Is there a way to know how the amplitude of a wave will evolve?

[SergioPL]

It’s very well know that Maxwell equations bring the solution for plane waves but as far as I know I cannot use them to detect how the field would evolve by only looking the E and B fields on the contour of the point under evaluation. Is that possible?
My question comes because on a plane wave, if we look at some point, all the points on the same plane of propagation that the point under study have the same phase, the same amplitude and they propagate on the same direction so this must be the reason the field doesn’t decrease as it propagates.
On the other hand, on a spherical wave, the contour of a point in the plane parallel to the (local) direction of propagation will have the same amplitude but different direction. I suppose this difference between these two types of waves is what makes spherical waves to decrease whereas plane waves don’t.
Does somebody know if there are some equations that can locally explain this?


Sergio

 

[SergioPL]

Ok, the matter gets closed by looking at the Pointing's vector definition :).

 

[SergioPL]

The equation is:

∂u/∂t = - \nabla · S - J · E

And looking how u evolves in the direction of propagation you can get how it's power decreases.

 

[chrisbaird]

Yes, this is essentially a statement of conservation of energy. If a wave spreads out, its energy density must decrease. The J dot E term accounts for the fact that energy can also be lost from the fields to accelerating charges that create currents.

 

[SergioPL]

Yes, the J·E is the power done on the charges for a distribution of charge.

J = qv where q is the density of charge and v the velocity, so J·E = qE·v = F·v, the classical definition of power, the decrease on the field's energy density is the (density of) power that is done on the current.


Sergio