Wave equations and plane waves
Seawater reflects almost all of a long-wave radio signal and swallows within a few meters whatever gets in. That is why a submerged submarine can only be reached on very low frequencies.
01From Maxwell to the wave equation
Combining the two curl equations of the previous unit in a source-free region, the electric field is governed by a single equation, the Helmholtz equation:
The simplest solution is the plane wave: a uniform field in every plane perpendicular to the direction of propagation, advancing as \( e^{-\gamma z} = e^{-\alpha z} e^{-j\beta z} \). It is the model for the wave from a distant antenna: many wavelengths away from the source, any spherical wave looks locally flat.
02The wave in a lossless medium
With \( \sigma = 0 \), the wave is not attenuated and its parameters come directly from \( \mu \) and \( \varepsilon \):
E and H are perpendicular to each other and to the direction of propagation, and are in phase. The power density is \( |E|^2 / 2\eta \): with an amplitude of 1 V/m in air, 1.33 mW per square meter. The direction in which the electric field oscillates is the polarization of the wave.
03The wave in a lossy medium
With conductivity, \( \gamma \) and \( \eta \) become complex. The two limiting cases are the useful ones:
| Low-loss dielectric, \( \sigma \ll \omega\varepsilon \) | Good conductor, \( \sigma \gg \omega\varepsilon \) | |
|---|---|---|
| Attenuation | \( \alpha \approx \dfrac{\sigma}{2}\sqrt{\mu/\varepsilon} \) | \( \alpha \approx \beta \approx 1/\delta \) |
| Impedance | Almost real, \( \approx \sqrt{\mu/\varepsilon} \) | \( (1 + j)/\sigma\delta \), very small, at 45° |
| Wavelength | That of the lossless dielectric | \( 2\pi\delta \): the wave vanishes within one wavelength |
Attenuation is expressed in decibels by multiplying by 8.686: one neper is 8.686 dB. In seawater at 1 kHz, \( \alpha = 0.126 \) Np/m, about 1.1 dB per meter; at 1 GHz, 673 dB per meter.
04Reflection at normal incidence
When a wave arrives perpendicular to the boundary between two media, part of it is reflected and part is transmitted. The boundary conditions give the coefficients for the electric field:
The incident and the reflected waves superpose and form a standing wave: fixed maxima and minima separated by half a wavelength. It is exactly the phenomenon that appears in badly terminated transmission lines, with the load impedance in the role of \( \eta_2 \).
A plane wave travels through air and strikes a medium perpendicularly. The attenuation, wavelength, impedance and reflection are calculated with the exact complex expressions, and the electric field inside the medium is plotted as a function of depth.
05Where it appears in electronics
| Case | The phenomenon |
|---|---|
| Radome of an antenna | A thickness of half a wavelength in the material so that the reflection cancels |
| Ground-penetrating radar | The attenuation of the ground limits the depth; low frequencies are used to see deeper |
| Microwave oven | The water in food absorbs at 2.45 GHz and heats up |
| Radio reception inside a building | Reflection and attenuation by walls and glass |
| Absorbing materials of an anechoic chamber | An impedance that grows gradually up to \( \eta_0 \) so as not to reflect |
06In the lab
With a 10 GHz transmitter aimed at a metal plate and a detector that moves between them, locate the minima of the standing wave and measure the wavelength.
Measure the signal of a 2.4 GHz link with water containers of different thickness between the antennas and estimate the attenuation per centimeter.
Wrap a phone in aluminum foil and check that it loses signal. Then try with a mesh with large holes and relate the result to the wavelength.
07Common mistakes
- Using \( c/\sqrt{\varepsilon_r} \) in a medium with large losses. The velocity and the wavelength change with the conductivity.
- Confusing nepers with decibels. 1 Np = 8.686 dB.
- Forgetting that \( \Gamma \) is for the field. The reflected power is \( |\Gamma|^2 \).
- Assuming that a good conductor absorbs the wave. Almost all of it is reflected; the little that enters is attenuated within a few \( \delta \).
08Self-assessment
What intrinsic impedance does a lossless dielectric with \( \varepsilon_r = 4 \) have?
\( 377/2 \approx 188 \) Ω.
What fraction of the power is reflected from air to a 188 Ω medium?
\( \Gamma = (188 - 377)/(188 + 377) \approx -0.33 \); 11% is reflected.
How many decibels is 0.5 Np?
\( 0.5 \times 8.686 \approx 4.3 \) dB.
Why does ground-penetrating radar use low frequencies?
Because the attenuation of the ground grows with frequency, and at low frequencies the wave reaches deeper.
09Further reading
- David K. Cheng. Field and Wave Electromagnetics. 2nd ed., Addison-Wesley, 1989. Plane waves in lossy media and reflection, with the notation of this page.
- Fawwaz T. Ulaby and Umberto Ravaioli. Fundamentals of Applied Electromagnetics. 7th ed., Pearson, 2015. Very graphic, with engineering applications in every chapter.
- Constantine A. Balanis. Advanced Engineering Electromagnetics. 2nd ed., Wiley, 2012. The complete treatment, for going deeper into media and oblique reflection.