Right then, setting the integrand to zero:
e^{-i\omega t} \left( \nabla^2 \tilde{\psi}\left( x,y,z,\omega \right) + c^{-2} \omega^2 \tilde{\psi} \left( x,y,z,\omega \right) \right) = 0
Which gives
\nabla^2\tilde{\psi}+\frac{\omega^2 \tilde{\psi}}{c^2}=0
which is...the Helmholtz equation...