11.1 Electromagnetic Wave Fundamentals, Polarisation, and Ground Waves

11.1 Electromagnetic Wave Fundamentals, Polarisation, and Ground Waves

All wireless telecommunications rely on the generation, radiation, propagation, and reception of electromagnetic waves. Formally unified by James Clerk Maxwell in 1865 and experimentally validated by Heinrich Hertz in 1887, electromagnetic waves represent self-sustaining oscillations of energy capable of travelling across both material media and the vacuum of space. For the advanced amateur radio operator, mastering the mechanics of wave polarisation, spatial geometric spreading, and terrestrial surface wave interaction is essential for designing effective antenna systems and predicting path reliability across the radio spectrum.

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The Physical Nature of Electromagnetic Waves

An electromagnetic (EM) wave is an unguided, transverse wave comprising two time-varying vector fields: an electric field ($\vec{E}$, measured in volts per meter, $\text{V/m}$) and a magnetic field ($\vec{H}$, measured in amperes per meter, $\text{A/m}$). These two fields oscillate in time and space at identical frequencies, in time phase, and at strict right angles ($90^\circ$) to each other. Furthermore, both field vectors are perpendicular to the wave's direction of propagation.

                    Electromagnetic Wave Orthogonal Field Vectors

          Electric Field (E)
             ^
             |       *               *
             |      / \             / \
             |     /   \           /   \
             |----+-----+---------+-----+--------> Direction of Propagation (z)
             |   /       \       /       \        (Poynting Vector S = E x H)
             |  *         *     *         *
             |
            /       .               .
           /       / \             / \
          /       /   \           /   \
         /-------+-----+---------+-----+--------
        /       /       \       /       \
       v       .         .     .         .
  Magnetic Field (H)

The Velocity of Light and Wave Parameters

In a vacuum, electromagnetic waves propagate at the universal speed of light ($c$), defined by the fundamental electromagnetic constants of free space—the electric permittivity ($\varepsilon_0 \approx 8.854 \times 10^{-12}\text{ F/m}$) and magnetic permeability ($\mu_0 = 4\pi \times 10^{-7}\text{ H/m}$):

$$c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}} \approx 2.99792 \times 10^8\text{ m/s} \approx 3.00 \times 10^8\text{ m/s}$$

The relationship between the velocity of propagation ($c$), carrier frequency ($f$ in Hertz), and wavelength ($\lambda$ in meters) is defined by the fundamental wave equation:

$$c = f \lambda \implies \lambda = \frac{c}{f} \approx \frac{300}{f_{\text{MHz}}}\text{ (meters)}$$

Wave Impedance of Free Space

The ratio of the electric field strength ($|\vec{E}|$) to the magnetic field strength ($|\vec{H}|$) at any point in a propagating transverse electromagnetic (TEM) plane wave equals the characteristic impedance of free space ($\eta_0$, also termed the intrinsic wave impedance):

$$\eta_0 = \frac{|\vec{E}|}{|\vec{H}|} = \sqrt{\frac{\mu_0}{\varepsilon_0}} \approx 120\pi \approx 376.73\ \Omega \approx 377\ \Omega$$

The flow of electromagnetic power per unit area carried by the wave is defined by the Poynting vector ($\vec{S}$, measured in watts per square meter, $\text{W/m}^2$):

$$\vec{S} = \vec{E} \times \vec{H}$$

Because $\vec{E}$ and $\vec{H}$ are in time phase and spatially orthogonal, the time-averaged power density ($S_{\text{avg}}$) in free space can be expressed directly in terms of peak electric field strength ($E_0$):

$$S_{\text{avg}} = \frac{E_0^2}{2 \eta_0} = \frac{E_{\text{RMS}}^2}{\eta_0} \quad (\text{W/m}^2)$$

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Wave Polarisation: Linear, Circular, and Mismatch Losses

The polarisation of an electromagnetic wave describes the orientation and trajectory of the electric field vector ($\vec{E}$) in space as a function of time, observed at a fixed point looking along the direction of propagation. By international convention, wave polarisation is defined exclusively by the electric field vector; the magnetic field is ignored because its orientation is invariably locked at $90^\circ$ relative to $\vec{E}$.

1. Linear Polarisation

In a linearly polarised wave, the electric field oscillates back and forth within a single stationary plane throughout propagation:

• Horizontal Polarisation: The electric field vector oscillates parallel to the Earth's horizontal plane. Horizontal half-wave dipoles, inverted-V antennas, and horizontally oriented Yagi arrays radiate horizontal polarisation. This mode is universally favored for amateur HF skywave operations and VHF/UHF weak-signal CW/SSB communications due to lower ground clutter pickup.
• Vertical Polarisation: The electric field vector oscillates perpendicular to the Earth's surface (along the local vertical axis). Vertical quarter-wave monopoles, collinear arrays, and ground-plane antennas radiate vertical polarisation. Vertical polarisation is the standard for VHF/UHF mobile FM communications because omnidirectional vertical whips are simple to mount on moving vehicles.

