10.04.02 · em-sr / maxwell-fields

EM waves and the wave equation

draft3 tiersLean: nonepending prereqs

Anchor (Master): Jackson, *Classical Electrodynamics*, 3e (1999), Ch. 7; Landau-Lifshitz Vol 2, Ch. 4

Intuition [Beginner]

Maxwell's equations 10.04.01 pending contain a feedback loop. Faraday's law says a changing magnetic field produces an electric field. The Maxwell-Ampere law says a changing electric field produces a magnetic field. In empty space — no charges, no currents — the two feed each other. A disturbance in $E$ creates a disturbance in $B$ , which creates a new disturbance in $E$ , and so on. This self-sustaining cycle propagates outward through space as an electromagnetic wave.

The wave travels at a fixed speed determined by two constants of nature: the electric constant $ϵ_{0}$ and the magnetic constant $μ_{0}$ . That speed is $c = 1/ ϵ_{0} μ_{0}$ . Plugging in the measured values gives approximately $3 \times 1 0^{8}$ m/s — the speed of light. Maxwell's equations predict that light is an electromagnetic wave.

Different frequencies of oscillation correspond to different parts of the electromagnetic spectrum: radio waves (low frequency), microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays (high frequency). They are all the same physical phenomenon — propagating $E$ and $B$ fields — distinguished only by how rapidly they oscillate.

In an electromagnetic wave the electric field $E$ and magnetic field $B$ oscillate perpendicular to each other and both are perpendicular to the direction the wave travels. If the wave moves in the $z$ -direction, $E$ might point along $x$ and $B$ along $y$ — all three at right angles.

Visual [Beginner]

Picture a wave crest moving to the right along the $z$ -axis. The electric field $E$ oscillates up and down along the $x$ -axis — a sinusoidal ripple in the vertical plane. The magnetic field $B$ oscillates in and out along the $y$ -axis — a sinusoidal ripple in the horizontal plane. The two ripples are in phase: $E$ is at its peak exactly where $B$ is at its peak, and both are zero at the same points.

The distance between successive crests is the wavelength $λ$ . The number of complete oscillations per second at any fixed point is the frequency $f$ . They are related by $λ f = c$ — higher frequency means shorter wavelength. A snapshot of the wave at one instant looks like two perpendicular sine curves, one in the $x z$ -plane and one in the $y z$ -plane, both marching along $z$ .

Worked example [Beginner]

A radio station broadcasts at frequency $f = 100$ MHz ( $= 100 \times 1 0^{6}$ Hz). What is the wavelength?

Use the relation $λ = c / f$ :

λ = \frac{3 \times 1 0 ^{8} m/s}{100 \times 1 0 ^{6} Hz} = \frac{3 \times 1 0 ^{8}}{1 0 ^{8}} = 3 m .

A 100 MHz radio wave has a wavelength of 3 metres — roughly the height of a room. This is why FM radio antennas are about 1.5 m long (half a wavelength is an efficient antenna length).

By the same formula, visible light at frequency $f \approx 5 \times 1 0^{14}$ Hz has wavelength $λ \approx 600$ nm — about a thousand times thinner than a human hair. X-rays at $f \approx 3 \times 1 0^{18}$ Hz have $λ \approx 0.1$ nm, comparable to atomic spacing in a crystal.

Check your understanding [Beginner]

Formal definition [Intermediate+]

In vacuum ( $ρ = 0$ , $J = 0$ ), Maxwell's equations 10.04.01 pending reduce to

\nabla \cdot E = 0, \nabla \times E = - \frac{\partial B}{\partial t},

\nabla \cdot B = 0, \nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t} .

Take the curl of Faraday's law: $\nabla \times (\nabla \times E) = - \partial (\nabla \times B) / \partial t$ . Substitute the Maxwell-Ampere law for $\nabla \times B$ on the right:

\nabla \times (\nabla \times E) = - μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}} .

The vector identity $\nabla \times (\nabla \times E) = \nabla (\nabla \cdot E) - \nabla^{2} E$ , together with $\nabla \cdot E = 0$ , gives

\nabla^{2} E = μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}} .

