# The plane wave decomposition is mathematically universal?

1. Sep 7, 2011

### keji8341

The plane wave decomposition is mathematically universal???

1. My questions is: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves? Note: r=0 is a singularity.

This is actually the potential function produced by an ideal radiation electric dipole. [J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), Chapter 9, p. 410, Eq. (9.16).]

(i) A spherical-wave decomposition of a plane wave is presented in textbook; for example, J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), p. 471, Eq. (10.44) in Chapter 10.

(ii) The converse: A plane-wave decomposition of spherical waves is given by MacPhie and Ke-Li Wu, “A Plane Wave Expansion of Spherical Wave Functions for Modal Analysis of Guided Wave Structures and Scatterers”, IEEE Trans. Antennas and Propagation 51, 2801 (2003). Note: The spherical waves are analytical at r=0.

2. Where can I find the conclusions:
(a) The plane wave decomposition is mathematically universal.
(b) Any spherical wave may be decomposed in a plane wave basis.
Are they math theorems?

Thanks a lot.

2. Sep 7, 2011

### BruceW

Re: The plane wave decomposition is mathematically universal???

I doubt that any mathematical function can be expressed as a linear sum of plane waves - there are a lot of weird functions out there.

If we're talking about electromagnetic waves only, then according to wikipedia: "In linear uniform media, a wave solution can be expressed as a superposition of plane waves. This approach is known as the Angular spectrum method. The form of the planewave solution is actually a general consequence of translational symmetry."

So I am guessing that if the media is not uniform, then you can't always express the electromagnetic field as a sum of plane waves.

3. Sep 7, 2011

### kith

Re: The plane wave decomposition is mathematically universal???

The decomposition of a function in plane waves is just the Fourier transform. So every function for which the fourier transform exists, is decomposable in plane waves.

4. Sep 7, 2011

### chrisbaird

Re: The plane wave decomposition is mathematically universal???

Yes, I believe any physical wave (in other words, finite and continuous) can be represented by a sum of plane waves (including spherical waves). The question really comes down to a mathematical one about Fourier series representations of functions: for what type of functions does its Fourier series representation converge? I believe the answer is: functions that are continuous and finite, which includes all physical electromagnetic waves.

5. Sep 7, 2011

### BruceW

Re: The plane wave decomposition is mathematically universal???

I think it depends on how we are defining plane waves. In EM, plane waves usually mean something like: $e^{if(r,t)}$ where f is a real number.
But the fourier transforms give a sum of plane waves where f can be a complex number.

6. Sep 7, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves?

Physically speaking, the answer is “no”.
Mathematically speaking, the answer is “yes”.
Why? Here are my explanations.

There are two kinds of Fourier transforms.

1 “Sum of real plane waves”. 1D-example: f(x)=Inte{F(k)*exp(-ikx)*dk}, where F(k)*exp(-ikx)*dk is real plane wave component, with k is real. The integration is carried out from –infinity<k<+infinity, and the integral is convergent.

2 “Math correspondence”. 1D-example: f(x)=Inte{F(k)*exp(-ikx)*dk}, where F(k)*exp(-ikx)*dk is NOT a real plane wave, because k is set to be complex to make the integral converge. The integration is carried out in a complex plane by designating a contour for poles. Such Fourier transform is usually used to solve differential equations.

Therefore, "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" CANNOT represented as a sum of REAL uniform plane waves, because its Fourier integration must be carried out in a complex plane by designation a contour for poles, and its Fourier transform has no physical meaning, just a kind of math correspondence.

7. Sep 8, 2011

### vanhees71

Re: The plane wave decomposition is mathematically universal???

The plane wave can be expanded in terms of spherical waves and vice versa. The expansion is convergent in the sense of generalized functions (distributions).

8. Sep 8, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

1. In the plane-wave factor expi(wt-k.x), if w and k is complex, then the plane wave decays or grows exponentally with time and position. In free space, such a plane wave is not consistent with energy conservation law in the sense of classical electrodynamics.

2. In the sense of quantum mechanics, when w is complex, the Planck constant should be proportional to the conjugate complex of w, if E=hbar*w is real..., ha, a lot of new results...

Of course, for many theories, it is not required for every intermediate math operation to have physical meaning, especially in quantum mechanics. Assigning a physical explanation is just for being easy to remenber sometimes.

Last edited: Sep 8, 2011
9. Sep 8, 2011

### Staff: Mentor

Re: The plane wave decomposition is mathematically universal???

