An electrical signal described by the function u(x,t) is transmitted in the x direction. The signal satisfies the wave equation,

$\displaystyle u_{tt} - c^2u_{xx} = 0, x > 0, t > 0 $

subject to the conditions

$\displaystyle u(x,0) = q(x), u_t(x,0) = 0 $ for x > 0,

$\displaystyle u(0,t) = 0, $ for t > 0, where q(x) is a known function.

Find the solution for u(x,t).

So far I have done...

The product solution is:

$\displaystyle u(x,t) = \varphi(x)h(t) $

Plugging into boundary conditions gives:

$\displaystyle \varphi(0) = 0 $

Plugging the product solution into the DE and separating gives:

$\displaystyle \frac{\partial^2}{\partial t^2}(\varphi(x)h(t)) = c^2\frac{\partial^2}{\partial x^2}(\varphi(x)h(t)) $

$\displaystyle \varphi(x) \frac{d^2h}{dt^2}= c^2h(t)\frac{d^2 \varphi}{dx^2} $

$\displaystyle \frac{1}{c^2h}\frac{d^2h}{dt^2} = \frac{1}{\varphi}\frac{d^2\varphi}{dx^2} = -\lambda $

From separting variables we get 2 ODES:

$\displaystyle \frac{d^2h}{dt^2} + c^2\lambda h = 0 $

$\displaystyle \frac{d^2\varphi}{dx^2} + \lambda \varphi $

And then I'm really really stuck about how to do the rest, I think I need to get some eigenfunctions and eigenvalues, but then don't know how to get the solution u(x,t) from there...

Any help would be so appreciated, I need to know how to do loads fo these questions for my exam but I'm really struggling

Thanks in advance!