The same calculation starting from the curl of the Maxwell-Ampere law produces the identical equation for $B$ :

\nabla^{2} B = μ_{0} ϵ_{0} \frac{\partial ^{2} B}{\partial t ^{2}} .

Each component of $E$ and $B$ satisfies the wave equation with propagation speed $v = 1/ μ_{0} ϵ_{0} = c$ . This is the central prediction: vacuum Maxwell equations imply that electromagnetic disturbances propagate as waves at speed $c$ .

Defining the d'Alembertian operator $□ := \nabla^{2} - μ_{0} ϵ_{0} \partial^{2} / \partial t^{2} = \nabla^{2} - c^{- 2} \partial^{2} / \partial t^{2}$ , the vacuum equations read

□ E = 0, □ B = 0, \nabla \cdot E = 0, \nabla \cdot B = 0.

The divergence-free conditions are constraints: not every solution of the scalar wave equation for each component gives a valid electromagnetic wave. The components must combine into divergence-free vector fields.

Plane-wave solutions. A monochromatic plane wave propagating in the direction of the unit vector $\hat{k}$ with angular frequency $ω$ and wave vector $k = (ω / c) \hat{k}$ has the form

E (r, t) = E_{0} e^{i (k \cdot r - ω t)}, B (r, t) = B_{0} e^{i (k \cdot r - ω t)},

where $E_{0}$ and $B_{0}$ are constant complex amplitudes (physical fields are the real parts). Substituting into the divergence conditions forces $\hat{k} \cdot E_{0} = 0$ and $\hat{k} \cdot B_{0} = 0$ — the fields are transverse. Faraday's law relates the amplitudes:

B_{0} = \frac{1}{c} \hat{k} \times E_{0}, ∣ B_{0} ∣ = \frac{∣ E _{0} ∣}{c} .

The $E$ and $B$ fields of a plane wave have equal magnitudes (up to the factor $1/ c$ ), are perpendicular to each other, and are both perpendicular to the propagation direction.

Polarization

For a plane wave propagating along $\hat{z}$ , the transversality condition restricts $E$ to the $x y$ -plane. Write $E_{0} = E_{x} \hat{x} + E_{y} \hat{y}$ where $E_{x}, E_{y}$ are complex.

Linear polarization. If $E_{x}$ and $E_{y}$ have the same phase (their ratio is real), the electric field oscillates along a fixed direction in the $x y$ -plane. At any fixed point the tip of $E$ traces a line segment.

Circular polarization. If $E_{x}$ and $E_{y}$ have equal magnitude and a phase difference of $\pm π /2$ — for instance $E_{x} = E_{0}$ , $E_{y} = \pm i E_{0}$ — the tip of $E$ traces a circle. The sign determines the handedness: $E_{y} = + i E_{0}$ gives right-hand circular polarization (the field rotates clockwise as viewed facing the oncoming wave in the standard optics convention; note that the physics and optics conventions for handedness differ), and $E_{y} = - i E_{0}$ gives left-hand circular.

General (elliptical) polarization. For arbitrary complex $E_{x}, E_{y}$ , the tip of $E$ traces an ellipse. Linear and circular are the degenerate special cases.

Energy and momentum

The energy density stored in the electromagnetic field is

u = \frac{1}{2} ϵ_{0} ∣ E ∣^{2} + \frac{1}{2 μ _{0}} ∣ B ∣^{2} .

For a plane wave with $∣ B ∣ = ∣ E ∣/ c$ and $c^{2} = 1/ (μ_{0} ϵ_{0})$ , the electric and magnetic contributions are equal: $u = ϵ_{0} ∣ E ∣^{2}$ .

The Poynting vector

S = \frac{1}{μ _{0}} E \times B

gives the energy flux (power per unit area carried by the wave). For a plane wave, $S = u c \hat{k}$ — energy flows in the propagation direction at speed $c$ .

The intensity (time-averaged power per unit area) of a monochromatic plane wave with electric-field amplitude $E_{0}$ is

I = ⟨ ∣ S ∣ ⟩ = \frac{1}{2} ϵ_{0} c E_{0}^{2} .

The factor $1/2$ comes from time-averaging $cos^{2} (ω t)$ .

