SUPERNUMERARY BOWS By Katrina Brubacher Asha Padmanabhan Rainbow

SUPERNUMERARY BOWS By Katrina Brubacher & Asha Padmanabhan

“Rainbow” commonly refers to a single circular arc of non repeating colors.

Is the rainbow a spot 42° above your head’s shadow?

Is the rainbow a spot 42° above your head’s shadow? A spherical raindrop will not prefer one direction to another. All locations that lie 42° from the shadow of your head are equally likely to send the concentrated rainbow to you.

Therefore the primary rainbow is a circle with radius 42° and it’s center at your head’s shadow.

Alexander’s Dark Band • Most of the rays that come out of a drop are concentrated at 138° from the sun, but some light is bent through all angles between 180° and 138°

SUPERNUMERARY BOWS • Supernumeraries are much more common than you’d think but the number that are seen vary. • Their colors also vary. The most common colors are pinks and bluish greens but yellow is sometimes also observed as well as violet.

Newton believed that the behavior of light was best explained as a series of small particles traveling from the light source to the eye but this does not explain the presence of supernumerary bows.

“Supernumeraries proved to be the midwife that delivered the wave theory of light to its place of dominance in the 19 th century. ” ~Rainbow Bridge

Young’s Theory

Young’s Theory • In the 1800’s most scientists agreed with Newton, but Robert Hooke and Christiaan Huygens believed that light behaved more like waves than particles.

Young’s Theory • In the 1800’s most scientists agreed with Newton, but Robert Hooke and Christiaan Huygens believed that light behaved more like waves than particles. • In 1803 Thomas Young asserted that supernumerary bows could be explained only if light were thought of as a wave phenomenon.

Interference • It is the interference of waves that explains supernumerary bows.

Interference cont. • It is the interference of waves that explains supernumerary bows. • If the crests of two waves coincide, they reinforce each other to make a larger wave. If a crest of one wave sits in the trough of another, the two disturbances cancel each other and the medium will be at it’s original level.

Interference cont. • It is the interference of waves that explains supernumerary bows. • If the crests of two waves coincide, they reinforce each other to make a larger wave. If a crest of one wave sits in the trough of another, the two disturbances cancel each other and the medium will be at it’s original level. • This is called constructive and destructive interference.

Supernumerary bows are not caused by the interference between two light waves, they are caused by the interference of two different portions of the same light wave.

Size of raindrops • Young used the wave theory to account for the color and brightness of the supernumerary bows and to estimated the sizes of raindrops that yielded supernumeraries.

Size of raindrops cont. • The size of the raindrops change the appearance of the supernumerary bows. – A smaller drop gives widely spaced bows, the larger drop gives more tightly spaced bows and each bow is narrower. – The first supernumerary for the smaller drop occurs at the same deviation angle as the second supernumerary of the larger drop.

Size of raindrops cont. • When the drops are small, each bow is broad, including the primary. Hence the bow’s colours overlap and appear pastel

• Young was able to estimate the raindrop size of a shower based on the spacing between supernumerary bows. The spacing decreases as the drop increases. • The reason for this is that the spacing of bright and dark bands in the folded wave front depends on the path length the wave has traversed within the drop.

Size of raindrops cont. In nature, drops with a radius that is greater than 0. 4 mm can make the supernumeraries brighter than the primary rainbow.

• Supernumeraries of the secondary rainbow?

• Young did not give a quantitative account of the interference theory of the rainbow.

• Young did not give a quantitative account of the interference theory of the rainbow. • For a numerical description we must look to Airy’s Integral.

George Biddel Airy (1801 -92) Airy’s theory of the rainbow extended and mathematically formalized Young’s largely empirical explanations of interference within a raindrop.

AIRY’S MATHEMATICS • The explanation for the supernumerary bows come from looking at light exiting a raindrop. • The light is sharply cut off in the direction of minimum deviation and the effects are similar to those of a shadow along a straight edge. This was first solved by Fresnel.

Fresnel’s Integral • Total disturbance given by: Ao sin pt ∫ cos δ dx + Ao cos pt ∫ sin δ dx where • ∫ cosδ dx = B ∫ cos( v²/2) dv • ∫ sinδ dx = B ∫ sin ( v²/2) dv

• In a rainbow the effects of diffraction are seen just inside the illuminated area. This area is cut off by the cone of minimum deviation. • This leads to bright and dark bands within the primary bow or outside the secondary bow : supernumerary bows.

AIRY’S INTEGRAL A = ( λa²/(4 kcosθ))^⅓ ∫ cos (( /2)(u³-zu) du

bow nu m be r z at max inte nsit y z at min inte nsit y 1 1. 085 2 3. 4669 4. 3631 3 5. 1446 5. 8922 4 6. 5782 7. 2436 5 7. 8685 8. 4788 6 9. 0599 9. 6300 10. 177 10. 716 4 1 11. 236 11. 749 7 2. 4955

• We’ve computed the values of the table by using a series developed from Airy’s Integral. Pochhammer

bow number z at max intensity z at min intensity 1 1 0. 2868 2. 3 0. 0007 2 3. 1 0. 1392 3. 6 0. 0016

Things that are NOT possible

T H E E N D