Consider the standard wave equation $c^{2}u_{xx}=u_{tt}$. Suppose that we are given $u(x,0)=0, u_{t}(x,0)=0$ for $x \geq 0$ and $u(0,t)=b(t)$ for $t \geq 0$. Find a solution to the boundary value problem.
I first assumed that $b(0)=0, b'(0)=0$ for consistency. Then, as the standard wave equation has a solution of the form $u(x,t)=f(x+ct)+g(x-ct)$ for $f, g$ functions of one variable, I obtain that (we can replace $x$ by a dummy variable $s$ as well):
$f(x)+g(x)=0 \quad cf'(x)-cg'(x)=0$
I derived that $f$ must be the constant function from the above equation, however, I found that if $g(-x)=f(x)$, then the two equations above are as well satisfied. What would be the issue in this case? I'm not sure on how to derive a non-trivial solution to this PDE. On the other hand, must this solution be necessarily unique? I'm very confused. Thanks for all the help!
The equations $f(x)+g(x)=f'(x)-g'(x)=0$ hold for $x\ge0$. This gives you that $f$ and $g$ are constant (that can be taken to be $0$ without loss of generality) for $x\ge0$. To find the solution you still have to find $g(x)$ for $x<0$ (the value of $f$ for $x<0$ is irrelevant, since $x+c\,t\ge0$.) For this you use the boundary condition: $$ f(c\,t)+g(-c\,t)=b(t),\quad t>0. $$