Electromagnetic waves also carry momentum. The momentum density is $g = S / c^{2} = ϵ_{0} E \times B$ , and the radiation pressure on a perfectly absorbing surface is $P = I / c$ .

Key theorem with proof [Intermediate+]

Theorem (Wave equation from Maxwell's equations in vacuum). In a region of space free of charges and currents, each Cartesian component of $E$ and of $B$ satisfies the wave equation $\nabla^{2} ψ = μ_{0} ϵ_{0} \partial^{2} ψ / \partial t^{2}$ , with propagation speed $c = 1/ μ_{0} ϵ_{0}$ . Furthermore, the transversality conditions $\nabla \cdot E = 0$ and $\nabla \cdot B = 0$ force the physical solutions to be transverse to the propagation direction.

Proof. Start with Maxwell's equations in vacuum:

\nabla \cdot E = 0 (i), \nabla \times E = - \frac{\partial B}{\partial t} (ii),

\nabla \cdot B = 0 (iii), \nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t} (iv) .

Step 1. Take the curl of (ii): $\nabla \times (\nabla \times E) = - \partial (\nabla \times B) / \partial t$ .

Step 2. Apply the vector identity $\nabla \times (\nabla \times V) = \nabla (\nabla \cdot V) - \nabla^{2} V$ to the left-hand side and substitute (i) and (iv):

\nabla (\nabla \cdot E) - \nabla^{2} E = - \frac{\partial}{\partial t} (μ_{0} ϵ_{0} \frac{\partial E}{\partial t}) .

The first term vanishes by (i), giving

\nabla^{2} E = μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}} .

Step 3. The identical argument starting from the curl of (iv) and using (iii) and (ii) produces $\nabla^{2} B = μ_{0} ϵ_{0} \partial^{2} B / \partial t^{2}$ .

Step 4. For a monochromatic plane wave $E = E_{0} e^{i (k \cdot r - ω t)}$ , the divergence condition $\nabla \cdot E = 0$ becomes $i k \cdot E_{0} = 0$ , so $E_{0} ⊥ \hat{k}$ . The dispersion relation from the wave equation is $∣ k ∣^{2} = ω^{2} / c^{2}$ . From Faraday's law, $B_{0} = (1/ c) \hat{k} \times E_{0}$ , which is automatically perpendicular to both $\hat{k}$ and $E_{0}$ . $□$

Counterexamples to common slips

The wave equation alone does not guarantee a valid electromagnetic wave. Any function satisfying $\nabla^{2} ψ = c^{- 2} \partial^{2} ψ / \partial t^{2}$ for each component is a wave-equation solution, but only divergence-free combinations satisfying Faraday's law and the Maxwell-Ampere law are physical EM waves. A longitudinal wave ( $E$ parallel to $\hat{k}$ ) satisfies the scalar wave equation componentwise but violates $\nabla \cdot E = 0$ .
The relation $∣ B ∣ = ∣ E ∣/ c$ holds for plane waves, not for arbitrary field configurations. Near an antenna or in a standing wave the instantaneous ratio $∣ E ∣/∣ B ∣$ can differ from $c$ .
Polarization is defined relative to the propagation direction. For a wave reflecting off a surface, the reflected wave has a different propagation direction and therefore a different transverse plane; polarization states do not transfer unchanged across reflections without applying the Fresnel equations.

Exercises [Intermediate+]

Exercise 6 (medium, symbolic).

Starting from Maxwell's equations in vacuum, show that the electromagnetic energy density $u = \frac{1}{2} ϵ_{0} ∣ E ∣^{2} + \frac{1}{2 μ _{0}} ∣ B ∣^{2}$ satisfies the continuity equation $\partial u / \partial t + \nabla \cdot S = 0$ where $S = μ_{0}^{- 1} E \times B$ .

Hint

Differentiate $u$ in time, substitute the time-derivatives of $E$ and $B$ from Maxwell's equations, and use a vector identity on the resulting terms.

Answer

$\partial u / \partial t = ϵ_{0} E \cdot \partial E / \partial t + μ_{0}^{- 1} B \cdot \partial B / \partial t$ . From Maxwell: $\partial E / \partial t = (μ_{0} ϵ_{0})^{- 1} \nabla \times B$ and $\partial B / \partial t = - \nabla \times E$ . Substituting:

\frac{\partial u}{\partial t} = \frac{1}{μ _{0}} (E \cdot (\nabla \times B) - B \cdot (\nabla \times E)) .

