An electrical signal described by the function u(x,t) is transmitted in the x direction. The signal satisfies the wave equation,
subject to the conditions
for x > 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:
Plugging into boundary conditions gives:
Plugging the product solution into the DE and separating gives:
From separating variables we get 2 ODES:
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!