I am trying to build understanding on the Helmholtz wave equation $\Delta p + k^2 p = 0$, where $p$ is the deviation from ambient pressure and $k$ the wave number, in order to use it in numerical simulations. However, I'm not sure how to interpret the results. Take as an example the figure 5 on this work
More precisely, my questions are:
Since we are not in time-domain anymore, does that mean the output will be a stationary wave?
Is it possible to draw an analogy between the frequency-domain form and its time-domain form? For example, I know what happens when a propagating wave hits an obstacle and is scattered in time-domain, but in frequency-domain will we have the same behaviour?
Does the real and imaginary parts have a physical meaning?
I'd be very thankful if someone could answer at least part of my questions. Thanks a lot!