Wave Fundamentals

In this section, we study the physics of waves in general. Understanding these mechanics is essential before we move on to the specific nature of electromagnetic (EM) waves in the following chapters.

0. The "Leap-Frog" Effect

Imagine a region of empty space. If you somehow create a changing electric field $\vec{E}$, what happens?

Ampère-Maxwell

A time-varying $\vec{E}$ is linked to a circulating $\vec{B}$.

\[ \nabla \times \vec{B} = \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} \]

Faraday

A time-varying $\vec{B}$ is linked to a circulating $\vec{E}$.

\[ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \]

These fields don't just "stay" where they are. They kick-start each other. A time-varying $\vec{E}$ appears in the Ampère-Maxwell law as circulation of $\vec{B}$ in the surrounding space. A time-varying $\vec{B}$ appears in Faraday's law as circulation of $\vec{E}$. In a wave, the coupled curl equations pass the disturbance onward, forming a "leap-frog" through space.

Deep Dive: The Wave Equation

1. The 1-D Wave Equation

The motion of any wave is governed by a second-order partial differential equation. If $f(x,t)$ represents the displacement at position $x$ and time $t$:

\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}

The most remarkable property of this equation is that any function of the form $f(x,t) = g(x - vt) + h(x + vt)$ is a valid solution. This means any shape—whether a pulse, a square wave, or a complex signal—can propagate as long as its variables are coupled in this way.

2. Sinusoidal Waves

While the wave equation accepts any function, Sinusoidal Waves are the most popular and useful type of solution. They are the building blocks for more complex disturbances and are mathematically described as:

f(x,t) = A \cos(kx - \omega t + \delta)

Where the parameters are defined as:

$A$ Amplitude. The maximum displacement from equilibrium.

$k$ Wave Number. Spatial frequency, given by $2\pi/\lambda$.

$\omega$ Angular Frequency. Temporal frequency, given by $2\pi f$.

$\delta$ Phase constant. Determines the initial state/shift of the wave.

The speed of the wave is given by the relation $v = \omega/k = \lambda f$.

Interactive: 1D Wave Propagation

Deep Dive: Mathematics of Waves