Abstract

Light falling from a point source on a ruffled surface produces a pattern of images, which move about over the surface. The image points correspond to the maxima, minima, and saddle points of a certain function. It is shown that the images are generally created in pairs, a maximum with a saddle point or a minimum with a saddle point, and that the total numbers of maxima, minima, and saddle points satisfy the relation

Nma+Nmi=Nsa+1.

The process of creation or annihilation of images is studied in detail, and also the tracks of the image points, in certain special cases. It is shown that closed tracks may be common. This is confirmed by photography of the sea surface.

© 1960 Optical Society of America

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References

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  1. C. Cox and W. Munk, J. Opt. Soc. Am. 44, 838 (1954).
    [CrossRef]
  2. M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. 56, 234 (1956).
    [CrossRef]
  3. H. Shenck, J. Opt. Soc. Am. 47, 653 (1957).
    [CrossRef]
  4. M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. London A247, 321 (1957).
  5. H. S. M. Coxeter, Regular Polytopes (Methuen and Company, Ltd., London, 1948), p. 321.
  6. L. Euler, Nov. Comment. Acad. Sci. Imp. Petropol. 4, 109 (1752–1753).
  7. D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions (Methuen and Company, Ltd., London, 1929), p. 196.
  8. At an ordinary point the total brightness is proportional to |Ωf|−1, but when Ωf vanishes this approximation breaks down.
  9. This is for gravity waves. For surface-tension waves the reverse is true, but a similar argument applies.

1957 (2)

M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. London A247, 321 (1957).

H. Shenck, J. Opt. Soc. Am. 47, 653 (1957).
[CrossRef]

1956 (1)

M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. 56, 234 (1956).
[CrossRef]

1954 (1)

Cox, C.

Coxeter, H. S. M.

H. S. M. Coxeter, Regular Polytopes (Methuen and Company, Ltd., London, 1948), p. 321.

Euler, L.

L. Euler, Nov. Comment. Acad. Sci. Imp. Petropol. 4, 109 (1752–1753).

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. London A247, 321 (1957).

M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. 56, 234 (1956).
[CrossRef]

Munk, W.

Shenck, H.

Sommerville, D. M. Y.

D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions (Methuen and Company, Ltd., London, 1929), p. 196.

J. Opt. Soc. Am. (2)

Nov. Comment. Acad. Sci. Imp. Petropol. (1)

L. Euler, Nov. Comment. Acad. Sci. Imp. Petropol. 4, 109 (1752–1753).

Phil. Trans. Roy. Soc. London (1)

M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. London A247, 321 (1957).

Proc. Cambridge Phil. Soc. (1)

M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. 56, 234 (1956).
[CrossRef]

Other (4)

H. S. M. Coxeter, Regular Polytopes (Methuen and Company, Ltd., London, 1948), p. 321.

D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions (Methuen and Company, Ltd., London, 1929), p. 196.

At an ordinary point the total brightness is proportional to |Ωf|−1, but when Ωf vanishes this approximation breaks down.

This is for gravity waves. For surface-tension waves the reverse is true, but a similar argument applies.

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Figures (9)

Fig. 1
Fig. 1

The full lines indicate contours of f(x,y,t) in the neighborhood of an ordinary specular point: (a) a maximum, (b) a saddle point, (c) a minimum. The broken lines and arrows indicate directions of steepest ascent.

Fig. 2
Fig. 2

Configurations of stationary points. (●=maximum, ○=minimum, ×=saddle point).

Fig. 3
Fig. 3

Contours of the function

Fig. 4
Fig. 4

Modifications of the pattern of stationary points by the addition of a maximum and a saddle point.

Fig. 5
Fig. 5

Modifications of the pattern of stationary points by the addition of a minimum and a saddle point.

Fig. 6
Fig. 6

The formation of specular lines on a moving waveform.

Fig. 7
Fig. 7

The formation of specular points by two intersecting wave systems.

Fig. 8
Fig. 8

Possible tracks of specular points (the arrows indicate directions of motion).

Fig. 9
Fig. 9

A time exposure of the sea surface, showing tracks formed by images of the sun. The photograph was taken at midday, the camera being inclined at about 45° to the horizontal. (Triex XXX plate film was used, with a red filter.)

