### From a Wide Wake to the Width of the World

The finite angular width of a ship’s turbulent wake at the horizon enables the Earth’s radius to be estimated.

By ignoring evidence, Flat-Earthers remain secure in their delusions. The rest of us benefit greatly from accurate geodesy. Satellite communications, GPS navigation, large-scale surveying and cartography all require precise knowledge of the shape and form of the Earth and a precise value of its radius.

The turbulent wake of a ship is commonly observed streaming aft from the stern. The wake can be used to prove that the Earth is round. Look at the left and right edges of the wake in the Figure below. They form two fairly sharp lines that converge in the distance toward the horizon as a result of perspective. But they don’t meet at a point, as they would if the Earth were flat and the line-of-sight infinite. Rather, they are separated by a small angle at the horizon, where the observer’s finite line of sight is tangent to the spherical Earth.

Ancient Ideas

Aristotle (384–322~BC) gave several arguments in support of a spherical Earth, although he did not provide an estimate of its size. Eratosthenes (276–194 BC) — a pen-pal of Archimedes — deduced the circumference from simultaneous measurements of the solar angle at Alexandria and Aswan (REF). Abu Rayhan al-Biruni (AD~973–1050), the Persian polymath, devised a method of determining the radius of the Earth by observing the horizon from a mountain whose height is known. He carried out measurements in what is now the Punjab in Pakistan, and obtained an accurate estimate of the Earth’s circumference.

Ship Wakes

A different method of measuring the size of the Earth that could potentially have been used by the ancients, was described by David K. Lynch in a paper in the journal Applied Optics (Lynch, 2005). It uses simple observations of a ship’s turbulent wake. The wake is of roughly constant width ${W}$, but perspective gives it a triangular profile. On a flat Earth, the edges would converge to a point on the horizon. On the round Earth, the wake has a perceptible angular width ${\Theta}$ at the horizon. This can be measured with a simple optical instrument. As a first guess, the wake width may be set equal to the beam-width of the ship. Then the angular width at the horizon is ${\Theta = W / D}$, where ${D}$ is the distance to the horizon. This distance, in turn, can be estimated using the well-known approximation ${D = \sqrt{2 R H }}$ where ${R}$ is the radius of the Earth and ${H}$ is the elevation of the observer, the height of the ship deck (see Figure below).

Eliminating ${D}$ between the two equations above, we get $R = \frac{W^2}{2H\Theta^2}$

(more accurate formulae are given in the source reference). For example, for a ship of beam ${W = 20}$m, observer height ${H=12}$m and angle ${\Theta = 0.1^\circ = 1/600\,}$rad, we get $R = \frac{(20\times20)\times(600\times600)}{2\times 12} = 6\times 10^{6} = 6{,}000\,\mathrm{km}\,.$

This is reasonably close to the true value ${R = 6{,}371}$km. Other factors like optical refraction and widening of the wake must be considered if more precise results are required.

The greatest uncertainty is in measuring the angular width ${\Theta}$ of the wake at the horizon. Averaging of several measurements could reduce the error here. Analysis of photographs such as in the figures above is also useful although, obviously, such an option was not available to ancient mariners. In the source paper, D. K. Lynch remarks that “any sailor could determine all the necessary quantities … with relative ease”.

Sources ${\bullet}$ David K. Lynch, 2005: Turbulent ship wakes: further evidence that the Earth is round. Applied Optics, 44 (27), 5759–5762. https://doi.org/10.1364/AO.44.005759