Dispersion in Dispersive Media: MIT OCW Lecture on Omega and Dispersion Relations

MIT OpenCourseWare presents a lecture on dispersion in dispersive media, focusing on the frequency omega as a function of the wavevector k and how the dispersion relation governs wave propagation. The talk explains why omega depends on k in many media, how phase velocity and group velocity are derived from the relation, and how to extract frequency information from observed dispersion. The transcript outlines general strategies for analyzing omega(k), including special cases where the frequency can be expanded around a reference wavevector, and discusses the implications for signal propagation, information transfer, and the design of experiments to measure dispersion. The content illustrates the connection between mathematical dispersion relations and physical wave behavior in real materials.

Overview

The MIT OCW lecture centers on dispersion relations in dispersive media, introducing the core idea that the frequency omega is a function of the wavevector k. This omega versus k relationship determines how different frequency components propagate through a medium and thereby shapes both phase velocity and group velocity.

Core Concepts

Key quantities include the phase velocity vp = omega/k and the group velocity vg = d omega/d k evaluated at a reference k. In non dispersive media these speeds coincide, but in dispersive media they differ and lead to phenomena such as wavepacket spreading and information transfer limits.

Mathematical Framework

The lecture outlines a general approach to working with dispersion relations, showing how one can obtain omega from measurements of k, or vice versa, and how the curvature of the dispersion relation encodes higher order dispersive effects.

Expansions and Special Cases

A central technique described is expanding omega around a reference wavevector k0, so that omega(k) approximately equals omega0 plus the slope times (k minus k0) plus higher order terms. This expansion clarifies how small changes in k affect omega and how to interpret omega0, omega0 plus, and omega0 minus in practical calculations.

Applications and Implications

The discussion connects these concepts to wave packet propagation, information transfer in communications, and the interpretation of experimental dispersion data. The MIT OCW context emphasizes that the material is part of a course on fundamental wave physics and electromagnetism and can be applied to a wide range of media.