# Is the imaginary part needed in this problem?

Before jumping into my question, let me contextualize it.

I'm doing numerical simulations of a Helmholtz scattering problem

$$\Delta p + \kappa^2 p = 0\, .$$

The incident pressure wave $$p^{inc}$$ will be scattered by some obstacle on the boundary $$\Gamma_{obstacle}$$ such that $$p = p^{inc} + p^{sc}$$.

From the literature, an incident plane wave in the $$\mathbf{x}$$ direction is commonly described by $$p^{inc}(\mathbf{x}) = e^{-i \kappa \mathbf{x}}$$.

Due to some constraints of the Finite Element platform I'm using, I have to split $$e^{-i \kappa \mathbf{x}}$$ into its real and imaginary parts. That is $$\cos(\kappa \mathbf{x}) - i \sin(\kappa \mathbf{x})$$.

Well, according to this discussion, that means I'll have to split the pressure into its real and imaginary parts as well $$p_r + ip_i = p^{inc}_r + p^{sc}_r + i(p^{inc}_i + p^{sc}_i)$$ and rewrite the weak form of the problem.

My question is, since I'm only interested in the real part of the solution, would it be wrong to account only for the real part of the whole problem by making $$p^{inc}(\mathbf{x}) = \cos(\kappa \mathbf{x}) + 0i$$?

Can somebody explain me the implications of doing so, if any?

• Complex numbers are used only for simplifying the algebra, it is implied that the real part of a complex number is used for describing a physical quantity. Jul 26 '20 at 18:39
• @MaximUmansky, I think that amplitude and phase are both physical quantities. Jul 26 '20 at 18:54

## 3 Answers

The system of equations after discretization with the FEM can be written in real algebra as

$$\begin{bmatrix} A_{R} &-A_{I}\\ A_{I} &A_{R} \end{bmatrix} \begin{Bmatrix}p_R\\ p_I\end{Bmatrix} = \begin{Bmatrix}f_R\\ f_I\end{Bmatrix}\, ,$$

where the subindices $$R$$ and $$I$$ refer to the real and imaginary parts of the impedance matrix, pressure vector and source vector.

In general, you would have contributions from the imaginary parts of your incoming field in the impedance and load vector. This implies that the real part of the pressure vector depends on these imaginary parts. If you can guarantee that you have $$A_I = 0$$, I think that you could just use the real part.

• Spot on! I was imagining some kind of interdependence between the real and imaginary parts along these lines. As for now, I am doing some simulations accounting for different scenarios of the imaginary part and I'll present them in a short while. Thank you. Jul 26 '20 at 19:27
• From this answer, it looks to me $\exp(i k x -i \omega t)$ dependence is assumed here for all quantities, so the FM code solves for the spatial distribution of the complex amplitude - is this correct? So this FM solution is not a general solution of Maxwell's equations but specifically for modeling EM linear waves propagation in a medium? Jul 26 '20 at 21:05
• @MaximUmansky, not really. You could say that you have a harmonic behavior in time, and that's why you have the Helmholtz equation and not the wave equation. I don't think that it is not an electromagnetic wave but an acoustic wave. Jul 26 '20 at 21:41
• If you only impose Dirichlet or Neumann boundary conditions, then $A_I=0$ and the problem decomposes into two independent problems. The typical way the two would couple is if you have absorbing boundary conditions, which are often of the form $p = \alpha \partial p / \partial n$ with a complex-valued $\alpha$ -- in this case, the boundary condition couples real and imaginary parts, and $A_I\neq 0$. Jul 27 '20 at 19:37

I'd like to share some tests I did in order to see if there's any penalty by not taking the imaginary part into account.

Normally, the weak form of the Helmholtz equation is written as

$$\int_{\Omega}(\nabla p \cdot \nabla \overline{q} - \kappa ^2 p \overline{q})d \Omega = \int_{\Gamma}g \overline{q}ds$$

But when we account for the real ($$p_r$$) and imaginary ($$p_i$$) parts, it becomes

$$\int_{\Omega}(\nabla p_r \cdot \nabla \overline{q}_r + \nabla p_i \cdot \nabla \overline{q}_i + \nabla p_r \cdot \nabla \overline{q}_i + \nabla p_i \cdot \nabla \overline{q}_r - \kappa^2 p_r \overline{q}_r - \kappa^2 p_i \overline{q}_i - \kappa^2 p_r \overline{q}_i - \kappa^2 p_i \overline{q}_r)d\Omega = \int_{\Omega}(g \overline{q}_r + g \overline{q}_i)d\Omega$$

So I ran 3 simulations in a 2D square domain where the left boundary is subjected to the referred incident wave. In the first one I used the original weak form. In the second one I split $$p$$ and the domain into their real and imaginary parts, but kept $$0i$$. Finally, for the third one I took $$isin(\kappa x_1)$$ into account. The results are presented below, where I plot $$p$$ for the first simualtion and $$p_r$$ for the others.

Unless I made a mistake somewhere, the results are identical. So I believe I may proceed with the simpler simulation.

• What's the shape of the scatterer? I don't see any perturbation. Jul 27 '20 at 0:52
• Yes, you're right, I haven't put any scatterer for I wanted to see only the behavior of the wave. In reality, my final goal is to come up with a code to deal with the acoustic-elastic interaction on FEniCS (I presume you know me from FEniCS community), and not only the scattering. I find it easier to break down the problem into its main components: Helmholtz problem, Navier problem and perfectly-matched-layers. Then I should be able to assemble it all together. But sometimes I get stuck with FEniCS syntax and other times with the math behind the problem. Well, at least I'm learning a lot! Jul 27 '20 at 1:25
• I would add a scatterer, it does not has to be elastic, because I think that there maybe the results don't match. Learning is the way to go ;) Jul 27 '20 at 2:33

As pointed out by @nicoguaro, I did some more tests with a scatterer. The overall set up of these new simulations is the same as in my previous answer, as well as the conclusion. There must be some way of proving mathematically that $$A_I = 0$$, but unfortunately I don't have the time to investigate it right now.

I'll leave an image for further reference.