# Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e.,

$$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) & =Ae^{\frac{(x-x_{0})2}{\sigma^{2}}}\\ u_{t}(0,t) & =u_{x}(0,t)\\ u_{t}(1,t) & =-u_{x}(1,t) \end{cases}$$ I need to get the exact solution, to verify a program that I'm making.

what I know is that :$$u(x,t)=f(x-t)+g(x-t)$$

and I should proceed defining: $$u_{t}(0,x)=\phi(x)$$

and make $\phi$ in such way that the boundary condition are met.

I have try this but I'm really stuck :(.

Some help or reference will be very nice.

Thanks

• The boundary conditions you are using, are called "Transparent boundary condition". In fact you'r equation is the tipical IVP wave equation. Thus I dont really know how to deduce those "boundary condition". – Porufes Feb 28 '15 at 15:32