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- 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

Is it that what it means at first place when it says that these waves are solutions

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##.

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##.

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