# Homework Help: Harmonic Waves on a String

1. Jan 20, 2010

### LeakyFrog

1. The problem statement, all variables and given/known data
Two very long strings are tied together at the point x = 0. In the region x < 0, the wave speed is v1, while in the region x > 0, the speed is v2. A sinusoidal wave is incident on the knot from the left (x < 0); part of the wave is reflected and part is transmitted.

For x < 0, the displacement of the wave is described by...
y(x,t) = Asin(k1x - wt) + Bsin(k1x + wt)

For x > 0, the displacement of the wave is described by...
y(x,t) = Csin(k2x - wt)

where,
w/k1 = v1
w/k2 = v2

a) if we assume that both the wave function y and its first spatial derivative dy/dx must be continuous at x = 0, show that:
C/A = 2v2/(v1 + v2),
and that:
B/A = (v1 - v2)/(v1 + v2)

b) Show that
B2 + (v1/v2)C2 = A2

2. Relevant equations
y(x,t) = Asin(kx +- wt)
This shows that A, B, C are the amplitudes of their portions.

3. The attempt at a solution
I'm really unsure how to go about his problem. I am getting stuck on the idea of getting C/A or B/A because I don't really see how you can solve for them.