Space wave over plane and actual earth

Space wave propagation is the mode in which the signal reaches the receiver by the direct wave plus the ground-reflected wave through the troposphere. It is used at VHF, UHF and microwaves. The ground affects it in two ways: by reflection (interference of the two waves) and, for real earth, by curvature.

(a) Over a plane (flat) earth

   Tx                          Rx
   |\ ------ direct r1 -------> |
 ht|  \                      / | hr
   |    \  reflected r2    /   |
 ==|======\______P______/======|==
   |<------------ d ---------->|

• Path lengths: r₁ = √(d² + (ht − hr)²), r₂ = √(d² + (ht + hr)²).
• For d ≫ ht, hr: path difference Δr = r₂ − r₁ ≈ 2ht·hr/d.
• Phase difference = (2π/λ)(2ht·hr/d) + π (reflection at grazing angle gives coefficient ≈ −1).
• Resultant field (E₀/d = field of direct wave at distance d):

E = (2E₀/d) · sin(2π ht hr / λd)
For large d (small angle):  sin x ≈ x
E ≈ 4π E₀ ht hr / (λ d²)
In practical form: E = 88 √P · ht · hr / (λ d²) V/m
  (P in W, heights, λ and d in metres)

Effects:

• Field strength varies as 1/d² (not 1/d) at large distance, so it falls fast.
• Near the transmitter, maxima and minima (lobes) occur as d changes, due to interference.
• Raising either antenna increases the field directly (E ∝ ht·hr), so high masts are used.
• Horizontal and vertical polarisation behave similarly at grazing incidence over most ground.

(b) Over the actual (curved) earth

1. Radio horizon: curvature limits the line-of-sight range. With standard refraction (effective radius 4R/3):

d = √(2kR ht) + √(2kR hr) ≈ 4.12 (√ht + √hr) km   (h in m)

2. Effective antenna heights: the curved ground rises between the antennas, so heights measured from the tangent plane at the reflection point are reduced:

h't = ht − d1²/(2kR),   h'r = hr − d2²/(2kR)

These reduced heights are used in the field formula, giving a weaker field than for plane earth. 3. Divergence factor: reflection from a convex surface spreads the reflected rays, so the reflected wave is weaker; the reflection coefficient is multiplied by a divergence factor D < 1. 4. Shadow zone: beyond the horizon, only weak diffracted fields exist; the field falls rapidly. 5. Tropospheric refraction: decrease of refractive index with height bends rays down and extends the horizon by about 15% (k = 4/3); abnormal conditions give super-refraction, ducting or sub-refraction. 6. Terrain, obstacles and roughness: hills and buildings block or diffract the wave; rough ground gives diffuse rather than mirror reflection (Rayleigh criterion); Fresnel-zone clearance (≥ 0.6 of first zone) is needed for a clean link.

Conclusion of comparison: flat-earth analysis fits short links; for longer links, curvature reduces effective heights, limits range to the radio horizon, and weakens the reflected wave.