Solving the Wave Equation in semi-infinite domain with easy ICs

[Gengar]

Hi, so the problem is this:

I am trying to solve (analytically) the wave equation with c=1:

$$u_{xx}=u_{tt}$$

on x,t>0 given the initial conditions

$$u(x,0)=u_{t}(x,0)=0, u(0,t)=sin(wt)$$

I know how to solve on semi-infinite domains for quite a few cases using Green's Functions, Fourier Transforms, D'Alembert's solution and separation of variables. But I keep getting u=0 with these familiar methods due to the initial conditions of u being 0 and unmoving at t=0.

I feel like this is easier than I'm making it! Anyway, any help would be appreciated!

 

[Gengar]

Yeh... After solving numerically I realized that I just hadn't thought about the fact that the waves must propagate at speed 1 and u=0 for all x>t. So a quick bit of algebra gives:

$$u(x,t)=H(t-x)sin(w(t-x))$$

simples