A strange but reasonable solution for Helmholtz equation

Click For Summary

Discussion Overview

The discussion revolves around a solution to the 3D Helmholtz equation derived from Maxwell's equations, specifically exploring the possibility of different wave vector components for the electric field components Ex, Ey, and Ez while maintaining equal magnitudes. Participants examine the implications of this approach and its compatibility with established electromagnetic theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a solution where the wave vector components for Ex, Ey, and Ez can differ as long as their magnitudes are equal, questioning the uniqueness of plane wave solutions.
  • Another participant asks whether the proposed solution has been verified against Maxwell's equations, particularly regarding the magnetic field.
  • A participant claims to have checked the magnetic field and asserts that differing k vectors do not violate Maxwell's equations, expressing skepticism about traditional textbooks.
  • One participant suggests that the proposed solution could be expressed as a superposition of plane waves with varying amplitudes, phases, and directions.
  • Another participant counters that such a superposition may not yield the proposed vortex-like wave due to the nature of the phase factors involved.
  • A later reply acknowledges that two plane waves can indeed form a vortex-like wave that satisfies Maxwell's equations, indicating a shift in understanding.
  • One participant mentions the possibility of using Fourier transformation to analyze the components of the wave further.

Areas of Agreement / Disagreement

Participants express differing views on the validity and implications of the proposed solution, with some supporting the idea of superpositions while others remain skeptical about the uniqueness of plane wave solutions. The discussion does not reach a consensus.

Contextual Notes

Participants reference the need for verification against Maxwell's equations and the potential limitations of existing literature, indicating that assumptions about wave solutions may not be fully explored.

qilong
Messages
9
Reaction score
0
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!
 

Attachments

Physics news on Phys.org
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.
 
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...:cry:
 
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.
 
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.
 
@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!
 

Attachments

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

Similar threads

Undergrad EM equations - am I missing something?
  • · Replies 4 ·
Replies
4
Views
1K
Graduate Maxwell’s Equations Wave Solutions
  • · Replies 28 ·
Replies
28
Views
5K
Graduate Electric and magnetic field lines in a plane wave of finite extent
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Instantaneous electric field solved by extended electrodynamics?
  • · Replies 1 ·
Replies
1
Views
907
Graduate Angularly Propagating Modes In a Cylindrical Waveguide
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Are spherical transverse waves exact solutions to Maxwell's equations?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Vector field and Helmholtz Theorem
  • · Replies 20 ·
Replies
20
Views
4K
Graduate Solutions of wave equation but not Maxwell equations
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Voltage and current waves in a transmission line
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Maxwell's equations and time-harmonic solutions
  • · Replies 4 ·
Replies
4
Views
1K