Alexander Haussmann searched specifically for a reflected rainbow at the fountain of the palace pond at Dresden. And with his precise look he found it – but it was faint. With a polarising filter he could increase its contrast. The reflected bow changed in brightness, occasionally it was obvious but then it became faint again. The visibility was most probably influenced by the wind and the smoothness of the water surface that was to some extent disturbed by the falling drops. This example shows that it is not difficult to find a (artificial) reflected rainbow with help of a fountain.
Saturday, May 20, 2006
Reflected rainbow at a fountain
Alexander Haussmann searched specifically for a reflected rainbow at the fountain of the palace pond at Dresden. And with his precise look he found it – but it was faint. With a polarising filter he could increase its contrast. The reflected bow changed in brightness, occasionally it was obvious but then it became faint again. The visibility was most probably influenced by the wind and the smoothness of the water surface that was to some extent disturbed by the falling drops. This example shows that it is not difficult to find a (artificial) reflected rainbow with help of a fountain.
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Some time ago, based on this picture the general chances of seeing reflected-light rainbows in fountains were discussed in the german "Meteoros" forum. I thought further about several features of the phenomenon, and it might be useful to present here a translation of my posting.
1. Elevation of the sun
With rising elevation, the intensity of the reflected light decreases rapidly, according to Fresnel's formulas. Therefore, only low positions of the sun are promising for reflected-light bow sightings. On May 3rd, 2006 at 19.25 CEST the sun's elevation in Dresden was about 8.8°, corresponding to a reflectivity of 0.5 for TE- and 0.3 for TM-polarization.
2. Planar water surface
Even small water ripples tend to disturb a well-defined reflexion of the sunlight. Such ripples or small waves are generated by wind or downfalling water from the fountain, whereas the latter might not disturb the whole water area, just some circle around the fountain.
3. Positions of reflection on the water surface
It was difficult for me to guess ad hoc the location on the water from which the light is reflected for a certain portion of the bow. These "reflection-positions" can be calculated by using some geometry, but this is too complicated for the practical use. Therefore its helpful to keep some rules-of-thumb in mind while looking for a sighting at a certain place.
Firstly, one can simplify the fountain's drop spread to a plane perpendicular to the water surface (useful for reflected-light bows which are not generated by drops parallel to the water surface, as it was reported in some dew bow observations). At this, all reflection-positions form a more or less strung-out ellipse. The reflecting point for the bow's vertex lies, depending on the sun's elevation, rather far "behind" the observer (that means towards the sun). In the book "Rainbows, Halos, and Glories" by Robert Greenler an illustrating drawing of this effect can be found. At the Dresden sighting, in this direction no water was located, so that the vertex could not be seen.
Another remarkable point is the crossing between the ordinary and the reflected-light bow. It is always located at the same level above the water like the observer's eye - and the corresponding reflection-position lies in a distance of cotangens(solar elevation) * eye level in the direction towards the sun away from the point on the water surface just beneath the crossing. Taking an eye level of roughly 2m (the road along the pond lies a bit higher than the water), this distance turns out to be about 13m, so its certainly outside the disturbed water surface around the fountain. Otherwise, the part of the bow beneath the crossing would require a reflection nearer to the fountain and can't bee seen due to the disturbance.
Obviously, the reflection-positions on the water must be exposed to direct sunlight, which becomes more and more difficult in a park with buildings, trees, etc. while the sun sinks further. To illustrate the geometry in Dresden I modified a picture from "Google Maps":
http://www.teundem.de/test/files/gg1.jpg
The red arrows show the direction of the incident sunlight, the blue points represent (roughly estimated) the reflection-positions. Thus, the upper limit of the reflected-light bow is likely to be determined by the north-western pond shore.
4. Presence of water drops in reflected sunlight at the reflected-light bow position
The bow's vertex is also invisible because the fountain's drops do not reach the required height above the water level (even the normal rainbow's vertex could not be seen in the actual observing distance). For the lateral distribution of drops some wind is quite useful.
Conclusion: For both "artificial" and natural reflected-light bows the net result of solar elevation, mirror surface quality and geometry as well as the spatial drop spread is crucial. The geometric conditions can be estimated in general. A thought of further interest might be the reflection from a vertical plane (house fronts made from glass etc.), whereas the drops have to be located beneath the reflection and near to the surface, since the elevation angle would remain constant in such a case. For a fountain this situation could be possible - and the crossing between the bows would then be found in the vertical.
Best regards,
Alexander Haussmann
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