Abstract

Inferior mirages over sun-exposed roads often appear in isolated strips at their near sides and the reflected scenery exhibits multiple images. This effect is explained as due to slight undulations of the road’s surface. At the same time, some of these images, although they are reflections, are not inverted. Photographic material illustrates this phenomenon and a ray tracing study is presented that confirms these conclusions.

© 2011 Optical Society of America

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References

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  1. For parameterizations as a function of wavelength, see, e.g., D. R. Lide, Handbook of Chemistry and Physics, 81st ed. (CRC Press, 2000).
  2. S. Y. van der Werf, “Ray tracing and refraction in the modified US1976 atmosphere,” Appl. Opt. 42, 354–366 (2003).
    [CrossRef] [PubMed]
  3. S. Y. van der Werf, “Comment on ‘Improved ray tracing air mass numbers model’,” Appl. Opt. 47, 153–156 (2008).
    [CrossRef] [PubMed]
  4. E. Tränkle, “Simulation of inferior mirages observed at the Halligen Sea,” Appl. Opt. 37, 1495–1505 (1998).
    [CrossRef]
  5. H. Fakhruddin, “Specular reflection from a rough surface,” Phys. Teach. 41, 206–207 (2003).
    [CrossRef]
  6. M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252–260 (2004).
    [CrossRef]

2008 (1)

2004 (1)

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252–260 (2004).
[CrossRef]

2003 (2)

1998 (1)

Ebadi, Z.

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252–260 (2004).
[CrossRef]

Fakhruddin, H.

H. Fakhruddin, “Specular reflection from a rough surface,” Phys. Teach. 41, 206–207 (2003).
[CrossRef]

Lide, D. R.

For parameterizations as a function of wavelength, see, e.g., D. R. Lide, Handbook of Chemistry and Physics, 81st ed. (CRC Press, 2000).

Nahal, A.

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252–260 (2004).
[CrossRef]

Tavassoly, M. T.

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252–260 (2004).
[CrossRef]

Tränkle, E.

van der Werf, S. Y.

Appl. Opt. (3)

Opt. Commun. (1)

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252–260 (2004).
[CrossRef]

Phys. Teach. (1)

H. Fakhruddin, “Specular reflection from a rough surface,” Phys. Teach. 41, 206–207 (2003).
[CrossRef]

Other (1)

For parameterizations as a function of wavelength, see, e.g., D. R. Lide, Handbook of Chemistry and Physics, 81st ed. (CRC Press, 2000).

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Figures (6)

Fig. 1
Fig. 1

Road mirage, showing its near side fragmented into strips. Note that the reflection of the white arrow of the “keep right” sign on the traffic island, about 400 m away, seems noninverted. Photograph: Siebren van der Werf, April 20, 2011, Roden, The Netherlands.

Fig. 2
Fig. 2

(a) Same road sign as in Fig. 1. Left: a white pole has been put up against a nearby street light. (b) Same scenery from a distance of about 400 m , close to the position from where the picture of Fig. 1 was taken. Camera height: 40 cm . (c) Idem, camera height 50 cm . (d) Idem, camera height 60 cm . Photographs: Siebren van der Werf. (b) April 11, 2011; (a), (c), (d) April 24, 2011.

Fig. 3
Fig. 3

Ray tracing example over a road surface, which has a sinusoidal undulation with amplitude 2 cm and a period of 50 m . The observer is at 50 cm above the road. The temperature jump is 20 ° C , decaying exponentially with a decay constant d = 2 cm .

Fig. 4
Fig. 4

Height of the ray at a distance of 400 m away from the observer as a function of observation angle.

Fig. 5
Fig. 5

Simulation of the direct image and the reflection of a 2.5 m long skewed pole over a (a) perfectly flat road and an (b) undulating road as in the present analysis. Note that the nearest reflection zone is at closer distance in the presence of undulations.

Fig. 6
Fig. 6

Illustrating the lens formula for paraxial reflection off a concave road element. Objects beyond a distance L + L away from the observer are seen upright.

Equations (9)

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n = 1 + A P / T ,
n ( h ) cos ( β ) = n ( 0 ) .
L = h T ( 0 ) T ( h ) 2 A P [ T ( 0 ) T ( h ) ] .
g ( x ) = a sin ( 2 π x / D )
T ( z ) = T 0 + Δ T exp [ ( z g ) / d ]
L = ( h R / 2 L ) 1 ( 2 L / h ) d g / d x 1 ( h R / 2 L 2 ) .
1 L + 1 L = 2 L h R .
1 L + 1 L 2 R T 2 2 A P Δ T ,
1 D = 1 2 [ 1 L 1 L ]

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