The vector identity $\nabla \cdot (E \times B) = B \cdot (\nabla \times E) - E \cdot (\nabla \times B)$ gives $E \cdot (\nabla \times B) - B \cdot (\nabla \times E) = - \nabla \cdot (E \times B)$ . Therefore

\frac{\partial u}{\partial t} = - \frac{1}{μ _{0}} \nabla \cdot (E \times B) = - \nabla \cdot S .

This is Poynting's theorem in vacuum: energy is locally conserved, with $S$ as the energy current.

Exercise 8 (hard, symbolic).

Derive the wave equation for $E$ in a conducting medium with conductivity $σ$ (so $J = σ E$ ). Show that the resulting equation is a damped wave equation and find the skin depth $δ$ — the distance over which the amplitude decays by $1/ e$ — in the high-frequency limit $ω ≫ σ / (ϵ_{0})$ .

Hint

In a conductor the Maxwell-Ampere law gains a $J$ term: $\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \partial E / \partial t = μ_{0} σ E + μ_{0} ϵ_{0} \partial E / \partial t$ . Repeat the curl-of-curl derivation with this extra term.

Answer

With $J = σ E$ , taking the curl of Faraday's law and substituting as before yields

\nabla^{2} E = μ_{0} σ \frac{\partial E}{\partial t} + μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}} .

The first term is a damping/attenuation term. For a plane-wave ansatz $E = E_{0} e^{i (k z - ω t)}$ , substitution gives $- k^{2} = - i μ_{0} σ ω - μ_{0} ϵ_{0} ω^{2}$ , so $k^{2} = μ_{0} ϵ_{0} ω^{2} (1 + iσ / (ϵ_{0} ω))$ . Writing $k = k_{R} + i k_{I}$ with $k_{I} > 0$ , the wave amplitude decays as $e^{- k_{I} z}$ , giving skin depth $δ = 1/ k_{I}$ . In the high-frequency limit $σ / (ϵ_{0} ω) ≪ 1$ :

k \approx \frac{ω}{c} (1 + \frac{iσ}{2 ϵ _{0} ω}), k_{I} \approx \frac{σ}{2} \frac{μ _{0}}{ϵ _{0}} \cdot \frac{1}{c} = \frac{σ}{2} \frac{μ _{0}}{ϵ _{0}} \frac{1}{c} .

More precisely: $δ = 1/ k_{I} \approx 2 ϵ_{0} c / (σ μ_{0} / ϵ_{0} \cdot c) = 2/ (σ μ_{0} / ϵ_{0}) = 2/ (σ μ_{0} c)$ . In the low-frequency limit (the usual skin-depth regime for metals), $k \approx (1 + i) / δ_{0}$ with $δ_{0} = 2/ (μ_{0} σ ω)$ , the standard skin-depth formula.

Exercise 9 (hard, symbolic).

Show that for any vacuum electromagnetic wave (not necessarily a plane wave), the time-averaged Poynting vector equals the time-averaged energy density times $c$ times the propagation direction, provided the wave is monochromatic — that is, show $⟨ S ⟩ = ⟨ u ⟩ c \hat{k}$ for a monochromatic wave propagating in the $\hat{k}$ direction.

Hint

Use the complex representation and the relation $B_{0} = (1/ c) \hat{k} \times E_{0}$ . For time averages, use $⟨ Re (A e^{- iω t}) Re (B e^{- iω t})⟩ = \frac{1}{2} Re (A B^{*})$ .

Answer

Write $E = Re (\tilde{E} e^{- iω t})$ and $B = Re (\tilde{B} e^{- iω t})$ where $\tilde{E} = E_{0} e^{i k \cdot r}$ and $\tilde{B} = B_{0} e^{i k \cdot r}$ . Using the time-average identity:

⟨ u ⟩ = \frac{1}{2} ϵ_{0} ∣ E_{0} ∣^{2} + \frac{1}{2 μ _{0}} ∣ B_{0} ∣^{2} = \frac{1}{2} ϵ_{0} ∣ E_{0} ∣^{2} + \frac{1}{2 μ _{0}} \frac{∣ E _{0} ∣ ^{2}}{c ^{2}} = ϵ_{0} ∣ E_{0} ∣^{2} .

