# right-running waves

Can you help me please? The problem is:

1.Solve the wave equation in finite interval with Dirichlet boundary condition at he right and Neumann boundary condition at the left .

2.Choose the initial conditions for right-running waves.

3.Show the phase difference of wave reflection at the boundaries.

I solved the 1'st question by assuming $u_{tt}=c^{2}u_{xx}$

for $0<x<l$ with the B.C. $u_x(0,t)=0$; $u(l,t)=0$

$u(x,t)=\sum_{n=0}^{\infty }\left [ A_n \cos \frac{(n+\frac{1}{2})\pi ct}{l}+B_n \sin \frac{(n+\frac{1}{2})\pi ct}{l} \right ]\cos \frac{(n+\frac{1}{2})\pi x}{l}$

But i stuck with part 2. and 3. of the question.

What conditions i should to choose in order to get right-running waves? Thanks for you help!

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Left- and right-running waves take the form $u(x,t)=g(x+ct)$ and $u(x,t)=g(x-ct)$, respectively. Differentiating a right-running wave with respect to $t$ and $x$ yields
$$\frac{\partial u}{\partial t}=-cg'(x-ct)=-c\frac{\partial u}{\partial x}\;,$$
so to specify initial conditions for a right-running wave, you can specify $u(x,0)=f(x)$ with any $f$ and then require $u_t(x,0)=-cf'(x)$.