# Phase and Group Velocity Problem

• coltin.walsh
In summary, phase velocity and group velocity are two different measures of the speed of a wave. Phase velocity describes the movement of the wave's peaks and troughs, while group velocity describes the overall movement of the wave's energy or information. The phase velocity can sometimes exceed the speed of light, which is known as superluminal phase velocity. The group velocity is calculated by taking the derivative of the wave's frequency with respect to its wave number. The phase and group velocity problem is an important concept in the study of wave propagation, with practical applications in various fields. While they can be equal in some cases, the phase and group velocities are typically different and it is important to understand their difference when studying wave behavior.

## Homework Statement

see:

http://www.flickr.com/photos/55153239@N03/6371848305/ [Broken]

vp = w/k

vg = dw/dk

## The Attempt at a Solution

I know the relationship of:

vg = vp(1+lambda/n * (dn/dlambda))

but i don't know where to go from here..

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1. given n(λ), what is phase velocity vp of light of wavelength λ?
2. as you state, w/k = vp. So how about rewriting w in terms of λ and n(λ), using (1) above?
3. use chain rule dw/dk = dw/dλ*dλ/dk

## 1) What is the difference between phase velocity and group velocity?

Phase velocity refers to the speed at which the phase of a wave travels, while group velocity refers to the speed at which the energy or information of the wave travels. In other words, phase velocity describes the movement of the peaks and troughs of a wave, while group velocity describes the overall movement of the wave.

## 2) Why is the phase velocity sometimes greater than the speed of light?

According to classical physics, nothing can travel faster than the speed of light. However, in certain situations, the phase velocity of a wave can exceed the speed of light. This is known as superluminal (faster than light) phase velocity and it is a result of the interaction between different media through which the wave is traveling.

## 3) How is the group velocity calculated?

The group velocity is calculated by taking the derivative of the wave's frequency with respect to its wave number. In other words, it is the slope of the wave's dispersion curve, which plots the frequency of the wave against its wave number.

## 4) What is the significance of the phase and group velocity problem?

The phase and group velocity problem is an important concept in the study of wave propagation. It helps us understand the behavior of different types of waves, such as electromagnetic waves and sound waves, and how they interact with different media. It also has practical applications in fields such as optics, telecommunications, and seismology.

## 5) Can the phase and group velocities be equal?

Yes, in some cases, the phase and group velocities can be equal. This occurs when the wave is traveling through a medium with a constant refractive index, such as in a vacuum. In this case, the phase and group velocities are both equal to the speed of light. However, in most cases, the phase and group velocities are different and it is important to understand the difference between the two when studying wave propagation.