Dear Sirs,
Everybody knows plane and spherical wave solutions of Maxwell equations, e.g for decaying plane wave E=E0*exp(-kx)*cos(w(t-x/v)). But seems to me they give the unreal situation that the wave amplitude is nonzero at different points of space at given time moment. Could you advise the experiment or natural phenomenon which produces such a wave in nature?
Maybe we have infinte speed of the EM interaction? Do you know any real solution of Maxwel equations which exists only in one space point at the given time moment? Maybe using delta function? Or maybe there is my mistake?
Dear Anatoly,
We don't have infinite speed. You are talking about basic solutions and we can construct any solution as their superposition. What does it mean - real solution? Do you mean solution for zero current and charge?
Dear Eugene,
I mean that electric (magnetic) field of the wave exists in some space volume in any point of time. So EM interaction propagates infinitely fast in this space volume.
Real wave means we have not the above infinite interaction speed.
We know that our understanding of wave motion is not always the same based on the available data. This is the philosophy of the difference between the equations. we use an equation based on data. Then select the equation based on the available data
Anatoly A Khripov "I mean that electric (magnetic) field of the wave exists in some space volume in any point of time. So EM interaction propagates infinitely fast in this space volume."
That is a non-sequitur. In Maxwell's equations, the (front) speed of electromagnetic waves is always (smaller than or) equal to c. The phase speed of a wave has nothing to do with its speed of information propagation. In vacuum it is c, too.
K. Kassner , I agree with you. The is thought experiment: rotate flashlight such a way, that it will have light spot on the Moon. This spot will have speed > c.
In fact Maxwell equations has infinite number of solutions. You should choose one, which satisfy border and initial conditions for your task.
Dear K. Kasner, E. Prokhorenko,
Is there a EM wave solution which exists only in one point of space at a given time moment?
Dear Anatoly,
Yes, it exists as mathematical formula, but nobody knows how to realized it in physics. BTW, it's not smooth function, but distribution and it has infinite energy.
Dear Eugene,
Thank for your reply. Could you give me the reference or www address? Is it a solution when initial condition is a rectangular pulse of zero width, or something similar to it?
Dear Anatoly,
This is like delta-function and it's generalized derivatives. Do you need only 3+1 case? In N dimensions it's too complicated.
theta(t)*delta(t^2-x^2-y^2-z^2) see Владимиров. Обобщенные функции в математической физике. стр. 203.
Dear Eugene,
Thank you very much for your reply. It is very interesting. I will study it.
Is E=E0*exp(-kx)*cos(w(t-x/v)) a solution of Maxwell equations in vacuum ? Only if k is purely imaginary and the relation c/v = 1 + ikc/w holds. Otherwise this can be a solution of a Proca-like equation.
Joaquin Diaz-alonso , your solution corresponds to nonphysical boundary conditions. You can play with different formulas. Mostly I like this: take ANY EM field Am and insert it in Maxwell equations. Then you will find charge density ro and current Jm for which your Am is solution! :)
Dear Eugene: The field E=E0*exp(-kx)*cos(w(t-x/v)) is not my solution. It is the one proposed by Anatoly A Khipov in his initial question "Does plane (spherical) EM wave expression mean infinite EM interaction speed?". This is the reason I mention it my commentary. This damped field (if k is real and positive) can never be a solution of VACUUM Maxwell equations. Obviously, it is a solution of Maxwell equations with appropriate sources.
Short note: This "solution" in fact is Fourier transformation of D'Alamber equation: exp(i*(x*kx+y*ky+z*kz+t*kt)) with constrain kx^2+ky^2+kz^2=kt^2 for any complex kx, ky, kz, but meaning can be understand in theory of distributions only.
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