3. The EM Wave Equation

We have all the ingredients. Time to put them together into the single most important result of nineteenth-century physics: light is an electromagnetic wave.

3.1 Deriving the wave equation from Maxwell

Work in free space: no charges ($\rho = 0$), no currents ($\vec{J} = 0$). Maxwell's equations become

$\nabla\cdot\vec{E} = 0, \quad \nabla\cdot\vec{B} = 0$ $\nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t}, \quad \nabla\times\vec{B} = \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}$

Take the curl of Faraday's law:

$\nabla\times(\nabla\times\vec{E}) = -\frac{\partial}{\partial t}(\nabla\times\vec{B})$

The vector identity $\nabla\times(\nabla\times\vec{E}) = \nabla(\nabla\cdot\vec{E}) - \nabla^2\vec{E}$ collapses (because $\nabla\cdot\vec{E} = 0$ in free space) to

$-\nabla^2\vec{E} = -\frac{\partial}{\partial t}\left(\mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}\right)$

which rearranges to the vector wave equation:

$\nabla^2\vec{E} = \mu_0\varepsilon_0\frac{\partial^2\vec{E}}{\partial t^2}$

Identical procedure on Ampere-Maxwell yields

$\nabla^2\vec{B} = \mu_0\varepsilon_0\frac{\partial^2\vec{B}}{\partial t^2}$

Compare with the standard wave equation $\nabla^2\psi = (1/v^2)\partial^2\psi/\partial t^2$. The wave speed in free space is

$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 2.998\times 10^8\text{ m/s}$

This is the speed of light. Light is an electromagnetic wave. Maxwell did not even know he was about to merge optics with electromagnetism when he wrote his equations down; the speed just fell out, matched the experimental value of $c$, and nobody could ignore it.

Why this is staggering. Two seemingly unrelated constants of nature, $\mu_0$ from magnetism and $\varepsilon_0$ from electrostatics, combine to give the speed of light. Maxwell's equations are not a description of light. They are the cause of light, in the sense that the wave nature of light is a logical consequence of the equations.

3.2 Plane wave solutions

The simplest solutions are plane waves propagating along one direction, say $+z$:

$\vec{E} = E_0\hat{x}\cos(\omega t - kz)$ $\vec{B} = B_0\hat{y}\cos(\omega t - kz)$

Three observations.

1. $\vec{E}$ and $\vec{B}$ are both perpendicular to the direction of propagation $\hat{z}$ and to each other. This is a Transverse Electromagnetic (TEM) wave.
2. The wavenumber $k = \omega/c = 2\pi/\lambda$, where $\lambda$ is the wavelength.
3. The ratio $E_0/B_0 = c$, equivalently $E_0/H_0 = \eta$, the intrinsic impedance of the medium.

    Direction of propagation: +z
 
      y ↑                ↑ E
        │ ╲      ╱╲     │
        │  ╲    ╱  ╲   │
        ─────────────────→ z
        │  ╱  ╲  ╱
        │╱    ╲╱
        ↓ B (out of page when E up)
      x
 
    E and B perpendicular, both perpendicular to z.

3.3 Intrinsic impedance

The ratio of $E$ to $H$ in a plane wave is the intrinsic impedance of the medium:

$\eta = \sqrt{\frac{\mu}{\varepsilon}}, \quad \eta_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} \approx 376.73\text{ Ω}$

Free space has an "impedance" of about 377 Ω. This number will appear later in antenna theory (Chapter 13), where antennas are essentially impedance transformers between feedline impedance (typically 50 Ω) and free-space impedance (377 Ω). It is also why you cannot match a 50 Ω transmitter directly to vacuum; you need an antenna structure to do the matching gracefully.

3.4 Wave propagation in different media

Real media are not free space. The propagation constant in a general medium is

$\gamma = \alpha + j\beta = j\omega\sqrt{\mu\varepsilon}\sqrt{1 - j\frac{\sigma}{\omega\varepsilon}}$

where $\alpha$ is the attenuation constant (Np/m) and $\beta$ the phase constant (rad/m). The wave then takes the form $E(z, t) = E_0 e^{-\alpha z}\cos(\omega t - \beta z)$. Three regimes:

• Lossless dielectric ($\sigma = 0$): $\alpha = 0$, $\beta = \omega\sqrt{\mu\varepsilon}$. The wave travels with no attenuation at speed $v = 1/\sqrt{\mu\varepsilon} = c/\sqrt{\varepsilon_r\mu_r}$. In FR-4 PCB ($\varepsilon_r\approx 4.5$, $\mu_r = 1$), $v \approx 1.4\times 10^8$ m/s, about $c/2.1$, or roughly 14 cm per nanosecond. This is the speed your high-speed digital signals actually travel on a board.
• Lossy dielectric (small $\sigma$, $\sigma/(\omega\varepsilon)\ll 1$): wave attenuates exponentially as $e^{-\alpha z}$, with $\alpha \approx \sigma\eta/2$.
• Good conductor ($\sigma \gg \omega\varepsilon$): the wave attenuates dramatically. Skin depth, defined below, is the characteristic 1/$e$ penetration distance.

