Skip to main content

Two-ray model and breakpoint calculator

Why doubling the distance costs 12 dB, not 6

Over ground, two rays reach the receiver: the direct one and the reflected one. Up to a certain distance they alternately add and subtract, and the level pulsates. Beyond that distance — the breakpoint — the reflected ray steadily cancels the direct one, and the loss begins to grow twice as fast as in free space.

h₁ 30 mh₂ 1.5 m
Before the breakpoint the rays interfere; beyond it the loss grows twice as fast
Frequency (MHz)
First antenna height (m)
Second antenna height (m)
Distance (km)
Result
Wavelength
Breakpoint
Loss from the two-ray model
Free-space loss
Path difference between the rays

📐 The formulas​

The breakpoint:

d_b = 4 · h₁ · h₂ / λ

Loss:

d < d_b :  FSPL applies, exponent 2
d > d_b :  L[dB] = 40·log d − 20·log h₁ − 20·log h₂

Path difference between the rays:

Δ = 2·h₁·h₂ / d

A check point:​

h₁ = 30 m, h₂ = 1.5 m, f = 900 MHz (λ = 0.3331 m) → breakpoint at 540 m. At 5 km the loss is 114.9 dB against 105.5 dB in free space.

Where this calculator stops:
  • The model assumes a flat surface and a reflection coefficient close to −1: over woodland or rolling country it is inaccurate
  • Below the breakpoint it shows FSPL, whereas in reality there are interference maxima and minima — the level can differ by ±10 dB
  • Polarisation is not accounted for: vertical has a different reflection coefficient, and the breakpoint moves
  • Diffraction over obstacles and the curvature of the Earth are not accounted for — long paths need other models

🎯 In practice​

  • A VHF path between two masts — a loss estimate without modelling.
  • Explaining why at 10 km the signal is weaker than “the calculation” said, by twice as much as expected.
  • Choosing antenna height: the breakpoint is proportional to the product of the heights, so doubling both pushes it four times further out.
Practical advice:
  1. Below the breakpoint, raising an antenna by a metre can gain or lose you signal — there is no guessing without a measurement.
  2. Beyond the breakpoint height works unambiguously: +6 dB for every doubling.
  3. Horizontal polarisation over ground gives sharper nulls, vertical blurrier ones.
  4. Compare with the Hata model: in built-up areas it will give greater loss still.

© 2026 UR3PKI · CyberDev.Space · Content licensed under CC BY-NC-SA 4.0.

How to cite this calculator
UR3PKI. «Two-ray model and breakpoint calculator». CyberDev.Space. https://cyberdev.space/en/radio/calculators/others/two_ray (licence CC BY-NC-SA 4.0).

The licence lets you use this material freely, including in teaching materials, but only with attribution to the author and a link to the source.

Last updated on