⟨ S ⟩ = \frac{1}{2 μ _{0}} Re (\tilde{E} \times \tilde{B}^{*}) = \frac{1}{2 μ _{0}} Re (E_{0} \times B_{0}^{*}) .

Since $B_{0} = (1/ c) \hat{k} \times E_{0}$ and $\hat{k} \times E_{0}$ is real for linear polarization (and the relation holds for general polarization), $E_{0} \times B_{0}^{*} = (1/ c) E_{0} \times (\hat{k} \times E_{0}^{*}) = (1/ c) (∣ E_{0} ∣^{2} \hat{k} - (E_{0}^{*} \cdot \hat{k}) E_{0})$ . By transversality $\hat{k} \cdot E_{0} = 0$ , this simplifies to $(∣ E_{0} ∣^{2} / c) \hat{k}$ . Therefore $⟨ S ⟩ = ∣ E_{0} ∣^{2} \hat{k} / (2 μ_{0} c) = \frac{1}{2} ϵ_{0} ∣ E_{0} ∣^{2} c \hat{k} = ⟨ u ⟩ c \hat{k}$ .

Exercise 10 (hard, symbolic).

A standing wave is formed by the superposition of two counter-propagating plane waves: $E = E_{0} [cos (k z - ω t) + cos (k z + ω t)] \hat{x}$ . Simplify using a trigonometric identity. Show that the resulting Poynting vector has zero time-average, and explain why.

Hint

Use $cos A + cos B = 2 cos ((A + B) /2) cos ((A - B) /2)$ . Compute $B$ for each traveling wave, add them, and form $E \times B$ .

Answer

$cos (k z - ω t) + cos (k z + ω t) = 2 cos (k z) cos (ω t)$ . So $E = 2 E_{0} cos (k z) cos (ω t) \hat{x}$ .

The forward wave has $B_{+} = (E_{0} / c) cos (k z - ω t) \hat{y}$ ; the backward wave has $B_{-} = - (E_{0} / c) cos (k z + ω t) \hat{y}$ . Adding: $B = (E_{0} / c) [cos (k z - ω t) - cos (k z + ω t)] \hat{y} = (2 E_{0} / c) sin (k z) sin (ω t) \hat{y}$ .

The Poynting vector is $S = μ_{0}^{- 1} E \times B = (4 E_{0}^{2} / (μ_{0} c)) cos (k z) sin (k z) cos (ω t) sin (ω t) \hat{z}$ . Using $sin (2 ω t) = 2 sin (ω t) cos (ω t)$ :

S = \frac{2 E _{0}^{2}}{μ _{0} c} sin (2 k z) \cdot \frac{1}{2} sin (2 ω t) \hat{z} .

Time-averaging $sin (2 ω t)$ over a period gives zero. A standing wave does not transport energy on average — energy oscillates back and forth between electric and magnetic forms at each point, but the net flow is zero. This contrasts with a traveling wave where $⟨ S ⟩ \neq = 0$ .

Complex representation and wave packets [Master]

The complex plane-wave ansatz $E = E_{0} e^{i (k \cdot r - ω t)}$ with the physical field as the real part is a bookkeeping device for linear operations. Products (energy density, Poynting vector) require care: for two complex fields $A$ and $B$ representing real quantities, $Re (A) Re (B) \neq = Re (A B)$ in general. The correct product rule for time averages of monochromatic fields is $⟨ Re (A e^{- iω t}) Re (B e^{- iω t})⟩ = \frac{1}{2} Re (A B^{*})$ .

Wave packets. A monochromatic plane wave extends over all space and carries infinite energy. Physical waves are wave packets: superpositions of plane waves centered on a central wave vector $k_{0}$ with a spread $Δ k$ . Write

E (r, t) = \int \tilde{E} (k) e^{i (k \cdot r - ω (k) t)} d^{3} k,

where $ω (k)$ is the dispersion relation. In vacuum $ω = c ∣ k ∣$ , which is linear: the phase velocity $v_{p} = ω / k = c$ equals the group velocity $v_{g} = \partial ω / \partial k = c$ . A vacuum wave packet propagates without distortion — all frequency components travel at the same speed.

