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neelakash
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Is there any unconventional method to find out the relation between group velocity and phase velocity? know there is a method employing Fourier tarnsforms and another easier method as well
That is a very good point. The equation v_g=dk/dw comes from keeping only the first term of a Taylor expansion. Many of the papers claiming funny group velocities for light miss the point you mention.Fizik said:one thing to notice is that, for a group velocity to be meaningful, the wave packet has to be merely composed of wavenumber in a narrow range.
The group velocity is the speed at which the energy or information of a wave propagates, while the phase velocity is the speed at which the phase of the wave propagates. In simple terms, the group velocity is the speed of the "envelope" of a wave, while the phase velocity is the speed of the individual peaks and troughs.
Group velocity and phase velocity are related by the dispersion relation of a wave. In most cases, the group velocity is equal to the phase velocity only at the center frequency of the wave. As the frequency of the wave deviates from the center frequency, the difference between the two velocities becomes more significant.
The group velocity and phase velocity have different physical significances. Group velocity is important in determining the speed at which information is transmitted through a medium, while phase velocity is more relevant in understanding the behavior of individual wave peaks and troughs.
The relationship between group velocity and phase velocity can have significant effects on wave propagation. In some cases, such as in dispersive media, the group velocity can be slower than the phase velocity, leading to a phenomenon known as group velocity dispersion. This can result in the distortion of signals and information transmitted through the medium.
In certain cases, such as in non-dispersive media, the group velocity and phase velocity can be the same. In these cases, the wave is said to be traveling at the "normal" speed of the medium. However, in most cases, there will be a difference between the two velocities due to the varying behavior of the wave at different frequencies.