# Thread: boundary & initial conditions for wave equation

1. ## boundary & initial conditions for wave equation

Fro the 1D wave equation where the string is clamped at x=0 and x=L

u(x,0) = 0

the initial velocities are given by u'(x,0) = 2x/L for 0 < x < L/2, u'(x,0) = 2(L-x)/L, L/2 < x < L (derivatives wrt t)

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So the clamp conditions mean that u(0,t) = 0 = u(L,t).

We could integrate the initial velocities to give something, but I'm not sure how that translates to a solution. I'm also confused because if you integrate the given expressions, you'll get something that depends on t, but if we integrate u'(x,0) we'll get u(x,0), so it doesn't seem to make sense to have a t term..

2. Originally Posted by harbottle
Fro the 1D wave equation where the string is clamped at x=0 and x=L

u(x,0) = 0

the initial velocities are given by u'(x,0) = 2x/L for 0 < x < L/2, u'(x,0) = 2(L-x)/L, L/2 < x < L (derivatives wrt t)

---

So the clamp conditions mean that u(0,t) = 0 = u(L,t).

We could integrate the initial velocities to give something, but I'm not sure how that translates to a solution. I'm also confused because if you integrate the given expressions, you'll get something that depends on t, but if we integrate u'(x,0) we'll get u(x,0), so it doesn't seem to make sense to have a t term..
For this problem, you'll need to solve the 1D wave equation

$u_{tt} = c^2 u_{xx},\; 0 < x < L,\; t > 0$ subject to your IC's and BC's.

3. I know that the general solution is F(x+ct) + G(x-ct), where the functions F and G are defined by the ICs and BCs.

I do not know how to find F and G.