Abstract

The angular diameter of Snell’s window as a function of maximum wave slope is calculated. For flat water the diameter is 97° and increases up to about 122° when the wave slope is about 16°. Steeper waves break and disrupt the smooth surface used in the analysis. Breaking waves produce a window almost 180° wide. The brightness of the dark area around Snell’s window is heavily influenced by turbidity and upwelling radiation, especially in shallow water.

© 2014 Optical Society of America

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References

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  1. Like the rainbow, Snell’s window has certainly been observed for tens and perhaps hundreds of thousands of years. While the first mention of it we could find was in Minnaert [2], it has probably been discussed and named many times, the identity of the first person to do so or publish it being long lost.
  2. M. Minnaert, The Nature of Light and Colour in the Open Air (G. Bell, 1940).
  3. G. Horváth and D. Varjú, “Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell’s window of the flat water surface,” Vis. Res. 35, 1651–1666 (1995).
    [Crossref]
  4. S. Sabbah, A. Barta, J. Gal, and G. Horvath, “Experimental and theoretical study of skylight polarization transmitted through Snell’s window on a flat surface,” J. Opt. Soc. Am. A 23, 1978–1988 (2006).
    [Crossref]
  5. Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011).
    [Crossref]
  6. P. Laven, http://www.philiplaven.com/p20.html (retrieved May2014).
  7. Water waves are not sinusoidal but in most cases are close enough to sinusoids that the analysis presented here is indicative of real-world waves. Indeed, regardless of the wave shape, it is the maximum wave steepness that determines the size of Snell’s window.
  8. D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge University, 1995).
  9. M. S. Longuet-Higgins, “The instabilities of gravity waves of finite amplitude in deep water II: subharmonics,” Proc. R. Soc. A 360, 489–505 (1978).
    [Crossref]
  10. W. K. Melville, “The instability and breaking of deep-water waves,” J. Fluid Mech. 115, 165–185 (1982).
    [Crossref]
  11. But as any surfer can tell you, steep, shallow water waves can show smooth surfaces when breaking, e.g., “the curl,” “the tube,” etc.
  12. http://commons.wikimedia.org/wiki/File:Snell's_window,_St._Louis_Zoo.jpg .

2011 (1)

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011).
[Crossref]

2006 (1)

1995 (1)

G. Horváth and D. Varjú, “Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell’s window of the flat water surface,” Vis. Res. 35, 1651–1666 (1995).
[Crossref]

1982 (1)

W. K. Melville, “The instability and breaking of deep-water waves,” J. Fluid Mech. 115, 165–185 (1982).
[Crossref]

1978 (1)

M. S. Longuet-Higgins, “The instabilities of gravity waves of finite amplitude in deep water II: subharmonics,” Proc. R. Soc. A 360, 489–505 (1978).
[Crossref]

Barta, A.

Gal, J.

Horvath, G.

Horváth, G.

G. Horváth and D. Varjú, “Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell’s window of the flat water surface,” Vis. Res. 35, 1651–1666 (1995).
[Crossref]

Livingston, W.

D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge University, 1995).

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “The instabilities of gravity waves of finite amplitude in deep water II: subharmonics,” Proc. R. Soc. A 360, 489–505 (1978).
[Crossref]

Lynch, D. K.

D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge University, 1995).

Melville, W. K.

W. K. Melville, “The instability and breaking of deep-water waves,” J. Fluid Mech. 115, 165–185 (1982).
[Crossref]

Minnaert, M.

M. Minnaert, The Nature of Light and Colour in the Open Air (G. Bell, 1940).

Sabbah, S.

Shen, L.

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011).
[Crossref]

Varjú, D.

G. Horváth and D. Varjú, “Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell’s window of the flat water surface,” Vis. Res. 35, 1651–1666 (1995).
[Crossref]

Voss, K. J.

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011).
[Crossref]

Xu, Z.

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011).
[Crossref]

Yue, D. K. P.

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011).
[Crossref]

J. Fluid Mech. (1)

W. K. Melville, “The instability and breaking of deep-water waves,” J. Fluid Mech. 115, 165–185 (1982).
[Crossref]

J. Geophys. Res. (1)

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011).
[Crossref]

J. Opt. Soc. Am. A (1)

Proc. R. Soc. A (1)

M. S. Longuet-Higgins, “The instabilities of gravity waves of finite amplitude in deep water II: subharmonics,” Proc. R. Soc. A 360, 489–505 (1978).
[Crossref]

Vis. Res. (1)

G. Horváth and D. Varjú, “Underwater refraction-polarization patterns of skylight perceived by aquatic animals through Snell’s window of the flat water surface,” Vis. Res. 35, 1651–1666 (1995).
[Crossref]

Other (7)

Like the rainbow, Snell’s window has certainly been observed for tens and perhaps hundreds of thousands of years. While the first mention of it we could find was in Minnaert [2], it has probably been discussed and named many times, the identity of the first person to do so or publish it being long lost.

M. Minnaert, The Nature of Light and Colour in the Open Air (G. Bell, 1940).

P. Laven, http://www.philiplaven.com/p20.html (retrieved May2014).

Water waves are not sinusoidal but in most cases are close enough to sinusoids that the analysis presented here is indicative of real-world waves. Indeed, regardless of the wave shape, it is the maximum wave steepness that determines the size of Snell’s window.

D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge University, 1995).

But as any surfer can tell you, steep, shallow water waves can show smooth surfaces when breaking, e.g., “the curl,” “the tube,” etc.

http://commons.wikimedia.org/wiki/File:Snell's_window,_St._Louis_Zoo.jpg .

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Figures (7)

Fig. 1.
Fig. 1. Snell’s window. Note the ragged edges due to waves and the faint blue glow around the window. (Photograph and copyright by Simon Higton.)
Fig. 2.
Fig. 2. Optics of Snell’s window for flat water.
Fig. 3.
Fig. 3. Optical geometry of Snell’s window in the presence of surface waves.
Fig. 4.
Fig. 4. Width of Snell’s window as a function of maximum wave slope.
Fig. 5.
Fig. 5. Snell’s blanket in shallow water surrounding Snell’s window.
Fig. 6.
Fig. 6. Transmission of flat water producing Snell’s window (solid line) and reflectivity of the under surface of water producing Snell’s blanket. The angle from zenith is what an underwater observer would measure.
Fig. 7.
Fig. 7. For steeply inclined wave faces in breaking waves and turbulent, bubble-filled water, skylight can reach an underwater observer from a zenith angle of 90°. Thus skylight can be seen that originates from the entire upper celestial sphere (angular diameter 180°).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

W=Asin(2πx/λ),
α=tan1(2πA/λ).
httan1(4A/3λ).
R=180°htα.
it=90°htα.
St=α+rt=α+sin1(sin(90°αtan1[2tanα/(3π)])n).

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