The harmonic oscillator coupled to a sinodial external source

$$\frac{\partial^2 x(t)}{\partial t^2}+\omega\_0^2 x(t)=F\_0\sin(\omega\_\text{ext}\ t),$$

has the solution

$$x(t)=x(0)\cos(\omega\_0 t)+C \sin(\omega\_0 t) +\frac{F\_0}{\omega\_0^2-\omega\_\text{ext}^2}\sin(\omega\_\text{ext} t),$$

with a constant of integration $C$. So here we have a resonance for $\omega\_\text{ext}=\omega\_0$. 

If we consider the Klein-Gordon equation with non-vanishing coupling and with a source function $F$

$$\frac{\partial^2 \phi(t,x)}{\partial t^2}-c^2\nabla\_x^2\ \phi(t,x) +\omega\_0^2\ \phi(t,x)=F(s,x,t),$$

(one dimensional, three dimension, whatever you can deal with)

is there also an family of functions $F(s,t,x)$, (family w.r.t. the parameter s), such that we have a singularity at $s=\omega\_0$?

Maybe you go about similar as with the Helmholtz equation, except than you get a shifted coefficient $\omega\_0+k^2$.

I think that could be the case as a damping term $\delta\cdot x'(t)$ also destroys the resonance, at least for a sinusoidal source.