What is the relationship between group and phase velocity?

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Group velocity refers to the speed of the peak intensity of a wave group, while phase velocity is the speed of individual waveforms, such as sine waves. In non-dispersive media, both velocities are equal, but in dispersive media, they differ, causing wave shapes to change over distance. Group velocity is always less than or equal to phase velocity and is significant for understanding energy propagation and information transfer. An interesting phenomenon occurs in thin metal foils where light can exceed the speed of light in a vacuum due to the properties of group velocity. This relationship is crucial in the study of photons as energy pulses, reinforcing the importance of wave behavior in various media.
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undefinedundefinedWhat are group velocity and phase velcity? What are it's physical implications? :smile:
 
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Phase velocity is the speed in say degrees per sec of a single repeated waveform such as a sinewave which is traveling ( like a water wave ) past a point, 360 degrees being equivalent to one wavelength.
If not a sinewave then wavelength here just refers to the distance between like points of a repetitive wave .
However a waveform non sinusoidal can be viewed as made up of several or many other sine waves ( Fourier Analysis ) , if the medium (vaccuuo , air, glass etc , water )
is non-dispersive , then all these waves travel at the same speed , and the wave shape is maintained -- but many materials ARE dispersive and differing wave lengths travel at different speed .
This means that the wave shape will change with distance -- sometimes spreading out and sometimes being compressed.
Group velocity refers to the speed of the peak intensity of that group of waves ( i.e.
the points where they tend to reinforce ) , is therefore the speed of maximum energy.
Group velocity is always less than or equal to the phase velocity.
One slightly bizarre effect occurs in very thin ( hence transparent ) metal foils such as gold.
Here the refractive index is the opposite of say glass , the implication being that light travels at > 'c' the velocity in vaccuuo. The explanation is given in terms of the 'group' or energy speed rather than individual sinewaves .
Group velocity can also be taken to mean the information rate for the same reason.
It has particular significance in the concepts of the photon as a 'pulse' of energy
because again a pulse can be mathematically represented as a group of sinewaves which reinforce at only one point in space .
Hope this helps Ray.
 
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