
#1
Jul1811, 10:53 AM

P: 205

Hello, I have a problem understanding wave propagation in dispersive medium because the prescription to solve the wave equation for electromagnetics waves is this :
a) suppose [itex]\mu,\epsilon[/itex] are function of frequency only b) solve the wave equation [itex]\Delta\textbf{E}\mu\epsilon\frac{\partial^2\textbf{E}}{\partial t^2}=0[/itex], for each Fourier component c) use superposition to get (sum each solution) to obtain the general solution,ie, [tex]\textbf{E}(\textbf{r},t)=\int_{\infty}^{\infty}\textbf{E}(\textbf{r},\omega)e^{i\omega t}dw[/tex] How can this superposition be valid since there is no linearity?, or in any case, what is the wave equation this prescribed expression is solution to? It must be simple and clear since this is the ussual way to do it, but I'm not seeing it. 



#2
Jul1811, 11:02 AM

P: 882

[itex]\mu[/itex] and [itex]\epsilon[/itex] result from charges (electrons) and magnetic momenta of atoms/grains of the medium, responding to the fields calculated "in vacuum". All those responses (unless you go to very strong fields) are linear. So it is always true that sum of solutions is also a valid solution. But (for dispersive media) they may vary with frequency at which we force them to oscilate. You may build your final solution as a sum of any partial solutions chosen at any scheme as you like. As it is easy to compute solution for single frequency, Fourier decomposition is a convenient approach to build general solution. 



#3
Jul1811, 11:52 AM

P: 617

The term "linear" is overloaded, so we have to be careful. Maxwell's equations are linear. Electromagnetic fields obey linear superposition, meaning that they sum together. But, there are materials that have a significant nonlinear electromagnetic response to applied fields. These materials are rare. However, the wave equation in its textbook form requires linear materials in order to be derived. So if you see the wave equation, it is understood that linear materials is already implied.




#4
Jul1811, 12:31 PM

P: 205

Don't understand dispersive medium[tex]\textbf{E}(\textbf{r},t)=\int_{\infty}^{\infty}\textbf{E}(\textbf{r},\omega)e^{i\omega t}dw[/tex] is solution to? The expression under the integral symbol is solution for the equation [itex]\Delta\textbf{E}\mu\epsilon\frac{\partial^2\textbf{E}}{\partial t^2}=0[/itex] where [itex]\mu,\epsilon[/itex] are function of frequency. 



#5
Jul1811, 01:22 PM

P: 882

There is no simple representation of such equation, as in dispersive medium you can't write down Maxwell's equations abstracting from frequency. That is why you had to split your problem into possibletoformulateeasytosolve Fourier composition of monochromatic waves.




#6
Jul1811, 02:00 PM

P: 205

How do we justify the the superposition of such solucions is a solution when we don't even know what equation is to be satisfied? 



#7
Jul1811, 03:06 PM

P: 882

We can't write down the formula, but we know the formula is linear regarding electric field strength. The unwritten formula is equivalent to Maxwell's equations in vacuum, but in presence of additional charges and currents, resulting from atoms responding to the field. Maxwell's equations in vacuum are linear. We assume now that atoms in our medium respond linearily to attached field. Thus their response is also linear, and the final unwritable equation must also be linear.
Now you'll ask why we assumed that atoms in our medium respond linearily to attached field... That is the first order approximation, valid for most materials and light densities present in everyday life. This assumption about linearity had already been done implicitely by your teacher, who told you that [itex]\mu,\epsilon[/itex] are function of frequency (but not of E). Such model ([itex]\mu,\epsilon[/itex] depends only on frequency) results from modelling the medium as an ensamble of harmonic oscillators of various own frequencies, responding to attached field. Harmonic oscillators are linear. 



#8
Jul1811, 04:01 PM

P: 617





#9
Jul1811, 06:24 PM

P: 205





#10
Jul1811, 11:47 PM

P: 166

In Fourier space, you are referring to the equation
[tex]\left[ \nabla^2 +\mu(\omega)\epsilon(\omega) \omega^2 \right] \mathbf E(\mathbf r, \omega)=0[/tex] To find out what that equation is in terms of time, you would have to take its inverse Fourier transform. Then, the functions of [itex]\omega[/itex] will become complicated functions involving time derivatives. 



#11
Jul1911, 05:08 AM

P: 205





#12
Jul1911, 05:25 AM

P: 882

You are confused because you mix two meanings of 'nonlinearity'
Take a mechanical example: harmonic oscillator (e.g. guitar string) driven by external force (e.g. sound from nearby loudspeaker). It is strongly nonlinear regarding frequency (response chages as the frequency of driving force changes), but (unless you go to very high amplitudes) it is linear regarding amplitude: if you apply twice bigger force (regardless of its freq. composition) the string responds with vibrations of twice bigger amplitude. Actually that is our case, as a classical way to introduce [itex]\epsilon[/itex] of real media is to treat them as a set of harmonic oscillators: massive electric charges at the springs (see e.g. great lecture in Feynman's Lectures on Physics) 



#13
Jul1911, 06:02 AM

P: 205

Thank you folks! 



#14
Jul1911, 09:33 AM

P: 617

The term "linearity" in electrodynamics typically refers to the fact that there is a linear material response between the induced and total field:
D = ε E and B = μ H as opposed to a nonlinear response: D = ε_{1} E + ε_{2} E^{2} x+ ... and B = μ_{1} H + μ_{2} H^{2} x + ... The linearity of the material is what allows us to create the standard wave equation in the first place. The term "dispersive" is what we use to describe a material that has a permittivity that is significantly nonconstant as a function of frequency: D = ε(ω) E and B = μ(ω) H Dispersive materials can still be linear, and thus propagate waves according to the standard wave equation. The frequencydependent response means that different frequency components of a wave packets travel at different speeds in the material. This causes a wave packet to disperse, or spread out. Physically speaking, the wave equation as a function of time, and the wave equation as a function of frequency are the same fundamental equation, directly deduced from Maxwell's equations. They are just different mathematical representations. 


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