A strange but reasonable solution for Helmholtz equation

[qilong]

Hi guys,
I have a question when solving 3D Helmholtz equation derived from Maxwell equations. Normally I will get a plane wave solution. But when I used the method of separation variables for the three components Ex Ey and Ez, I found that the vector k in these three can be different as long as their lengths are the same. So Ex has a phase of vector(kx)*vector(r), Ey has a phase of vector(ky)*vector(r) and Ez has a phase of vector(kz)*vector(r), where |kx|= |ky|=|kz|=Constant. Besides this solution meets all the conditions in Maxwell equations(as least I have calculated). Attached is two MATLAB codes describing the normal plane wave solution and this vortex-like solution in 2D situation. I haven't found any material describing this wired solution in EM theory. All of them give out plane wave solutions. Is there anything wrong with this solution? THANKS ALL!

 

[mfb]

Did you check that solution with Maxwell's equation? In particular, did you calculate the magnetic field, and put that into the equations?
The electric field looks interesting.

 

[qilong]

yes. I have check the magnetic field in the 3D case. The result shows that there is no need that the k vectors in these 3 components are the same. It is easy to verify this. And still I don't know where I got a mistake. Does the plot I attached seem unreasonable? Now I am a little skeptical about all the textbooks. Maybe they have hidden this result for some reason...

 

[mfb]

Plain waves are just one class of solutions - all superpositions of them are solutions, too. Maybe it is possible to write your solution as sum of different plain waves with different amplitude, phase and direction.

 

[qilong]

hmm...But I believe that the superstition of some plane waves cannot form a wave like this. 'cause that the phase factor is in the power part of exp. Even if a superstition cannot behave like this.

 

[qilong]

@mfb, Thanks, I've got it. Two plane wave can form a vortex-like wave which satisfies Maxwell equation. I have thought that the plane wave solution was the unique form. Attached are two plane wave solutions and another one is the superstition of them. From the plot you can see that two plane wave make a vortex-like wave which is of course compatible with Maxwell equation. Thank you so much!

 

[mfb]

Well, it has to be possible. You can perform a Fourier transformation to get the components.
At least it is an interesting superposition ;).