In a medium the dispersion relation becomes nonlinear ( $ω = ω (k)$ depends on the material's electromagnetic response), and $v_{p} \neq = v_{g}$ in general. The packet envelope moves at the group velocity $v_{g} = \nabla_{k} ω$ , while the individual crests move at the phase velocity $v_{p} = ω / k$ . Group velocity is the signal velocity (energy and information transport) in normal-dispersion regions; near resonances the picture is subtler and the front velocity, not the group velocity, is the causally relevant quantity.

Dispersion in media [Master]

In a linear, isotropic medium characterized by frequency-dependent permittivity $ϵ (ω)$ and permeability $μ (ω)$ , the wave equation becomes

\nabla^{2} E = μ (ω) ϵ (ω) \frac{\partial ^{2} E}{\partial t ^{2}},

with refractive index $n (ω) = c μ (ω) ϵ (ω)$ and dispersion relation $k = n (ω) ω / c$ . The frequency dependence of $n$ is dispersion: different frequencies travel at different speeds, causing pulse broadening.

The ** Sellmeier equation** and Cauchy formula are empirical models for $n (ω)$ in transparent regions away from absorption resonances. Near resonances, $n (ω)$ develops an imaginary part $κ (ω)$ — the extinction coefficient — and the refractive index becomes complex: $\overset{n}{^} = n + iκ$ . The wave attenuates as $e^{- κω z / c}$ .

Normal dispersion ( $d n / d ω > 0$ , the typical situation away from resonances) gives $v_{g} < v_{p} < c$ . Anomalous dispersion ( $d n / d ω < 0$ , near absorption lines) can give $v_{g} > c$ or even $v_{g} < 0$ , but this does not violate causality: the front velocity and the information velocity never exceed $c$ .

Fresnel equations [Master]

At a planar interface between two media with refractive indices $n_{1}$ and $n_{2}$ , Maxwell's equations plus boundary conditions (continuity of the tangential components of $E$ and $H$ , continuity of the normal components of $D$ and $B$ ) determine the reflected and transmitted amplitudes.

For a wave incident at angle $θ_{i}$ from medium 1, Snell's law gives the refraction angle: $n_{1} sin θ_{i} = n_{2} sin θ_{t}$ . The Fresnel equations for the amplitude reflection and transmission coefficients depend on the polarization of the incident wave relative to the plane of incidence:

s-polarization ( $E$ perpendicular to the plane of incidence):

r_{s} = \frac{n _{1} cos θ _{i} - n _{2} cos θ _{t}}{n _{1} cos θ _{i} + n _{2} cos θ _{t}}, t_{s} = \frac{2 n _{1} cos θ _{i}}{n _{1} cos θ _{i} + n _{2} cos θ _{t}} .

p-polarization ( $E$ in the plane of incidence):

r_{p} = \frac{n _{2} cos θ _{i} - n _{1} cos θ _{t}}{n _{2} cos θ _{i} + n _{1} cos θ _{t}}, t_{p} = \frac{2 n _{1} cos θ _{i}}{n _{2} cos θ _{i} + n _{1} cos θ _{t}} .

At the Brewster angle $θ_{B} = arctan (n_{2} / n_{1})$ , $r_{p} = 0$ : p-polarized light is fully transmitted with no reflection. This is why reflected light from a dielectric surface is predominantly s-polarized — the principle behind polarizing sunglasses.

At normal incidence ( $θ_{i} = 0$ ), both reduce to $r = (n_{1} - n_{2}) / (n_{1} + n_{2})$ and $t = 2 n_{1} / (n_{1} + n_{2})$ . The reflectance (power fraction reflected) is $R = ∣ r ∣^{2} = (n_{1} - n_{2})^{2} / (n_{1} + n_{2})^{2}$ . For glass ( $n \approx 1.5$ ) in air, $R \approx 4%$ per surface.

