Plane wave

A plane wave is the simplest solution of the source-free Maxwell’s equations in a homogeneous medium: an electromagnetic wave whose phase fronts (surfaces of constant phase) are infinite parallel planes, and whose amplitude is uniform across each plane.

For a plane wave propagating in $+\hat{\mathbf{z}}$ with electric field along $\hat{\mathbf{x}}$:

$\mathbf{E}(\mathbf{r},t)=\hat{\mathbf{x}}{E}_0\cos (\omega t-\beta z+{\varphi }_0).$

Field is constant on any plane $z=$ const, oscillates sinusoidally in time and space, and propagates at phase velocity $\omega /\beta$.

The accompanying magnetic field is in $\hat{\mathbf{y}}$ (perpendicular to both $\mathbf{E}$ and the propagation direction):

$\mathbf{H}(\mathbf{r},t)=\hat{\mathbf{y}}\frac{E_0}{\eta }\cos (\omega t-\beta z+{\varphi }_0).$

The ratio $|\mathbf{E}|/|\mathbf{H}|=\eta =\sqrt{\mu /ϵ}$ is the intrinsic impedance of the medium. In free space: ${\eta }_0=\sqrt{{\mu }_0/{ϵ}_0}\approx 377$ Ω.

Why it’s “transverse electromagnetic” (TEM)

Both $\mathbf{E}$ and $\mathbf{H}$ are perpendicular to the direction of propagation, and to each other. The triad $(\mathbf{E},\mathbf{H},\hat{\mathbf{k}})$ forms a right-handed orthogonal set, where $\hat{\mathbf{k}}$ is the propagation direction.

This is what “transverse” means in TEM: there are no field components along the wave’s travel direction. Different geometries of guided waves (TE, TM, hybrid) can have non-transverse components, but uniform plane waves in unbounded space are always TEM.

The wave equation

Plane waves are solutions of the wave equation derived in source-free space:

${\nabla }^2\mathbf{E}-\mu ϵ\frac{{\partial }^2\mathbf{E}}{\partial t^2}=0,$

with propagation speed

$u_p=\frac{1}{\sqrt{\mu ϵ}}=\frac{c}{\sqrt{{\mu }_r{ϵ}_r}}.$

In free space: $u_p=c=3\times 10^8$ m/s.

The frequency $f$, wavelength $\lambda$, angular frequency $\omega$, and propagation constant $\beta$ are related by:

$\omega =2\pi f,\beta =\frac{\omega }{u_p}=\frac{2\pi }{\lambda },u_p=f\lambda .$

Phasor form

In sinusoidal steady state, drop the time dependence and the cosine:

$\tilde{\mathbf{E}}(\mathbf{r})=\hat{\mathbf{x}}{E}_0{e}^{-j\beta z}.$

The complex amplitude $E_0$ can carry a phase. The full time-dependent field is $\mathbf{E}=\text{Re}[\tilde{\mathbf{E}}{e}^{j\omega t}]$.

In phasor form, Maxwell’s equations become algebraic:

$\nabla \times \tilde{\mathbf{E}}=-j\omega \mu \tilde{\mathbf{H}},\nabla \times \tilde{\mathbf{H}}=j\omega ϵ\tilde{\mathbf{E}},$

and the wave equation is

${\nabla }^2\tilde{\mathbf{E}}+{\beta }^2\tilde{\mathbf{E}}=0\text{(Helmholtz equation)},$

with $\beta =\omega \sqrt{\mu ϵ}$. The propagation constant $\beta$ is the “spatial frequency” of the wave.

Lossy media

In a medium with finite conductivity $\sigma$, replace $ϵ$ by the complex permittivity ${ϵ}_c=ϵ(1-j\sigma /\omega ϵ)$. The propagation constant becomes complex: $\gamma =\alpha +j\beta$, with:

• $\alpha$ — attenuation constant (Np/m): how fast the wave decays.
• $\beta$ — phase constant (rad/m): the spatial frequency.

The field becomes

$\tilde{\mathbf{E}}=\hat{\mathbf{x}}{E}_0{e}^{-\alpha z}{e}^{-j\beta z},$

with exponential decay $e^{-\alpha z}$. The intrinsic impedance is also complex, and $\mathbf{E}$ and $\mathbf{H}$ have a phase lag relative to each other.

Why plane waves are useful

Real EM waves are not infinite plane waves (no such source exists). But:

1. Far from a localized source, the wavefront curvature is small, and locally the wave looks like a plane wave. The plane-wave solution is the standard starting point.
2. By Fourier decomposition, any wave field can be written as a superposition of plane waves with different propagation directions and frequencies. Solving for plane waves gives a building block for general solutions.
3. The plane-wave solution captures all the essential physics — propagation speed, polarization, energy flow (via Poynting vector), reflection and transmission at interfaces — without geometric complications.

This is why textbooks introduce plane waves first, even though they’re an idealization.