Abstract

When viewed from the stern, a ship’s turbulent wake appears as a narrow strip of bubble-whitened water converging toward the horizon. The wake does not reach a sharp point on the horizon but has a finite angular width, indicating that the Earth is not flat, but rather round. A simple analysis of the geometry of the observations shows that the radius of the Earth can be estimated using only simple instruments and observations.

© 2005 Optical Society of America

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References

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  1. D. K. Lynch, W. C. Livingston, “Mountain shadow phenomena,” Appl. Opt. 18, 265–269 (1979).
    [CrossRef] [PubMed]
  2. P. Gorman, Pythagoras: A Life (Routledge & Kegan Paul, London, 1979).
  3. J. Barnes, ed., The Cambridge Companion to Aristotle (Cambridge University, New York, 1995).
  4. T. L. Heath, A History of Greek Mathematics (2 vols.) (Oxford University, 1921).
  5. N. Bowditch, The American Practical Navigator: An Epitome of Navigation (National Imagery and Mapping Agency, 1995).
  6. A. P. French, “How far away is the horizon,” Am. J. Phys. 50(9) 795–799 (1982).
    [CrossRef]
  7. http://www.1728.com/cubic.htm and http://www3.telus.net/thothworks/Quad3Deg.html .
  8. C. Bohren, A. Fraser, “At what altitude does the horizon cease to be visible?,” Am. J. Phys. 54, 222 (1986).
    [CrossRef]
  9. S. Shi, A. Smirnov, I. Celik, “Large-eddy simulations of turbulent wake flows,” in Twenty-Third Symposium on Naval Hydrodynamics (National Academics Press, 2001) pp. 579–598.
  10. E. H. Buller, J. K. E. Tunaley, “The effect of the ships screws on the ship wake and its implication for the radar image of the wake,” Digest 1989 IEEE International Geo-science and Remote Sensing Symposium (IEEE, 1989), pp. 362–365.

1986 (1)

C. Bohren, A. Fraser, “At what altitude does the horizon cease to be visible?,” Am. J. Phys. 54, 222 (1986).
[CrossRef]

1982 (1)

A. P. French, “How far away is the horizon,” Am. J. Phys. 50(9) 795–799 (1982).
[CrossRef]

1979 (1)

Bohren, C.

C. Bohren, A. Fraser, “At what altitude does the horizon cease to be visible?,” Am. J. Phys. 54, 222 (1986).
[CrossRef]

Bowditch, N.

N. Bowditch, The American Practical Navigator: An Epitome of Navigation (National Imagery and Mapping Agency, 1995).

Buller, E. H.

E. H. Buller, J. K. E. Tunaley, “The effect of the ships screws on the ship wake and its implication for the radar image of the wake,” Digest 1989 IEEE International Geo-science and Remote Sensing Symposium (IEEE, 1989), pp. 362–365.

Celik, I.

S. Shi, A. Smirnov, I. Celik, “Large-eddy simulations of turbulent wake flows,” in Twenty-Third Symposium on Naval Hydrodynamics (National Academics Press, 2001) pp. 579–598.

Fraser, A.

C. Bohren, A. Fraser, “At what altitude does the horizon cease to be visible?,” Am. J. Phys. 54, 222 (1986).
[CrossRef]

French, A. P.

A. P. French, “How far away is the horizon,” Am. J. Phys. 50(9) 795–799 (1982).
[CrossRef]

Gorman, P.

P. Gorman, Pythagoras: A Life (Routledge & Kegan Paul, London, 1979).

Heath, T. L.

T. L. Heath, A History of Greek Mathematics (2 vols.) (Oxford University, 1921).

Livingston, W. C.

Lynch, D. K.

Shi, S.

S. Shi, A. Smirnov, I. Celik, “Large-eddy simulations of turbulent wake flows,” in Twenty-Third Symposium on Naval Hydrodynamics (National Academics Press, 2001) pp. 579–598.

Smirnov, A.

S. Shi, A. Smirnov, I. Celik, “Large-eddy simulations of turbulent wake flows,” in Twenty-Third Symposium on Naval Hydrodynamics (National Academics Press, 2001) pp. 579–598.

Tunaley, J. K. E.

E. H. Buller, J. K. E. Tunaley, “The effect of the ships screws on the ship wake and its implication for the radar image of the wake,” Digest 1989 IEEE International Geo-science and Remote Sensing Symposium (IEEE, 1989), pp. 362–365.

Am. J. Phys. (2)

A. P. French, “How far away is the horizon,” Am. J. Phys. 50(9) 795–799 (1982).
[CrossRef]

C. Bohren, A. Fraser, “At what altitude does the horizon cease to be visible?,” Am. J. Phys. 54, 222 (1986).
[CrossRef]

Appl. Opt. (1)

Other (7)

P. Gorman, Pythagoras: A Life (Routledge & Kegan Paul, London, 1979).

J. Barnes, ed., The Cambridge Companion to Aristotle (Cambridge University, New York, 1995).

T. L. Heath, A History of Greek Mathematics (2 vols.) (Oxford University, 1921).

N. Bowditch, The American Practical Navigator: An Epitome of Navigation (National Imagery and Mapping Agency, 1995).

S. Shi, A. Smirnov, I. Celik, “Large-eddy simulations of turbulent wake flows,” in Twenty-Third Symposium on Naval Hydrodynamics (National Academics Press, 2001) pp. 579–598.

E. H. Buller, J. K. E. Tunaley, “The effect of the ships screws on the ship wake and its implication for the radar image of the wake,” Digest 1989 IEEE International Geo-science and Remote Sensing Symposium (IEEE, 1989), pp. 362–365.

http://www.1728.com/cubic.htm and http://www3.telus.net/thothworks/Quad3Deg.html .

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Figures (5)

Fig. 1
Fig. 1

Ship’s stern wake in the Indian Ocean on a clear day. The turbulent wake appears to converge toward the horizon but does not come to an absolutely sharp point. There is a noticeable angular width on the horizon. The wake is caused by bubbles from the ships propellers and turbulence from the ship. It has a nearly constant width and remains essentially fixed with respect to the water. (Apparent curvature of horizon is an optical effect due to the camera lens and is not representative of the Earth’s curvature.)

Fig. 2
Fig. 2

Simulation of a ship wake on a flat Earth. The wake converges to a point in the true horizon. 90 degrees from the zenith (no refraction).

Fig. 3
Fig. 3

Geometry of a ship wake on a round Earth. An observer O standing a distance H above a spherical (or cylindrical) Earth whose radius is R, will have a line of sight that is tangent to the Earth’s surface at a distance DT and an angle αT below the geometrical horizon.

Fig. 4
Fig. 4

Simulation of a ship wake on a round Earth. The wake converges but does not come to a point. It has a finite width of the apparent horizon, which is below the geometrical horizon by an amount αT, the so-called “dip.”

Fig. 5
Fig. 5

ω(W) for various values of H. The dot marks the location of the calculations based on Fig. 1.

Equations (4)

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D T = ( 2 R H + H 2 ) 1 / 2 ,
α T = sin - 1 ( D T / ( R + H ) ) ,
ω = W / D T ,
R = ( W 2 - ω 2 H 2 ) / ( 2 H ω 2 ) ,

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