Waveguides [Master]

A waveguide is a hollow conducting tube (typically rectangular or circular cross-section) that confines and guides electromagnetic waves. The conducting walls impose boundary conditions ( $E_{∥} = 0$ and $B_{⊥} = 0$ at the walls) that quantize the allowed transverse wave vectors.

For a rectangular waveguide with dimensions $a > b$ (width, height), the modes separate into TE modes (transverse electric: $E_{z} = 0$ ) and TM modes (transverse magnetic: $B_{z} = 0$ ). Each mode has a cutoff frequency

ω_{mn} = c (mπ / a)^{2} + (nπ / b)^{2},

below which the mode does not propagate (the wave vector becomes imaginary, yielding exponential attenuation). The dominant mode (lowest cutoff) is TE $_{10}$ with $ω_{10} = c π / a$ .

Above cutoff the propagation constant is $β = (ω / c)^{2} - (ω_{mn} / c)^{2}$ , giving phase velocity $v_{p} = ω / β > c$ and group velocity $v_{g} = d ω / d β = c^{2} / v_{p} < c$ . The product $v_{p} v_{g} = c^{2}$ — neither velocity exceeds the constraint imposed by causality; the superluminal phase velocity is an artifact of the modal structure and does not carry information.

Causality and the Kramers-Kronig relations [Master]

The frequency-dependent response functions $ϵ (ω)$ and $μ (ω)$ are not arbitrary: causality (the medium cannot respond before the field arrives) imposes powerful constraints. Writing the electric susceptibility $χ_{e} (ω) = ϵ (ω) / ϵ_{0} - 1$ , its Fourier transform $χ_{e} (t)$ must vanish for $t < 0$ .

If $χ_{e} (ω)$ is analytic in the upper half-plane (no poles there, which causality enforces), then the real and imaginary parts of $χ_{e} (ω) = χ_{e}^{'} (ω) + i χ_{e}^{''} (ω)$ are related by the Kramers-Kronig relations:

χ_{e}^{'} (ω) = \frac{1}{π} P \int_{- \infty}^{\infty} \frac{χ _{e}^{''} ( ω ^{'} )}{ω ^{'} - ω} d ω^{'},

χ_{e}^{''} (ω) = - \frac{1}{π} P \int_{- \infty}^{\infty} \frac{χ _{e}^{'} ( ω ^{'} )}{ω ^{'} - ω} d ω^{'},

where $P$ denotes the Cauchy principal value. These are Hilbert transform pairs. The physical content: absorption ( $χ_{e}^{''} > 0$ ) and dispersion ( $χ_{e}^{'}$ varying with $ω$ ) are not independent — you cannot have one without the other. Any medium that absorbs at some frequencies must disperse at nearby frequencies. This is why anomalous dispersion always accompanies absorption lines.

The Kramers-Kronig relations follow from the analyticity of $χ_{e} (ω)$ in the upper half $ω$ -plane, which is itself a consequence of causality. Their proof uses contour integration: the function $χ_{e} (ω^{'}) / (ω^{'} - ω)$ integrated around a semicircular contour in the upper half-plane (closed above, since $χ_{e}$ is analytic there) picks up only the pole at $ω^{'} = ω$ , and the principal-value prescription handles this pole. The semicircle at infinity contributes zero if $χ_{e} (ω) \to 0$ as $∣ ω ∣ \to \infty$ .

Connections [Master]

