Dispersion Relation

Table of Contents

Introduction

A dispersion relation describes how the phase speed or group speed of a wave depends on its wavelength, or equivalently, how the energy of a particle depends on its momentum. The term “dispersion” refers to the phenomenon where the speed of a wave depends on its frequency.

Basic Form of the Dispersion Relation

In the simplest case, the dispersion relation for a wave can be written as:

$\omega = v k$

where $\omega$ is the angular frequency of the wave, $v$ is the phase velocity of the wave, and $k$ is the wave number.

Phase and Group Velocity

The phase velocity of a wave, which is the rate at which the phase of the wave propagates in space, is given by:

$v_p = \dfrac{\omega}{k}$ .

The group velocity of a wave, which is the velocity with which the overall shape of the wave’s amplitudes—known as the modulation or envelope of the wave—propagates through space, is given by:

$v_g = \dfrac{d\omega}{dk}$ .

Dispersion in Different Media

In many physical media, the speed of wave propagation changes with frequency, a phenomenon known as dispersion. The dispersion relation in such media is not a simple linear relation but instead can take on more complex forms.

For example, in an electromagnetic wave propagating in a dielectric medium, the dispersion relation is:

$\omega^2 = c^2 k^2 + \omega_p^2$

where $c$ is the speed of light in vacuum, $\omega_p$ is the plasma frequency of the medium, and $k$ is the wave number.

Applications

Dispersion relations are fundamental to the study of waves in physics and engineering, including light waves, sound waves, and quantum mechanical waves. They allow us to understand and predict how waves behave in different media, which is critical to the design of optical fibers, lenses, and many other technologies.

Conclusion

The concept of a dispersion relation is fundamental to understanding wave propagation in various physical systems. By providing a link between the wave’s frequency and its wave number, the dispersion relation serves as an essential tool in wave analysis and the study of systems ranging from optics to quantum mechanics.

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Introduction

Basic Form of the Dispersion Relation

Phase and Group Velocity

Dispersion in Different Media

Applications

Conclusion

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Dispersion Relation

Introduction

Basic Form of the Dispersion Relation

Phase and Group Velocity

Dispersion in Different Media

Applications

Conclusion

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