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Electromagnetic Waves
Wave equation
For a homogeneous source-free medium,
\[\nabla^2\mathbf E-\mu\epsilon\frac{\partial^2\mathbf E}{\partial t^2}=0,\]with an analogous equation for $\mathbf H$.
The phase velocity in a lossless medium is
\[v_p=\frac{1}{\sqrt{\mu\epsilon}}.\]Frequency and wavelength
\[f=\frac{\omega}{2\pi},\qquad \lambda=\frac{v_p}{f}.\]In free space,
\[\lambda_0=\frac{c}{f}.\]A useful mental rule is
\[\boxed{\lambda_0(\text{cm})\approx\frac{30}{f(\text{GHz})}.}\]Propagation constant
\[\gamma=\alpha+j\beta.\]A wave varying along $z$ can be written
\[E(z)=E_0e^{-\alpha z}e^{-j\beta z}.\]Here $\alpha$ is attenuation constant and $\beta$ is phase constant. In a lossless medium,
\[\beta=\frac{2\pi}{\lambda}.\]Intrinsic impedance
In a general conducting medium,
\[\eta=\sqrt{\frac{j\omega\mu}{\sigma+j\omega\epsilon}}.\]For a lossless dielectric,
\[\eta=\sqrt{\frac{\mu}{\epsilon}}.\]In free space, $\eta_0\approx376.73\ \Omega$.
Plane-wave field relation
For propagation along $+z$ with $\mathbf E$ along $x$,
\[\mathbf H=\frac{1}{\eta}\hat{\mathbf z}\times\mathbf E.\]The time-average power density is
\[\langle\mathbf S\rangle=\frac12\operatorname{Re}\{\mathbf E\times\mathbf H^*\}.\]Lossy media
When $\sigma\neq0$, propagation attenuates. In a good conductor, fields penetrate only a skin depth
\[\delta\approx\sqrt{\frac{2}{\omega\mu\sigma}}.\]Thus the same wave equations connect free-space radiation to conductor shielding and skin effect.
Worked example — 100 MHz versus 10 GHz
At 100 MHz,
\[\lambda\approx3\ \text{m}.\]At 10 GHz,
\[\lambda\approx3\ \text{cm}.\]A 10-cm object is electrically tiny at 100 MHz but several wavelengths across at 10 GHz. The same physical object can therefore move from lumped-circuit behavior to strong scattering/resonance simply by changing frequency.
Measurement
Wave quantities are inferred from:
- phase/amplitude with VNA measurements;
- field probes;
- antenna measurements;
- time-domain reflectometry;
- material-property extraction from transmission/reflection;
- optical interferometry at much higher frequencies.
Engineering reality
References
- R. E. Collin, Foundations for Microwave Engineering.
- D. M. Pozar, Microwave Engineering.
- D. J. Griffiths, Introduction to Electrodynamics.
Related: Maxwell’s Equations · Polarization · Antennas · Orders of Magnitude
Advanced topic — evanescent spectra and dispersive propagation
Any sufficiently regular tangential field distribution on a plane can be decomposed into a transverse spatial spectrum. In a homogeneous isotropic medium,
\[k_z=\sqrt{k^2-k_x^2-k_y^2},\]with the outgoing/decaying branch chosen according to the time convention and radiation condition. Terms with (k_x^2+k_y^2>k^2) are evanescent in a lossless medium and decay away from the source plane. They contribute strongly to near fields but do not form propagating plane waves in the unbounded medium.
In a dispersive medium, phase velocity (v_p=\omega/\Re k) and group velocity (v_g=(d\Re k/d\omega)^{-1}) need not be equal. In absorptive bands, interpreting group velocity as energy or information velocity requires particular care.
Advanced exercise: Fourier-decompose a Gaussian aperture field and compare propagating versus evanescent spectral content as aperture width approaches a wavelength.
Further reading: W. C. Chew, Waves and Fields in Inhomogeneous Media.