Waves and Electromagnetism

1. Wave Basics

A wave is a disturbance that transfers energy through a medium (or vacuum) without transferring matter.

Fundamental Quantities

• Amplitude ($A$): The maximum displacement from the equilibrium position.
• Wavelength ($\lambda$): The distance between two consecutive identical points (e.g., crest to crest).
• Frequency ($f$): The number of oscillations per second (measured in Hertz, $\mathrm{Hz}$).
• Period ($T$): The time taken for one complete cycle ($T=1/f$).
• Wave Speed ($v$): The speed at which the wave propagates:$$v=f\cdot \lambda =\frac{\lambda }{T}$$

2. Electromagnetic (EM) Waves

Maxwell’s equations predict that oscillating electric and magnetic fields can propagate through space as a wave.

Key Characteristics

• Transverse Nature: The electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ are perpendicular to each other and to the direction of propagation.
• Speed of Light: In a vacuum, all EM waves travel at the speed $c$:$$c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}\approx 3\times {10}^8\mathrm{m}/\mathrm{s}$$
• Wave Equation: Derived from Maxwell's Equations (in vacuum):$${\mathrm{\nabla }}^2\mathbf{E}={\mu }_0{\epsilon }_0\frac{{\partial }^2\mathbf{E}}{\partial t^2}$$

3. Phase vs. Group Velocity

When a signal consists of a "packet" of different frequencies, we must distinguish between two types of speed.

Phase Velocity ($v_p$)

The speed at which the individual peaks (the "phase") of a single-frequency wave move.

(Where $\omega =2\pi f$ is angular frequency and $k=2\pi /\lambda$ is the wave number.)

Group Velocity ($v_g$)

The speed at which the overall envelope (the "wave packet" or signal) travels. This is the speed at which information and energy are usually transmitted.

Dispersion

• Non-dispersive media: $v_p=v_g$ (e.g., light in a vacuum).
• Dispersive media: $v_p\ne v_g$ (e.g., light in glass or water waves). The wave packet "spreads out" over time.

Relationship Summary

The general relationship between the two velocities is given by: