Trying to solve a PDE problem consisting of these equations:
$$ u_{tt} - c^2 u_{xx} = 0 \quad \quad(0 < x < L, t > 0)$$ $$u(0,t), \quad u(L,t)=Bsin(\omega_{0}t) \quad (t >0)$$ $$u(x,0) = 0, \quad u_{t}(x,0) = 0 \quad (0 < x < L)$$
$C,L,B,\omega_{0} > 0$ are constants
I know that the general solution to the wave equation is $$u(x,t) = \phi(x+ct)+\psi(x-ct)$$
And I have d'Alembert's forumla. The problem is that since these initial values are on the standard form, d'Alembert's should apply, right? But $u(x,0) = u_{t}(x,0) = 0$, giving that u(x,t) = 0 as well, which cannot be right. I can't even get to the point where the second row of I.V is relevant (u(0,t) and u(L,t)).
Is there something fundamental I am missing with the wave equation and d'Alembert? Is there a reason why d'Alembert would not apply here? Would appreciate any help on those questions so that I can try and actually solve the question by myself.
The d'Alembert formula solves the Cauchy problem for wave equation, but you have first mixed problem, and you should use another methods such as Laplace transform or separation of variables. You can also build a solution using general form of solution.
In this problem the d'Alembert formula, which implies $u \equiv 0$, correct only in triangle $(0,0)$ -- $(L, 0)$ -- $(L/2, L/(2a))$, because you have the Cauchy problem in this triangle.