I'm trying to understand what a correct procedure is for solving the following wave equation on a semi-infinite plane $-\infty < x < \infty$, $0 \le z < \infty$:
$$\nabla^2 \Psi(x,z,t) - \frac{1}{c^2}\frac{\partial^2 \Psi(x,z,t)}{\partial t^2}=f(z,t)H(t)\sin (q_0x)$$ $$\frac{\partial^2 \Psi(x,0,t)}{\partial x\partial z} = 0$$ $$\frac{\partial^2 \Psi(x,0,t)}{\partial x^2}-\frac{\partial^2 \Psi(x,0,t)}{\partial z^2}=0$$ $$\Psi(x,z,0)=\frac{\partial \Psi(x,z,0)}{\partial t}=0$$
where $H(t)$ is a Heaviside step function.
I started by separating the variables which leads to the solution:
$$\Psi(x,z,t)=T(t)(A\sin (kz)+B\cos(kz))(C\sin(qx)+D\cos(qx))$$
The first boundary conditions leads to $A=0$ and the second to $k=q$. To deal with the inhomogeneous part, one would generally do the expansion
$$\int\int T_{qk}(t)\cos(kz)(C\sin(qx)+D\cos(qx))dqdk = \int\int (C_s\sin(qx)+C_c\cos(qx))\cos(kz)$$
where $C_s$ and $C_c$ are the sine and cosine series coefficients of the source term. Now I'm struggling how to handle the fact that $k=q$. Moreover, I think that the inhomogeneous part of the equation implies that $q=q_0$, which if true, makes things much worse, since then we can't do any series expansion at all. I'm assuming I made an error somewhere in my reasoning and I would appreciate some help trying to figure this out.
Additionally, I can actually find the particular solution using the Fourier transform technique. It is not possible to get the homogeneous solution like that since I think its inverse Fourier transform is not analytically solvable. Is there any way knowing the particular solution can help me figure out the complete solution?