At what path length does earth bulge start to matter?

Around 10 km. Below that it stays under 1.5 m, smaller than the error in most elevation data. At 30 km it reaches 13 m, taller than the tree line you were worried about, and at 100 km it is 147 m, which is why long links need mast height rather than merely clear ground.

Why does the k-factor reduce the bulge?

The k-factor does not change the earth, it changes the model. Refraction bends radio rays slightly downward. Rather than curving the rays, link planning straightens them and inflates the earth radius to k·R. A larger radius is a flatter surface, so the computed bulge falls by a factor of k, a quarter less at k = 4/3.

Is earth bulge the same thing as the radio horizon?

Same geometry, different question. The horizon asks how far one antenna can see from a given height. The bulge asks how high the surface rises between two fixed points. Both come from the curvature term and both use the same k-factor, so they always agree: at exactly the horizon distance, the bulge just touches the sight line.

Do I add earth bulge to Fresnel zone clearance?

Yes. Add the bulge at the obstacle’s position to the obstacle’s own height, then require the sight line to clear that total by at least 60% of the first Fresnel zone radius. Both the bulge and the Fresnel radius peak near mid-path, which is why mid-path obstacles dominate link design.

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Earth bulge calculator

Earth bulge is the height by which the planet’s curved surface rises above the straight line joining two points on it. On a radio path it behaves like a hill in the middle: over 30 km it adds about 13 m to every obstacle beneath the sight line. The bulge peaks at mid-path and shrinks as refraction bends rays downward.

Path length

Position along path

Where the obstacle sits: 50% = mid-path, where the bulge is highest.

Atmosphere

k scales the earth radius to stand in for refraction.

Bulge at position

13.2 m

= 43.5 ft · d₁ = 15 km, d₂ = 15 km

Max bulge (mid-path)

13.2 m

= 43.5 ft

Practical formula

d₁·d₂ / (12.75·k)

meters, distances in km

The grey curve is the earth’s surface between sites A and B. The dashed line is the straight chord between them, the path a radio ray takes. The green arrow measures the gap at the chosen position. That gap is the bulge, tallest at mid-path and zero at both ends. Horizontal distances are to scale, while the vertical scale is exaggerated.

Smooth-earth term only, with a single k-factor for the whole path. Real terrain heights come from elevation data. Add this bulge on top of them.

How it works

01
See the chord, not the line

The straight line between two antennas is a chord of a sphere, so it cuts below the surface between them. Halfway along a 30 km path the ground sits about 13 m above that chord, even where the map shows it dead flat.
02
Compute the rise

The bulge at a point is h = d₁·d₂2·k·R, with d₁ and d₂ measured from the two ends. In practical units, h (m) ≈ d₁·d₂12.75·k with distances in km. Because it is a product, the bulge vanishes at both ends and peaks at mid-path: 1.5 m over 10 km, 13 m over 30 km, 147 m over 100 km.
03
Let refraction help

A standard atmosphere bends rays slightly downward, which link planning models by inflating the earth radius to k·R. At k = 4/3 that removes a quarter of the bulge, 13.2 m instead of 17.6 m over 30 km. Subrefractive weather (k ≈ 2/3) doubles it instead, which is why marginal paths fail in specific conditions.
04
Add it to every obstacle

An obstacle’s effective height is its own height plus the bulge at its distance along the path. Compare that against the sight line, then keep at least 60% of the first Fresnel zone clear on top of it. Under about 10 km the bulge stays below 1.5 m and usually disappears into the elevation data’s own error.

Formulas

Bulge at a point on the path

h (m) = 1000 · d₁ · d₂2 · k · R

h — height of the surface above the chord, m
d₁, d₂ — distances from the two ends, km
k — effective-earth-radius factor (4/3 in a standard atmosphere)
R — earth radius, 6371 km

Practical form

h (m) ≈ d₁ · d₂12.75 · k

d₁, d₂ — km
12.75 — 2·R / 1000, for R ≈ 6371–6375 km

Maximum, at mid-path

h_max (m) ≈ d²51 · k

d — total path length, km
follows from d₁ = d₂ = d/2

Worked example

30 km path, mid-point, k = 4/3

d₁ = d₂ = 15 km
h = 15 · 15 / (12.75 · 1.333) = 225 / 17.0 = 13.2 m
without refraction (k = 1): 225 / 12.75 = 17.6 m
at 5 km from one end: 5 · 25 / 17.0 = 7.4 m
→ the planet itself is a 13 m obstacle in the middle of this link

FAQ

At what path length does earth bulge start to matter?: Around 10 km. Below that it stays under 1.5 m, smaller than the error in most elevation data. At 30 km it reaches 13 m, taller than the tree line you were worried about, and at 100 km it is 147 m, which is why long links need mast height rather than merely clear ground.
Why does the k-factor reduce the bulge?: The k-factor does not change the earth, it changes the model. Refraction bends radio rays slightly downward. Rather than curving the rays, link planning straightens them and inflates the earth radius to k·R. A larger radius is a flatter surface, so the computed bulge falls by a factor of k, a quarter less at k = 4/3.
Is earth bulge the same thing as the radio horizon?: Same geometry, different question. The horizon asks how far one antenna can see from a given height. The bulge asks how high the surface rises between two fixed points. Both come from the curvature term and both use the same k-factor, so they always agree: at exactly the horizon distance, the bulge just touches the sight line.
Do I add earth bulge to Fresnel zone clearance?: Yes. Add the bulge at the obstacle’s position to the obstacle’s own height, then require the sight line to clear that total by at least 60% of the first Fresnel zone radius. Both the bulge and the Fresnel radius peak near mid-path, which is why mid-path obstacles dominate link design.

Bulge plus terrain is the real clearance question. Waveshed answers it with actual elevation data along your path, free in your browser.

Draw the path profile →

Related guides

Elevation profile & measuring tools — the simulator’s side view already adds this bulge to the terrain under the sight line
Accuracy & data sources — how far you can trust the elevation data this bulge gets added to

Sources & further reading

Earth bulge — Wikipedia ↗ — the d₁·d₂ / (2·k·R) form and its practical constants
ITU-R Recommendation P.530 — Propagation data for terrestrial line-of-sight systems ↗ — effective-earth-radius factor and path clearance criteria
ITU-R Recommendation P.526 — Propagation by diffraction ↗ — what happens once the bulge intrudes into the first Fresnel zone

Earth bulge calculator

How it works

See the chord, not the line

Compute the rise

Let refraction help

Add it to every obstacle

Formulas

Worked example

FAQ

Related guides

Sources & further reading

Related tools

Radio horizon calculator

Fresnel zone calculator

Knife-edge diffraction calculator

How far can you see? Horizon distance calculator