How to impose the boundary conditions on $u(x,t)=\int_{-\infty}^{\infty} dk\left[A(k)e^{ik(x-ct)}+B(k)e^{ik(x+ct)}\right]$?

Question

Using the method of Fourier transform, the solution of one-dimensional wave equation $$\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)u(x,t)=0$$ turns out to be (see this) $$u(x,t)=\int_{-\infty}^{\infty} dk\left[A(k)e^{ik(x-ct)}+B(k)e^{ik(x+ct)}\right].$$ Now if we are given a set of boundary conditions (i) $u(0,t)=u(L,t)=0$ and $u(a,0)=b$ and $\frac{\partial u}{\partial t}(x,0)=0$, how do we impose these efficiently? In particular, how to show the that the $k$-values become discrete?