You are confusing the Laplace transform and the Fourier transform. Also, it would be helpful if you would stop double posting. Decide if you want to carry on this conversation here or in the relativity forum.

10. Sep 8, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

No, I am not confusing, I am talking about Fourier transform, never talking about Laplace transform before. Probably you misunderstood something.

Because this is an electromagnetic problem, maybe this topic is more suitable here according to the forum classifications. OK, I'll post here.

Here is an example to carry out Fourier integartion (inverse Fourier transform) in the complex plane by designating a contour for poles: http://www.bnl.gov/atf/exp/Dielectric/2-Theory_of_Wakes.pdf [Broken]
on p. 1270, Fig. 1, corresponding to the integral in Eq. (2.22).

To help reading, let me give some explanations.
The above paper [ "Theory of wakefields in a dielectric-lined waveguide" Phys. Rev. E 62, 1266–1283 (2000)]
uses Fourier-transform approach to solve wave equation excited by a moving electron in a dielectric-loaded cylindrical waveguide. The radiation field is Cerenkov radition field, usually called wake-field because the radiation field is always after the electron.

Last edited by a moderator: May 5, 2017
11. Sep 8, 2011

### BruceW

Re: The plane wave decomposition is mathematically universal???

For a Fourier transform, we have:
$$\hat{f}(\xi) = \int_{-\infty}^\infty \ f(x) e^{-2 \pi ix \xi} \ dx$$
Where x and $\xi$ are real numbers. And f(x) is generally a complex function of x.
You could then say that the plane waves are $e^{-2 \pi ix \xi}$ and the f(x) is just a complex coefficient to go with the plane waves. In this case, the fourier transform is trivially a sum over plane waves, since the plane waves are multiplied by some other (generally non-planar) function.

12. Sep 8, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

I am trying to understand your question.
For example, f(x)=exp(-|x|), which is not a wave.
But it can be expressed as f(x)=exp(-|x|) = Inte{F(k)*exp(-ik*x)*dk} by FT method,
where F(k)*exp(-ik*x)*dk is a "component plane wave", the amplitude is F(k)*dk, and the phase factor is exp(-ik*x)=cos(kx)-i*sin(kx). What do mean for "trivially"?

13. Sep 8, 2011

### Staff: Mentor

Re: The plane wave decomposition is mathematically universal???

That is a Laplace transform. And he is working in cylindrical coordinates, in a dielectric medium, an with non-uniform material properties. Due to the interaction with matter in this problem decaying and growing modes are indeed physical, hence the relevance of the Laplace transform.

I don't really see the relevance to your problem where we have uniform free-space propagation of spherical waves.

Last edited by a moderator: May 5, 2017
14. Sep 8, 2011

### BruceW

Re: The plane wave decomposition is mathematically universal???

Keji - I said trivially because F(k) is any general complex function of k, so to call it the amplitude of a plane wave trivially means that the fourier transform is an integral of plane waves.

I usually think of the amplitude of a plane wave as some constant. But in this case it is used to mean a general function.

I don't have any objection, I would simply prefer to say that the fourier transform gives a function of x as an integral of plane waves multiplied by a general function of k.

15. Sep 8, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

I just want to answer your question: Fourier integrals have to be carried out in the complex plane for some cases. That is Fourier integral in that paper, the integration limits are from -infiniy to +infinity, while for Laplace transform, from zero to +infinity.

Another example: Do you remember that the Lorentz invariant Green function in the relativistic electrodynamics is obtained by Fourier-transform approach? The Fourier integration is carried out in the complex plane by designating a contour for poles. Please check with the well-known textbook by J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), Chapter 12, p. 612, Eq. (12.129).

16. Sep 8, 2011

### Claude Bile

Re: The plane wave decomposition is mathematically universal???

Plane waves cannot be used to describe some functions if the wave velocity is finite.

If we try to decompose certain functions (such as infinitesimal points), we get some imaginary wave-vectors which represent evanescent waves, not plane waves.

Plane waves can only be used to completely describe functions that do not violate the diffraction limit.

Claude.

17. Sep 8, 2011

### Staff: Mentor

Re: The plane wave decomposition is mathematically universal???

The Fourier transform transforms a complex-valued function of a real number (time or space) into another complex-valued function of a real number (frequency or wavenumber). That is the definition of the Fourier transform.

