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**1. Homework Statement**

Okay, I've got a question that's been bugging me for the longest time.

I've got a string attached to a wall at one end (the other end is free to move, though) and it has a sinusoidal wave traveling to the right and hits the wall at x = L and reflects. I'm supposed to prove that the sum of the leftward and rightward waves are

yR(x,t) + yL(x,t) = 2Asin(k(x - L))cos(kvt + phi)

which I did (unless there's a mistake somewhere), which makes a nice standing wave. but now they want me to solve for phi (the phase shift) in terms of k, L, and v. No numbers are given, it's all variables.

**2. Homework Equations**

yR(x,t) + yL(x,t) = 2Asin(k(x - L))cos(kvt + phi) <--- equation of the standing wave

yR(x,t) = Asin(k((x - L) - vt) - phi) <--- rightward wave equation (I hope)

yL(x,t) = Asin(k((x - L) + vt) + phi) <--- leftward wave equation (I hope)

k = 2pi / lambda

v = lambda / T

k = omega / v

k = wave number

v = velocity of wave

lambda = wavelength

phi = phase shift

y = height of section of wave at time t

T = period

**3. The Attempt at a Solution**

Okay, I know that at t = 0, parts of the wave actually hit the maximum height of the wave (2A). I found out that the parts of the rope hit this height are at x = L - n*lambda/4 = L - n*pi/2k, but putting this into the formula just gets rid of L completely, which is probably something I don't want to do. I did find out that phi = -kvt (set x to L - n*2pi/k and solve for phi), but I need L, not t.

I tried substituting x = 0 and t = 0 right away, but the problem is I don't know any initial values and just get y = 2Asin(-kL)cos(phi). This leads to cos(phi) = y(0,0)/2Asin(-kL), which I dunno helps or not. Besides, setting t=0 gets rid of the v term, which I think I need.

Differentiation gets me nowhere, so far. I tried to differentiate before adding and adding before differentiating and seeing if I can get an equation out of that, to no avail. Doing dy/dt brings out the v, but I dunno what to do with that (finding the max height means setting dy/dt to 0, which just gets rid of the constant coefficients, like v).

I'm really stuck and have been looking at this for about two days. Help!