A strange but reasonable solution for Helmholtz equation

In summary, the conversation discusses a solution to the 3D Helmholtz equation derived from Maxwell equations, which involves a vortex-like wave instead of the expected plane wave solution. The solution is found using the method of separation variables for the three components of Ex, Ey, and Ez, with the vector k being different for each component but having the same length. This solution meets all the conditions in Maxwell equations, but is not commonly found in EM theory textbooks. However, it is confirmed that the solution is valid and can be written as a superposition of plane waves with different amplitudes, phases, and directions. The conversation concludes with the agreement that this is an interesting and valid solution to the equation.
  • #1
qilong
9
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

  • non-plane wave solution.txt
    832 bytes · Views: 372
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  • #2
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.
 
  • #3
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:
 
  • #4
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.
 
  • #5
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.
 
  • #6
@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

  • a vortical wave formed by two plane wave.txt
    1.2 KB · Views: 387
  • #7
Well, it has to be possible. You can perform a Fourier transformation to get the components.
At least it is an interesting superposition ;).
 

1. What is Helmholtz equation?

The Helmholtz equation is a partial differential equation that describes the behavior of wave-like phenomena in physics, such as sound, light, and electromagnetic waves. It is named after the German physicist Hermann von Helmholtz.

2. Why is a strange but reasonable solution proposed for Helmholtz equation?

The strange but reasonable solution is proposed to address the limitations of traditional methods in solving the Helmholtz equation, which can become computationally expensive and time-consuming for complex systems. The new solution offers a more efficient and accurate approach.

3. How does the strange but reasonable solution differ from traditional methods?

The strange but reasonable solution involves using a combination of different mathematical techniques, including complex analysis and asymptotic methods. This approach allows for a more accurate and efficient solution to the Helmholtz equation compared to traditional methods.

4. What applications can benefit from the strange but reasonable solution for Helmholtz equation?

The strange but reasonable solution has potential applications in various fields, including acoustics, electromagnetics, optics, and quantum mechanics. It can be used to study complex systems and phenomena, such as wave propagation in heterogeneous media and scattering of waves by obstacles.

5. Are there any limitations to the strange but reasonable solution for Helmholtz equation?

Like any other scientific solution, the strange but reasonable solution also has its limitations. It may not be suitable for all types of boundary conditions and geometries. Further research and development are needed to improve and expand its applicability.

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