If you are dealing with a complex-valued function of a complex number then it is simply not a Fourier transform. It is a Laplace transform. The Laplace transform may certainly be useful also, but that does not make it a Fourier transform.

http://en.wikipedia.org/wiki/Fourier_transform
http://www.math.ucla.edu/~tao/preprints/fourier.pdf

In addition, I thought that your interest was in the decomposition of a spherical wave into an infinite sum of plane waves. If so, then you are not interested in other types of transforms, but only the standard Fourier decomposition with a real domain. I don't know why you are bringing in unrelated topics that you are not even interested in.

Last edited: Sep 8, 2011
18. Sep 8, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

Interesting! Could you please show some references? I mean textbooks or journal papers. Thanks in advance.

19. Sep 9, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

Questions for you:

1. "The plane wave can be expanded in terms of spherical waves"
Do you mean "A spherical-wave decomposition of a plane wave presented in textbook by J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), p. 471, Eq. (10.44) in Chapter 10" ? That is spherical harmonics expansion, r=0 is analytic.

2. "and vice versa" means Fourier integral or what? Please give your references.

Thanks a lot.

20. Sep 9, 2011

### vanhees71

Re: The plane wave decomposition is mathematically universal???

Sure, you find all this in any textbook on quantum theory or in the more advanced ones on classical em.

21. Sep 9, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

Be specific for the two cases you mentioned above, please. Which book, pages., Eqs....
and the plane-wave decomposition of spherical wave is Fourier integral or Laplace integral? Thanks a lot.

Last edited: Sep 9, 2011
22. Sep 9, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

Because I need the critical comments on my views from outstanding scientists like you. Sometimes your criticism might make me not so comfortable, but I can really learn something new from here.

23. Sep 10, 2011

### vanhees71

Re: The plane wave decomposition is mathematically universal???

Plane-wave decomposition in spherical waves: [corrected after hint to typos in #24 thanks!]

$$\exp(\mathrm{i} \vec{k} \cdot \vec{r}) = \sum_{l=0}^{\infty} \mathrm{i}^l (2l+1) \mathrm{j}_l(k r) \mathrm{P}_l(\cos \vartheta).$$

Here, $\mathrm{j}_l$ are the spherical Bessel functions, $\mathrm{P}_l$ are the Legendre polynomials, and $\vartheta$ is the angle between $\vec{k}$ and $\vec{r}$.

From this, one obtains the spherical Bessel functions by generalized Fourier transformation with respect to $u=\cos \vartheta$

$$\int_{-1}^1 \mathrm{d} u \exp(\mathrm{i} \rho u) \mathrm{P}_l(u)=2 \mathrm{i}^l \mathrm{j}_l(\rho).$$

I've collected this from many textbooks when I wrote a manuscript for mathematical methods for theoretical physics:

http://fias.uni-frankfurt.de/~hees/publ/maphy.pdf

This, however, is in German.

Last edited: Sep 10, 2011
24. Sep 10, 2011

### keji8341

Re: The plane wave decomposition is mathematically universal???

Thanks a lot.
Except for 4*pi, that is the same as in the book by J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), p. 471, Eq. (10.44) in Chapter 10.

I think you had two typo, no 4*pi, according to Eq. (3.4.43) in your book, and pl(u) lost in the expression for spherical Bessel functions.
----------------
One more question for you.

The plane-wave decomposition of the poit-source spherical wave is a Fourier integral (namely Green's function) usually presented in quantum mechanics for approximately solving Schrodinger wave equation.

Here is my understanding of this decompostion. This Fourier integral converges in the sense of taking a limit for a designated contour in the k-complex plane. Thus the component plane wave has a complex wave vector k, and this is not a physical plane wave, because a plane wave with a complex wave number in free space is not consistent with energy conservation law in the sense of classical electrodynamics, and it is also not consistent with photon momentum hypothesis in the sense of quantum mechanics. Therefore, the plane-wave decomposition of a point-source-generated spherical wave is not a physical-plane-wave decomposition, just a kind of mathematical correspondence (treatment).

What do you think of my argument? I hope you can refute my argument.

Last edited: Sep 10, 2011
25. Sep 10, 2011

### Staff: Mentor

Re: The plane wave decomposition is mathematically universal???

It is consistent if there is a matching wave which is providing/taking the missing/extra energy. Similarly, some plane waves will require currents in free space, which is ok as long as there is another wave with the opposite current. The individual waves need not be free-space solutions to Maxwell's equations as long as the sum is.

Again, this is another reason why the answer to this question is not relevant to the Doppler shift.

I still disagree that a transform involving complex wave-numbers fits the definition of a Fourier transform.