Simplification about the reflection of waves at a discontinuity.

[lele44]

The following reffers to the reflection of waves at a discontinuity. (incident, reflected, and transmitted waves)

Homework Statement

(a) Simplify Aicos(k1x-wt)+Arcos(k1x+wt) = Atcos(k2x-wt) by eliminating common trig factors (recall cos(-θ)=cos(θ)), and dividing terms by Ai to express the resulting equation in terms of r=Ar/Ai and τ=At/Ai.
(b) Substitute Aicos(k1x-wt)+Arcos(k1x+wt) = Atcos(k2x-wt) into

, eliminate common trig factors (recall sin(-θ)=sin(θ)), divide terms by k1Ai, and express the resulting equation in terms of r and τ as before. (the ratio k2/k1 should also appear in your result, the refracting index)
(c) By treating your resulting equations in parts (a) and (b) as two equations in two unknowns r and τ, find separate expressions for r and τ involving ONLY the refractive index n=k2/k1.

Homework Equations

yi(x,t)=Aicos(k1x-wt)
yr(x,t)=Arcos(k1x+wt)
yt(x,t) = Atcos(k2x-wt)
sin(-θ)=sin(θ)
cos(-θ)=cos(θ)

The Attempt at a Solution

This was a problem about reflection of waves at a discontinuity. (incident, reflected, and transmitted waves). I solved everything up to this point, and re-wrote the question.

 

[jbsteele]

I'm stuck on part (a) - to simplify Aicos(k1x-wt)+Arcos(k1x+wt) = Atcos(k2x-wt) by eliminating common trig factors (recall cos(-θ)=cos(θ)), and dividing terms by Ai to express the resulting equation in terms of r=Ar/Ai and τ=At/Ai.My attempt:Aicos(k1x-wt)+Arcos(k1x+wt) = Atcos(k2x-wt)Using cos(-θ)=cos(θ):Aicos(k1x-wt)+Arcos(k1x-wt) = Atcos(k2x-wt)Divide both sides by Aicos(k1x-wt):1 + r = τ cos(k2x-wt)Where r = Ar/Ai and τ = At/Ai