3.5 Skin depth and skin effect

In a good conductor,

$\alpha \approx \beta \approx \sqrt{\pi f\mu\sigma}$

The penetration depth (where the field falls to 1/$e$) is

$\delta = \frac{1}{\alpha} = \frac{1}{\sqrt{\pi f\mu\sigma}}$

Some numbers for copper ($\sigma = 5.8\times 10^7$ S/m, $\mu \approx \mu_0$):

At 1 GHz, current in a copper wire flows essentially in the outermost 2 µm. The interior carries almost nothing. Practical consequences:

• High-frequency PCB traces should be wide (more surface area means more usable copper).
• Coax cables sometimes use silver-plated outer conductors at GHz, since the outermost few microns dominate the loss.
• Litz wire, made of many thin insulated strands, breaks up the skin effect for moderate-frequency power applications.
• The hollow waveguide of a microwave radar is no worse a conductor than a solid silver bar, because the field never penetrates more than a few skin depths anyway.

Anticipating confusion. Why does skin depth have an imaginary part in the propagation constant? Because the same equations that produce attenuation $\alpha$ produce an equal phase shift $\beta$. In a good conductor, $\alpha = \beta$, which means the wave attenuates by a factor of $e^{-1}$ in exactly one radian of phase, an extreme case where the wave dies off in less than a wavelength. There is no "wave" left after a few skin depths. This is why a thin sheet of metal foil makes an excellent shield against high frequencies, and why a Faraday cage with even modest mesh stops a microwave oven leaking.

3.6 Phase velocity, group velocity, and dispersion

A pure sinusoid travels at the phase velocity $v_p = \omega/\beta$. A modulated signal (a packet of frequencies) travels at the group velocity $v_g = d\omega/d\beta$, which is the speed of energy and information.

In free space, a wave is non-dispersive; $v_p = v_g = c$. In a real waveguide, or in a dispersive medium (lossy dielectric, plasma), $v_p \neq v_g$, and the two can be very different. In some cases $v_p$ can exceed the speed of light. This sounds like a violation of relativity, but it is not, because $v_p$ is the speed of pure sinusoidal phase fronts, which carry no information. The information speed $v_g$ is always less than $c$ in any physical medium.

Confusion-busting picture. Imagine a long line of cars at a stoplight. When the light turns green, every car starts to move at almost the same time, but the front of the wave of motion propagates back through the line faster than any car actually moves. That backward-propagating "phase" is the phase velocity. Each car's actual progress is the group velocity. They are different things, and one can exceed the other.

3.7 Polarization

The direction of the $\vec{E}$ vector defines the polarization of a wave.

• Linear polarization: $\vec{E}$ stays along one axis. Vertical, horizontal, or any fixed angle.
• Circular polarization: $\vec{E}$ rotates uniformly as the wave propagates, always with constant magnitude. Right-hand circular if the rotation is clockwise as seen looking along the propagation direction; left-hand otherwise.
• Elliptical polarization: $\vec{E}$ rotates with varying magnitude.

Mathematically, circular polarization is two orthogonal linear waves of equal amplitude, 90° out of phase. Elliptical is two orthogonal waves of unequal amplitude or non-90° phase.

Real-world polarization examples:

• FM broadcast antennas are vertically polarized, matching car whip antennas.
• Old TV broadcast was horizontally polarized, matching horizontal Yagi rooftop antennas.
• Satellite TV uses circular polarization so dish orientation is not critical.
• 3D movie glasses (the modern, non-flickering kind) use opposite-handed circular polarization for left and right eyes.
• Radar systems often use circular polarization to penetrate rain, since spherical droplets reflect more weakly with circular than with linear polarization.
• GPS satellites transmit right-hand circular, and your phone's GPS antenna is matched to that polarization.

A polarization mismatch between transmit and receive antennas costs you signal. Cross-polarized linear-vs-linear: in theory, infinite loss; in practice, 20 to 30 dB. Linear vs circular: 3 dB loss, always. This is exploited in some interception attacks and counter-attacks: rotating a polarization filter near a transmitter can sometimes block direct emanations while letting a desired signal through.