How does the phase velocity can be greater than the speed of Light inside the waveguide?
This should not be taken as a surprise. There are many materials and also as you have mentioned some modes in a waveguide are characterized by phase velocities such that, v_p > c. At first glance this apparently looks like a violation of causality, but it is not!
You must realize that phase velocity and attenuation of a wave are coupled to each other through the Kramers-Kronig relationship due to linearity and the causality principle. Never infer from just one of them. Always consider both phase velocity dispersion and attenuation together before reaching a conclusion.
For modes which are characterized by v_p > c, their attenuation rises even faster. The ever increasing phase speed and attenuation compete against one another, and attenuation is the usual winner in such cases. To quantify this competition you can calculate penetration depth or skin depth of such modes. You will find that such modes could merely travel a distance less than a single (respective) wavelength before they undergo exponentially decay. Such modes are therefore called evanescent, implying the oscillatory motion of the "wave" ceases to exist. The "wave" motion just fades away within a travel distance of unit wavelength. There is effectively none propagation of such modes.
I had obtained similar result for acoustic waves. See, the red curve in figures 3, 4 and 5, in this link. Also read the respective discussion of the plots.
https://www.researchgate.net/publication/287249513_Connecting_the_grain-shearing_mechanism_of_wave_propagation_in_marine_sediments_to_fractional_calculus
It is the group velocity which matters as that corresponds to the information speed. The motion of the wave group, not the motion of the individual waves that make up the group, corresponds to the speed of information. Thus v_p > c , does not violate special relativity.
Hope that clarifies!
Article Connecting the grain-shearing mechanism of wave propagation ...
Remember that the phase velocity is just the apparent speed at which the humps of wave appear to travel. So if you interfere two plane waves propagating in opposite directions, but off by a fraction of a degree from being exactly opposite, the interference pattern will produce humps that propagate faster than the speed of light. However, the waves themselves are not travelling faster than light. This really is what is happening in a metallic waveguide with a higher order mode.
Dear Manoj,
The phase velocity of a guided wave in a wave guide is greater than the speed of light because the wavelength of the guided wave is basically greater then the corresponding wave length in free space. This is because of the guidance itself. It is so that waves in the wave guide can be transmitted only if they are inclined to the direction of the propagation along the wave guide. Visualizing this, the wave is propagated in the wave guide by multiple refections on the wave guide walls traversing longer distance than axial distance which would be the case in case of free space transmission. Therefore the guided wavelength lambdag is longer than the corresponding freesspace wavelength lambda0
Mathematically:
(1/lambdag)^2= (1/ lambda0)^2- (1/lambdac)^2,
where lambdac is the cut off wavelength.
Since the phase velocity is = lambda x f and f is the same then the guided phase velocity is greater than the free space wave velocity C, the speed of light.
for more information please follow the link:http://www.microwaves101.com/encyclopedias/waveguide-mathematics
Dear,
In a waveguide the phase velocity is not the energy velocity; this last one is always less than of speed of light. Therefore, the physical low is always satisfied!
https://www.researchgate.net/post/Phase_velocity_inside_a_waveguide
The exploitation presented by Dr Zekry is absolutely clear. The suggested site by him may give further elaboration for understanding the concept of free space, phase ,and group velocity.
If you go down to the beach at Kalpakkam and watch the waves coming in from the ocean, you may sometimes see them coming in at an angle so that the wave fronts are not parallel to the beach. You will then notice that the point at which the waves break moves along the beach. It moves at a speed which is greater than that of the waves coming in. This is precisely what happens in a waveguide. On occasions when the incoming waves are parallel to the beach they break all along simultaneously, which is to say the speed of the breaking point is infinite. This is is analogous to the case of a waveguide at cut-off.
The effect is sometimes called the scissors effect. If you close a pair of scissors the point where the two blades meet travels along the blades much faster than the speed at which the blades approach each other.
John ponsonby answer is more logical .when we tilt the sea shore parellel to the moving wave front ,all the time the wave front parellel to the sea shore . as per scissors effect the wave length (distance) adjusted to match the velocity and the velocity (distance by time) is maintained .
I would add a comment:
Phase velocity is merely velocity of the the phases of a harmonic wave.
Suppose we have two points displaced at distance delta_x, and at these points we measure appearance of the same phases (e.g. the same peaks of a sinusoid), let the time difference between them is delta_t, then phase velocity is delta_x/delta_t. Now, if these peaks occur at the same time with delta_t~0 (it can be at stationary waves), then the phase velocity becomes almost infinite. Thus, it should be noted, that the phase velocity does not mean speed of wave propagation.
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