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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 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##.
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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