Maxwell's equations in differential form 10.04.01 pending are the starting point for everything in this unit. The vacuum subset ( $ρ = 0$ , $J = 0$ ) is what produces the wave equation; the full equations with sources produce the retarded-potential solutions needed for radiation 10.07.01 pending.
Faraday's law 10.03.01 pending provides half the feedback loop — changing $B$ generates $E$ — that makes self-sustaining wave propagation possible. The other half comes from the displacement-current term in the Maxwell-Ampere law.
Radiation 10.07.01 pending (pending) picks up where this unit leaves off. The plane-wave solutions derived here are the far-field limit of the radiation fields produced by time-varying charge and current distributions. The retarded-potential framework for radiation takes the wave equation with sources as its starting point.
Spectroscopy 14.12.01 (pending) in chemistry measures absorption of electromagnetic waves at frequencies matching molecular energy-level transitions. The wave equation, polarization states, and the Poynting vector developed here are the classical substrate that the quantum treatment of light-matter interaction refines.
The d'Alembertian operator and special relativity connect through the observation that $□ = \nabla^{2} - c^{- 2} \partial^{2} / \partial t^{2}$ is the Lorentz-invariant wave operator on Minkowski spacetime. The wave equation $□ ψ = 0$ is preserved under Lorentz transformations, which is the first hint that Maxwell's equations are relativistic — a point developed systematically in the relativistic-electrodynamics chapter.
Group velocity and wave packets reappear in quantum mechanics 12.02.01 pending (pending): the de Broglie relations assign a wave vector and frequency to each momentum-energy eigenstate, and wave-packet propagation at the group velocity is how quantum-mechanical probability densities move. The mathematical framework is identical; the physical interpretation changes.
Kramers-Kronig relations are a special case of the causality constraints that appear throughout physics: the Titchmarsh theorem in signal processing, the optical theorem in scattering theory, and the dispersion relations of high-energy physics all have the same Hilbert-transform structure, all originating from the analyticity enforced by causality.

Historical & philosophical context [Master]

Maxwell's 1865 paper "A Dynamical Theory of the Electromagnetic Field" derived the wave equation from his equations and computed the propagation speed $c = 1/ μ_{0} ϵ_{0}$ . The agreement with Fizeau's measured speed of light ( $\approx 3.15 \times 1 0^{8}$ m/s) was the first quantitative evidence that light is electromagnetic. Maxwell wrote: "We have strong reason to conclude that light itself — including radiant heat and other radiations — is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field."

Heinrich Hertz's experiments (1886–1888) provided the direct experimental confirmation. Hertz generated electromagnetic waves with a spark-gap oscillator, detected them at a distance, verified their reflection, refraction, polarization, and standing-wave interference, and measured their speed as approximately $c$ . The wavelength of Hertz's waves was about 1 m — what we now call radio waves. Hertz's apparatus was the first radio transmitter and receiver; his experiments transformed Maxwell's theoretical prediction into a demonstrated physical phenomenon.

The wave-equation derivation carries a philosophical lesson: the speed of light is not a free parameter of nature but a consequence of the electromagnetic constants $ϵ_{0}$ and $μ_{0}$ . This means that $c$ is determined by the properties of the electromagnetic field itself, not by the motion of any medium. The failure of attempts to detect motion relative to a luminiferous ether (Michelson-Morley, 1887) confirmed this independence and set the stage for Einstein's 1905 identification of $c$ as a universal speed limit in special relativity. The wave equation, derived from Maxwell, was the first physical law to demand a relativistic spacetime structure.

Bibliography [Master]

Primary literature (cite when used; not all currently in reference/):

Maxwell, J. C., "A Dynamical Theory of the Electromagnetic Field", Phil. Trans. Roy. Soc. 155 (1865), 459–512. [Originator paper — derives wave equation and identifies light as EM.]
Hertz, H., "On Electromagnetic Waves in Air and Their Reflection", Annalen der Physik 34 (1888), 610–623. [Experimental verification.]
Fresnel, A., "Memoire sur la double refraction", Mem. Acad. Sci. 7 (1827), 45–176. [Fresnel equations originator.]
Kramers, H. A., "La diffusion de la lumiere par les atomes", Atti Cong. Intern. Fisici, Como 2 (1927), 545–557. [Kramers-Kronig relations.]
Kronig, R. de L., "On the Theory of Dispersion of X-rays", J. Opt. Soc. Am. 12 (1926), 547–557. [Independent derivation.]

Textbooks and monographs:

Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017). Ch. 9.
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999). Ch. 7.
Landau, L. D. & Lifshitz, E. M., The Classical Theory of Fields, 4th ed. (Course of Theoretical Physics Vol. 2, Pergamon, 1980). Ch. 4.
Zangwill, A., Modern Electrodynamics (Cambridge, 2013). Ch. 16–17.
Susskind, L. & Friedman, A., Special Relativity and Classical Field Theory (Basic Books, 2017). Lectures 5–6.
Tong, D., Electromagnetism (DAMTP Cambridge lecture notes), §4 "Electromagnetic Waves".