2. Circular and Elliptical Polarisation

If an antenna launches two orthogonal linear electric field components ($E_x$ and $E_y$) simultaneously, the resultant vector depends on their relative amplitude and phase displacement:

• Circular Polarisation (CP): Produced when two orthogonal linear components have equal amplitude ($|E_x| = |E_y|$) and exist in exact $90^\circ$ ($\pi/2\text{ radians}$) phase quadrature.
  • Right-Hand Circular Polarisation (RHCP): The electric field vector rotates clockwise as viewed looking along the direction of propagation (consistent with IEEE definitions).
  • Left-Hand Circular Polarisation (LHCP): The electric field vector rotates counter-clockwise.
• Elliptical Polarisation: Two nonzero orthogonal components generally trace an ellipse. If their phase difference is $0^\circ$ or $180^\circ$, the result is linear even when their amplitudes differ. Equal amplitudes in $90^\circ$ quadrature give the circular case; other combinations give an ellipse. The axial ratio, major-axis amplitude divided by minor-axis amplitude, is $1$ ($0\text{ dB}$) for a circle and tends to infinity for a line.

Circular polarisation is vital for satellite, space, and Earth-Moon-Earth (EME) communications because it overcomes antenna tumble and rotation fading.

Cross-Polarisation Loss on Line-of-Sight (LOS) Paths

On a direct line-of-sight path, matching the receiving antenna to the incoming wave’s polarisation maximizes captured power. For two linearly polarised fields whose axes differ by $\theta_{\text{pol}}$, the ideal coupling is:

$$P_{\text{rx}} = P_{\text{max}} \cos^2(\theta_{\text{pol}})$$

Ideal linear-to-linear polarisation coupling:

Axis angle       Coupled power          Power relative to aligned
   0 degrees         1.00                         0 dB
  30 degrees         0.75                        -1.25 dB
  45 degrees         0.50                        -3.01 dB
  60 degrees         0.25                        -6.02 dB
  90 degrees         0                            zero coupling

An ideal circular wave into a linear antenna also gives 0.50
coupling, regardless of the linear antenna's rotation. This is a
separate polarisation case, not a point on the linear-tilt curve.

• Orthogonal Linear Mismatch: Horizontal and vertical linear polarisations are orthogonal, so their ideal coupling is zero. Real antennas, reflections and scattering can introduce other polarisation components, making isolation finite and path-dependent. Matching polarisation remains important on a direct VHF/UHF link.
• Circular-to-Linear Coupling: Receiving a circularly polarised wave with a linear antenna (or a linearly polarised wave with a circularly polarised antenna) results in an exact, invariant $3\text{ dB}$ power penalty ($50\%$ power loss), irrespective of how the linear antenna is physically rotated in the plane of the wavefront.
• Opposite-Sense Circular Mismatch: Ideal RHCP and LHCP polarisations are orthogonal and have zero coupling. Real isolation depends on antenna axial ratio and propagation. Opposite circular senses can be used to reduce interference between channels, but they do not provide a universal isolation value.

Polarisation Transformation via Faraday Rotation in the Ionosphere

While polarisation alignment is critical on line-of-sight VHF/UHF paths, the rule changes fundamentally for High Frequency (HF) signals propagating via ionospheric skywave. When an electromagnetic wave traverses an ionised plasma permeated by Earth's geomagnetic field ($\vec{B}_0$), the free electrons are forced into orbital precession by the magnetic field's Lorentz force ($-e(\vec{v}\times\vec{B}_0)$). This magneto-ionic interaction makes the ionosphere birefringent (anisotropic).

The magnetised ionosphere supports ordinary and extraordinary characteristic modes with different phase velocities. A wave can excite both modes. Their relative phase changes during propagation, altering the received polarisation. In the circular-mode approximation, recombination rotates the plane of a linearly polarised wave; this Faraday rotation has the approximate dependence:

$$\Omega_F \approx \frac{e^3}{8\pi^2 \varepsilon_0 m_e^2 c f^2} \int N_e B_0 \cos \theta_B \, ds \quad (\text{radians})$$

where $N_e$ is the free electron density, $B_0$ is geomagnetic field strength, and $f$ is operating frequency. Because ionospheric electron density ($N_e$) fluctuates continuously due to atmospheric gravity waves, diurnal solar variations, and geomagnetic storms, the Faraday rotation angle changes dynamically over time.

