# Are spherical transverse waves exact solutions to Maxwell's equations?

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• Delta2
In summary: If you do not make these assumptions, then the solutions are not exact. This allows for the possibility of non-vanishing terms in the divergence and curl of the electric field, but they can be safely neglected in the limit of large r. The authors also state that the electric field must not depend on r, which helps in simplifying the equations and allows for the neglect of certain terms in the limit. Overall, spherical waves are not exact solutions, but they can be used as approximations in certain cases.
Delta2
Gold Member
TL;DR Summary
Spherical waves as solutions to Maxwell's equations in vacuum.
In this paper in NASA

https://www.giss.nasa.gov/staff/mmishchenko/publications/2004_kluwer_mishchenko.pdf

it claims (at page 38) that the defined spherical waves (12.4,12.5) are solutions of Maxwell's equations in the limit ##kr\to\infty##. I tried to work out the divergence and curl of ##\vec{E(r,t)}## and find out that for example the divergence of E contains a term $$\frac{e^{ikr}}{r}\nabla\cdot\vec{E_1(\hat r)}$$, which doesn't seem to vanish (given the extra conditions (12.6-12.9) unless of course we take the limit ##r\to\infty##.

Is it that what it means at first place when it says that these waves are solutions in the limit ##kr\to\infty##? Does this means that spherical waves are not exact solutions to Maxwell's equations in vacuum? (the paper considers the general case of a homogeneous medium present but vacuum is a special case of a homogeneous medium isn't it?)

P.S ##\vec{E_1(\hat r)}## cannot be a constant vector as that is implied by 12.6, that is it is always perpendicular to ##\hat r##.
P.S2 I find no easy way to prove that ##\nabla\cdot\vec{E_1}=0## from the 12.6-12.9 conditions

P.S3 I think I got it now. The authors of the paper say that ##\vec{E_1}## (and ##\vec{H_1}##) must not depend on r. If so then their divergence and curl have ##\frac{1}{r}## dependence which together with the other ##\frac{1}{r}## from the term ##\frac{e^{ikr}}{r}## make a term ##\frac{1}{r^2 }##which can safely be neglected for ##r\to\infty##.

Last edited:
Spherical waves are approximate solutions to Maxwell's equations. You have to make approximations based on the assumptions $r'<<r$ and kr>>1.

vanhees71 and Delta2

## 1. What are spherical transverse waves?

Spherical transverse waves are a type of electromagnetic wave that propagates through space in a spherical pattern. These waves have both electric and magnetic components that are perpendicular to the direction of propagation.

## 2. Are spherical transverse waves exact solutions to Maxwell's equations?

Yes, spherical transverse waves are exact solutions to Maxwell's equations, which describe the behavior of electromagnetic waves. These solutions can accurately predict the behavior of electromagnetic waves in free space.

## 3. How do spherical transverse waves differ from other types of waves?

Spherical transverse waves differ from other types of waves, such as plane waves, in their propagation pattern. While plane waves have a flat propagation pattern, spherical transverse waves have a spherical pattern. Additionally, spherical transverse waves have both electric and magnetic components, while some other types of waves may only have one component.

## 4. What are some real-world applications of spherical transverse waves?

Spherical transverse waves have various applications in the field of optics, such as in laser technology and holography. They are also used in wireless communication systems, as they can propagate through space without the need for a physical medium.

## 5. Are there any limitations to using spherical transverse waves?

One limitation of using spherical transverse waves is that they can only propagate through free space. They cannot pass through objects or materials, which can be a hindrance in certain applications. Additionally, the strength of the wave decreases as it travels further from its source, which can limit its range.

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