Equations (40)

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N m a + N m i = N s a + 1.
N m a + N m i = N s a + 1
z = ζ ( x , y , t ) ,
ζ / x = - κ x ,             ζ / y = - κ y ,
κ = 1 2 [ ( 1 / h 1 ) + ( 1 / h 2 ) ] ,
κ = ( μ 1 h 1 + μ 2 h 2 ) / ( μ 2 - μ 1 ) h 1 h 2 .
f / x = 0 ,             f / y = 0 ,
f ( x , y , t ) = ζ ( x , y , t ) + 1 2 κ ( x 2 + y 2 ) ,
f ( x , y ) = 1 2 ( a 20 x 2 + 2 a 11 x y + a 02 y 2 ) + R ,
Ω f = ( 2 f / x 2 ) ( 2 f / y 2 ) - ( 2 f / x y ) 2 = a 20 a 02 - a 11 2
N faces + N vertices = N edges + 2
N m a + N m i = N s a + 1 ,
f / x = 0 ,             f / y = 0 ,
( 2 f / x 2 ) d x + ( 2 f / x d y ) d y + ( 2 f / x t ) d t = 0 , ( 2 f / x y ) d x + ( 2 f / y 2 ) d y + ( 2 f / y t ) d t = 0.
Ω f , = ( 2 f / x 2 ) ( 2 f / y 2 ) - ( 2 f / x d y ) 2 0.
f ( x , y , t ) = i , j , k = 0 3 a i j k i ! j ! k ! x i y j t k + R ,
a i j k = ( i + j + k f / x i y j t k ) x = y = t = 0
a 000 = 0 ,
a 100 = a 010 = 0.
a 110 = 0.
a 200 a 020 = 0 ,
a 200 = 0.
a 001 = a 002 = a 003 = 0.
f ( x , y , t ) = 1 2 a 020 y 2 + 1 6 ( a 300 x 3 + 3 a 210 x 2 y + 3 a 120 x y 2 + a 030 y 2 ) + ( a 101 x + a 011 y ) t + 1 2 ( a 201 x 2 + 2 a 111 x y + a 021 y 2 ) t + 1 2 ( a 102 x + a 012 y ) t 2 + R .
1 2 ( a 300 x 2 + 2 a 210 x y + a 120 y 2 ) + a 101 t + = 0 , a 020 y + 1 2 ( a 210 x 2 + 2 a 120 x y + a 030 y 2 ) + a 011 t + = 0 ,
x = ± ( - 2 a 101 t a 300 ) 1 2 ,             y = a 210 a 101 - a 300 a 011 a 300 a 020 t .
( a 210 a 100 - a 300 a 010 ) x 2 + 2 a 100 a 020 y = 0 ,
d x d t = ± ( - a 101 2 a 300 t ) 1 2 ,             d y d t = a 210 a 101 - a 300 a 011 a 300 a 020 ,
Ω f , = ( 2 f / x 2 ) ( 2 f / y 2 ) - ( 2 f / x y ) 2
a 300 x + a 210 y + a 201 t = 0
tan - 1 ( a 300 / a 210 )
f ( x , y , 0 ) = 1 2 a 020 y 2 + 1 6 ( a 300 x 3 + 3 a 210 x 2 y + 3 a 120 x y 2 + a 030 y 3 ) + .
x + ( a 210 / a 300 ) y = ξ y = η
f ( x , y , 0 ) = 1 6 a 300 ξ 3 + 1 2 a 020 η 2 ( 1 + A ξ + B η ) + ,
f = 1 2 a 020 η 2 + 1 6 a 300 ξ 3 + .
f ( x , y , t ) = 1 2 y 2 + 1 6 ( x 3 + 3 x 2 y ) + x t ,
1 2 ( a 300 x 2 + 2 a 210 x y + a 120 y 2 ) + a 101 t + = 0 1 2 ( a 210 x 2 + 2 a 120 x y + a 030 y 2 ) + a 011 t + = 0 ,
ζ ( x , y , t ) = ζ 1 ( x , t ) + ζ 2 ( y , t ) ,
ζ 1 / x = - κ x ,             ζ 2 / y = - κ y ,
f ( x , y , t ) = 1 2 y 2 + 1 6 ( x 3 + 3 y 2 y ) + x t