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Free Space Path Loss (FSPL) and Geometric Spreading

As an electromagnetic wave propagates away from an antenna in an ideal, lossless vacuum, its wavefront expands outwards as a sphere of increasing radius ($d$). Because the total energy radiated by the transmitter is distributed across an ever-larger spherical surface area ($A = 4\pi d^2$), the power flux density ($S$) decreases proportionally to the square of the distance—a physical principle known as the inverse-square law:

$$S = \frac{P_t}{4\pi d^2} \quad (\text{W/m}^2)$$

where $P_t$ is the total power radiated by a theoretical lossless isotropic antenna ($G_t = 1$).

                   Isotropic inverse-square spreading

Distance from TX       Spherical area       Power flux density
       d                     A                      S
      2d                    4A                     S/4
      3d                    9A                     S/9

A = 4*pi*d^2. The same total radiated power crosses each sphere.
A fixed-size collecting patch intercepts less power as range grows.

The Friis Transmission Equation

To capture power from this expanding wavefront, a receiving antenna presents an effective capture area termed its effective aperture ($A_e$). For a theoretical lossless isotropic receiving antenna ($G_r = 1$), the effective aperture is directly related to wavelength:

$$A_e = \frac{\lambda^2}{4\pi} = \frac{c^2}{4\pi f^2}$$

Multiplying the incident power flux density ($S$) by the effective aperture ($A_e$) yields the total power received ($P_r$):

$$P_r = S \times A_e = \left( \frac{P_t}{4\pi d^2} \right) \times \left( \frac{\lambda^2}{4\pi} \right) = P_t \left( \frac{\lambda}{4\pi d} \right)^2$$

The ratio of transmitted power to received power between isotropic antennas is defined as the Free Space Path Loss ($\text{FSPL}$):

$$\text{FSPL} = \frac{P_t}{P_r} = \left( \frac{4\pi d}{\lambda} \right)^2 = \left( \frac{4\pi d f}{c} \right)^2$$

The Practical FSPL Formula in Decibels

Converting the power ratio into decibels and substituting standard engineering units—distance in kilometers ($d_{\text{km}}$), frequency in megahertz ($f_{\text{MHz}}$), and $c \approx 2.9979 \times 10^8\text{ m/s}$—yields the standard Friis path loss formula:

$$\text{FSPL (dB)} = 10 \log_{10}\left[ \left( \frac{4\pi (d_{\text{km}} \times 10^3)(f_{\text{MHz}} \times 10^6)}{2.9979 \times 10^8} \right)^2 \right]$$ $$\text{FSPL (dB)} = 20 \log_{10}(d_{\text{km}}) + 20 \log_{10}(f_{\text{MHz}}) + 20 \log_{10}\left( \frac{4\pi \times 10^9}{2.9979 \times 10^8} \right)$$

Evaluating the constant term:

$$20 \log_{10}(41.9169) \approx 32.44$$

Thus, the universal FSPL equation is:

$$\text{FSPL (dB)} = 20 \log_{10}(d_{\text{km}}) + 20 \log_{10}(f_{\text{MHz}}) + 32.44$$

Critical Path Loss Scaling Rules

From the logarithmic formulation, two foundational scaling rules emerge for RF engineers:

1. Distance Rule (Inverse-Square Law): Doubling the path distance ($2d$) adds $20 \log_{10}(2) \approx \mathbf{+6.02\text{ dB}}$ of path attenuation (a four-fold reduction in received power).
2. Frequency Rule: Doubling the operational frequency ($2f$) adds $20 \log_{10}(2) \approx \mathbf{+6.02\text{ dB}}$ of path loss between isotropic antennas. Moving from $144\text{ MHz}$ to $432\text{ MHz}$ (a factor of 3) increases path loss by $20 \log_{10}(3) \approx \mathbf{+9.54\text{ dB}}$.

Step-by-Step Worked Calculation: Free Space Path Loss

Problem: An amateur operator establishes a line-of-sight microwave link between two hilltops separated by $d = 35\text{ km}$. Calculate the Free Space Path Loss between isotropic antennas on:

1. The $144.20\text{ MHz}$ (2-meter) band
2. The $1296.20\text{ MHz}$ (23-centimeter) band

Step 1: Compute FSPL for 144.20 MHz:

$$\text{FSPL}_{144} = 20 \log_{10}(35) + 20 \log_{10}(144.20) + 32.44$$ $$\text{FSPL}_{144} = 30.88 + 43.18 + 32.44 = 106.50\text{ dB}$$

Step 2: Compute FSPL for 1296.20 MHz:

$$\text{FSPL}_{1296} = 20 \log_{10}(35) + 20 \log_{10}(1296.20) + 32.44$$ $$\text{FSPL}_{1296} = 30.88 + 62.25 + 32.44 = 125.57\text{ dB}$$

Step 3: Compare Path Losses:

$$\Delta \text{Loss} = 125.57\text{ dB} - 106.50\text{ dB} = 19.07\text{ dB}$$

The frequency ratio is $1296.2/144.2\approx8.989$, giving $20\log_{10}(8.989)\approx19.07\text{ dB}$ additional isotropic-antenna path loss. At the same distance the geometric power flux density is unchanged; the isotropic receiving aperture is smaller at the higher frequency. Maintaining SNR may require more transmitter power, more antenna gain or lower receiver noise, with the same reference bandwidth.

