmitted) with unknown phases, and use the boundary conditions to find the relation between the different phases.
I don't have a reference off hand, but most textbooks on acoustics, vibration transmission, etc. should work through details of the maths.