# Semi-infinite wave equation

1. Aug 26, 2010

### the_godfather

1. The problem statement, all variables and given/known data

solve the wave equation (dx^2 -dt^2)$$\Phi$$ = 0 on the semi-infinite line x<=0 with boundary conditions $$\Phi$$ at x=0 = 0 and initial conditions $$\Phi$$ at t=0 = tanh(x)

2. Relevant equations

solution of the wave equation is of the form $$\Phi$$ = f(x-t) + g(x+t).

$$\Phi$$ at t=0 = tanh(x)
tanh(x) is an odd function

3. The attempt at a solution

i know that $$\Phi$$ at t=0 = tanh(x) which i will call A
i know that $$\Phi$$ at x=0 is 0, which i will call B

am i correct in thinking that i can simply add the two equations together to get 2$$\Phi$$ = f(x-t) + g(x+t) = A + B
therefore giving $$\Phi$$ = [tanh(x+t) tanh(x-t)]/2
also what happens if tanh(x) is not an odd function? how can i turn it into an odd function?