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Ground Wave (Surface Wave) Propagation Mechanics

When a radio wave is radiated close to the Earth's surface, its propagation is divided into three distinct components:

1. Direct Wave: Line-of-sight propagation straight from transmitter to receiver.
2. Ground-Reflected Wave: Energy reflected from the surface, which interferes with the direct wave (forming the classic two-ray multipath model).
3. Surface Wave (True Ground Wave): An electromagnetic wave that remains guided along the boundary between the conductive Earth and the insulating troposphere.

                     Surface Wave Wavefront Tilt Mechanics

       Troposphere (Air, ε_r ≈ 1.0, σ ≈ 0)        Propagation Direction --->
       ---------------------------------------
                 \           \           \
                  \           \           \  Wavefronts tilt forward
                   \           \           \  due to boundary drag
       ~~~~~~~~~~~~~\~~~~~~~~~~~\~~~~~~~~~~~\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
       Earth Surface (Conductivity σ, Dielectric Constant ε_r)
            --> Current induced in soil (I_gnd) causes Ohmic dissipation (I^2 R)

The Wavefront Tilt and Earth Boundary Currents

A surface wave interacts with ground having finite conductivity $\sigma$ and permittivity $\varepsilon$. The fields penetrate the ground, and their boundary relationship produces a small forward tilt of the electric field. Its tangential component drives ground currents, so some of the wave's energy flows into the earth and is dissipated.

Both conductivity and dielectric loss affect this process. A real, lossy ground cannot generally be assigned the lossless-dielectric speed $c/\sqrt{\varepsilon_r}$; its propagation constant depends on its complex permittivity. The wavefront-tilt picture is a useful qualitative explanation of the attenuation, not a separate loss formula.

The Vertical Polarisation Imperative

For a perfect conductor the tangential electric field at the boundary is zero. Real ground has finite conductivity and permits a small tangential field associated with loss. The predominantly vertical surface-wave field can propagate along the boundary, whereas a horizontal antenna can still support direct, ground-reflected or skywave paths; those are different propagation components.

Terrestrial Electrical Parameters

The attenuation rate of a ground wave depends heavily on the electrical conductivity ($\sigma$) and dielectric constant ($\varepsilon_r$) of the terrain over which it travels:

These are comparative trends. For a numerical prediction, specify frequency, ground electrical properties, path length and antenna/system conditions. A terrain name alone does not establish a fixed dB loss or communication radius.

Because seawater exhibits high conductivity ($\sigma \approx 5\text{ S/m}$, five hundred times higher than good farmland), surface currents flow with minimal ohmic dissipation, permitting maritime ground-wave communications over several hundred kilometers.

Useful Frequency Bands for Ground Wave Propagation

Surface-wave attenuation generally increases with frequency and becomes greater over poorly conducting ground. Field penetration and loss depend on both conductivity and permittivity. Dielectric dissipation is associated with the loss component of permittivity, not its energy-storage component alone. Use a ground-wave prediction method for quantitative field strength; the good-conductor skin-depth approximation is not a universal soil-loss formula.

Amateur radio frequency allocations leverage ground wave propagation across specific operational bands:

• $136\text{ kHz}$ and $472\text{ kHz}$: LF/MF surface-wave operation can provide useful regional coverage. Range depends strongly on radiated power, antenna efficiency, ground and receiver noise; a band label alone does not guarantee hundreds of kilometres.
• $1.8\text{ MHz}$: Daytime D-region absorption usually restricts skywave, so surface wave and direct/reflected paths are important. Surface-wave range is generally greater over sea than land.
• $3.5\text{ MHz}$: Ground wave can serve local paths, while suitable high-angle skywave can provide regional coverage when the frequency lies within the usable ionospheric window.
• $7\text{ MHz}$ and above: Surface-wave coverage usually becomes shorter over land, but useful range still depends on the path and system. Direct/reflected, diffracted and ionospheric paths also contribute to HF communications. Quantitative ground-wave methods extend to $30\text{ MHz}$; they do not imply a fixed range on every band.