Ok!
However, as regards the first derivative, d\omega / dk = v_g and dk/d\omega = 1/v_g, so they are exactly reciprocal. If you take the unit measures, they are reciprocal too. So, here is still my doubt.
Considering the simplest case, the one regarding plane waves,
k = \omega / v
with v constant.
d\omega/dk = v = v_g is the group velocity and dk/d\omega = 1/v = 1/v_g is the reciprocal of the group velocity.
d^2 \omega/dk^2 = \alpha = 0 is the group velocity dispersion; so, the reciprocal of...
Hi!
Dealing about wave propagation in a medium and dispersion, wavenumber k can be considered as a function of \omega (as done in Optics) or vice-versa (as maybe done more often in Quantum Mechanics). In the first case,
k (\omega) \simeq k(\omega_0) + (\omega - \omega_0) \displaystyle \left...
Maybe it is a problem of phase shift: let z be the direction of propagation. Each wave propagates as described by the term
E(z) = E_0 e^{-j \beta z}
After the same distance z, waves with higher frequencies have necessarily a greater phase shift than waves at lower frequencies, because they...
Hello!
I still would like to thank those who participated to my previous thread about group velocity and dispersion. Now there is a (maybe) simpler question.
A sinusoidal, electro-magnetic plane wave in the vacuum propagates in a certain direction with the following wavenumber, which is supposed...
Thank you for having even read that book. I agree with all your considerations and (if you look back to my first post) they are almost equivalent to what I've written about a resonant line and the Z^{left} (x), Z^{right} (x) impedances.
But my question was slightly different: what is the...
It may be a trivial question, but how can the current be zero? Can you show me the steps?
Yes, it would.
It seems odd to me too, but I have to dig into it :frown:
The Transverse resonance method is used to determine the propagation constant of a wave in several waveguides, like the rectangular waveguide, or also dielectric waveguides.
It takes advantage of the fact that a standing wave is present along a certain direction (transverse with respect to the...
Hello!
By manipulating Maxwell's equation, with the potential vector \mathbf{A} and the Lorentz' gauge, one can obtain the following vector wave equation:
∇^2 \mathbf{A}(\mathbf{r}) + k^2 \mathbf{A}(\mathbf{r}) = -\mu \mathbf{J}(\mathbf{r})
The first step for the solution is to consider a...
Ok, I try to change the question:
The surface impedance on a conductor relates the *tangential* electric field to the *tangential* magnetic field, according to the preceeding expression. But what if a wave has an oblique incidence upon the conductor's surface? The components of the fields in the...
Hello!
Let a plane wave propagate towards the -y direction. It is normally incident upon the plane (x,z) (whose normal unit vector is the y-direction unit vector, \mathbf{\hat{u}}_y): the plane represents the interface between the free space (in y > 0) and a general lossy medium (in y < 0).
We...
Hello!
You can find in the picture attached a parallel plate waveguide which has an a1 height before the step and an a2 height after the step. The plates are perfect conductors and the step is ideal.
I can't determine which physical quantities are continuous across this discontinuity.
Suppose...