What is the meaning of the Helmholtz wave equation?

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:

1. Since we are not in time-domain anymore, does that mean the output will be a stationary wave?

2. 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?

3. 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!

• 1) Solution to Helmholtz equation corresponds to a single complex frequency. It is not necessarily a stationary (standing) wave. In general case it is a propagating and possibly also growing or decaying wave. 2) If the time domain it is a propagating wave with a periodic temporal part $exp(-i\omega t)$ then the solution of Helmholtz equation will show you the spatial part of it. 3) Physical quantities values are not complex, they are real. So taking the real part of a complex-values solution corresponds to physical observations. Complex numbers are used because they simplify the algebra. Jun 12, 2020 at 21:39
• Thank you for your reply @MaximUmansky, I have fully understood 1) and 3). But regarding 2), according to what I understood you are talking about a purely temporal wave $u(t) = exp(-i \omega t)$. Thus I don't see how Helmholtz eq. can show us the spatial part of u(t). Jun 12, 2020 at 23:02
• I'll give more details of my interpretation: When I think of a wave propagating I imagine the pressure gradient moving as time passes, i.e. for a given position where the pressure is positive, after a few moments it'll become negative, and so on. Now when I look to the Helmholtz eq. I can't see how the wave propagate because there's no time, and thus no movement. Looking at figure 5 of the referred work I don't understand why we have positive and negative pressure regions, because they are always changing in time-domain. Jun 13, 2020 at 1:04
• The solution is assumed to be of the form of a product of a spatial part and a temporal part, $f(x,t)=X(x) exp(-i \omega t)$ where $\omega$ is a complex frequency. So the time dependence is still there, actually. The Helmholtz equation solution yields the spatial part $X(x)$ but it is implied that the full solution is actually $X(x) exp(-i \omega t)$ where $\omega$ is also found from the Helmholtz equation because it is an eigenvalue problem, so solving it means you find both eigenfunctions and eigenvalues. Jun 13, 2020 at 1:33