Abstract

Ångstrom has proposed that rough absorbing materials are darker when wet because their diffuse reflection makes possible total internal reflection in the water film covering them, increasing the likelihood of the absorption of light by the surface. His model is extended here in two ways: the probability of internal reflection is calculated more accurately, and the effect on absorption of the decrease of the relative refractive index (liquid to material instead of air to material) is estimated. Both extensions decrease the albedo of the wetted surface, bringing the model into good agreement with experiment.

© 1988 Optical Society of America

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References

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  1. A. Ångstrom, “The Albedo of Various Surfaces of Ground,” Geogr. Ann. 7, 323 (1925).
    [CrossRef]
  2. C. F. Bohren, “Multiple Scattering at the Beach,” Weatherwise (Aug.1983).
  3. S. A. Twomey, C. F. Bohren, J. L. Mergenthaler, “Reflectance and Albedo Differences Between Wet and Dry Surfaces,” Appl. Opt. 25, 431 (1986).
    [CrossRef] [PubMed]
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  5. J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, Dordrecht, 1987).
  6. F. Stern, “Transmission of Isotropic Radiation Across an Interface Between Two Dielectrics,” Appl. Opt. 3, 111 (1964).
    [CrossRef]
  7. W. D. Sellers, Physical Climatology (U. Chicago Press, 1965).

1986 (1)

1983 (1)

C. F. Bohren, “Multiple Scattering at the Beach,” Weatherwise (Aug.1983).

1964 (1)

1925 (1)

A. Ångstrom, “The Albedo of Various Surfaces of Ground,” Geogr. Ann. 7, 323 (1925).
[CrossRef]

Ångstrom, A.

A. Ångstrom, “The Albedo of Various Surfaces of Ground,” Geogr. Ann. 7, 323 (1925).
[CrossRef]

Bohren, C. F.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Lekner, J.

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, Dordrecht, 1987).

Mergenthaler, J. L.

Sellers, W. D.

W. D. Sellers, Physical Climatology (U. Chicago Press, 1965).

Stern, F.

Twomey, S. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Appl. Opt. (2)

Geogr. Ann. (1)

A. Ångstrom, “The Albedo of Various Surfaces of Ground,” Geogr. Ann. 7, 323 (1925).
[CrossRef]

Weatherwise (1)

C. F. Bohren, “Multiple Scattering at the Beach,” Weatherwise (Aug.1983).

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, Dordrecht, 1987).

W. D. Sellers, Physical Climatology (U. Chicago Press, 1965).

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Figures (3)

Fig. 1
Fig. 1

Liquid layer over a rough surface. The coefficients represent the fraction of the incident light intensity which is transmitted along each path.

Fig. 2
Fig. 2

Ratio of average absorption by a wet surface to that of the dry surface, for varying refractive index of the wetting liquid. For the solid curves aw is given by (11), with nr = 2. The dashed curves are drawn for aw = ad(Ångstrom). In all cases we have set Rl = (nl − 1)2/(nl + l)2 (normal illumination).

Fig. 3
Fig. 3

Wet albedo as a function of dry albedo, for a layer of water over a rough surface. Normal illumination is assumed. Dry albedo is given by 1 − ad and wet albedo by 1 − A, the latter calculated using nr = 2. A is given by Eq. (1); the solid line has a = aw as given by (11), and p by Eq. (9). The dashed line has a = ad and p given by (2) (Ångstrom’s approximations). The experimental data are as described in the text.

Equations (13)

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A = ( 1 R l ) [ a + a ( 1 a ) p + a ( 1 a ) 2 p 2 + ] = ( 1 R l ) a 1 p ( 1 a ) .
p 2 π θ c π / 2 d θ sin θ cos θ 2 π 0 π / 2 d θ sin θ cos θ = cos 2 θ c = 1 1 / n 1 2 .
p = 0 π / 2 d θ sin θ cos θ R ( x , 1 / n l ) 0 π / 2 d θ sin θ cos θ = 0 1 d x R ( x , 1 / n l ) ,
p = 1 n 2 + 0 n 2 d x R ( x , n ) .
R ( x , n ) = R ( x / n 2 , 1 / n )
n 2 0 1 d y R ( n 2 y , n ) = n 2 0 1 d y R ( y , 1 / n ) = n 2 R ¯ ( 1 / n ) .
R ¯ ( n ) = 0 π / 2 d θ sin θ cos θ R ( x , n ) 0 π / 2 d θ sin θ cos θ = 0 1 d x R ( x , n )
R ¯ ( n ) = 3 n 2 + 2 n + 1 3 ( n + 1 ) 2 2 n 3 ( n 2 + 2 n 1 ) ( n 2 + 1 ) 2 ( n 2 1 ) + n 2 ( n 2 + 1 ) ( n 2 1 ) 2 log n n 2 ( n 2 1 ) 2 ( n 2 + 1 ) 3 log n ( n + 1 ) n 1 ( n > 1 ) .
p = 1 1 n l 2 [ 1 R ¯ ( n l ) ] .
a w a d [ 1 R ¯ ( n r / n l ) ] / [ 1 R ¯ ( n r ) ] a w ( 0 ) .
a w ( 1 a d ) a w ( 0 ) + a d a w ( 1 ) ,
a w a d ( 1 a d ) 1 R ¯ ( n r / n l ) 1 R ¯ ( n r ) + a d .
d A d a = ( 1 p ) ( 1 R 1 ) [ 1 p ( 1 a ) ] 2 > 0

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