sprinkler bow, MSU Bozeman
MT. As the jet rotated the bow width
changed and the colours varied from bright and saturated
colours to pastel hues. Image ©Les
A bow's appearance depends on drop size. The
best are narrow ones with intense colours and these are made by
large drops several mm in diameter. Look for them during very heavy
Smaller drops produce broader bows with less saturated colours. Very
small drops give nearly colourless cloudbows and fogbows.
The sprinkler's drops were largest near the main jet and they produced
the narrow colourful bow. After the jet had rotated away and the large
drops had fallen to earth the remaining smaller drops made the broader
optics cannot explain the the change in appearance
and dependence on drop size . We must invoke
the wave behaviour of light.
Classically, each angle of the rainbow is made
up of light that has traveled through drops along two different paths.
Again classically, the two emerging ray intensities simply
Waves behave differently. Here* we can represent
the waves as alternate positive and negative crests along the classical
each rainbow angle has two ray paths. Their intensities
of light waves along the paths. Positive and negative wave
crests are dark and light blue. The light intensity from
each path does not add because the wave crests are sometimes
out of step.
rainbows for two drop sizes and for blue monochromatic light
and sunlight***. As the drop
size decreases the phase difference between the two rainbow
forming ray (wave) paths oscillates more slowly with changes
in angle. The diffraction pattern including the main bow
broadens and the supernumeraries spread out.
A critical feature is that the paths have different
The crests start in step as they travel towards the drop but, as a
consequence of the different entrance
points and path lengths, they can emerge from the drop out of step
- there is a phase difference.
The emergent wave intensities cannot simply be added because the
waves interfere. When the crests are completely out of phase
their positive and negative amplitudes cancel and there is little light,
when in phase the light is intensified.
The phase difference and interference varies as the the rainbow deflection
angle changes. The result is a peak in intensity near the classical
rainbow angle with oscillatory fringes, supernumeraries, inside it.
The key to the changing appearance of the sprinkler bow is that as
the drop size decreases the phase difference changes
more slowly with
angle. The main rainbow peak therefore broadens and any supernumeraries
are more widely spaced.
Differences in colour with drop size? The sunlight bow is a superimposition
of monochromatic light bows. When the bows are broadened, the different
colours overlap more and the resultant hues are less saturated and
||This is diagrammatic
but a reasonable representation for rainbow forming droplets 0.3mm
dia. and upwards.
||Light waves are
transverse, the electric oscillations are at right angles to
the wave travel. The direction of the electric vibration,
the electric vector, determines the polarisation of the light.
Unpolarised light has, over a short time period, electric vectors
in all directions perpendicular to the ray direction. Pure plane
polarised light has the electric vectors in a single direction.
The intensity of light on being reflected or refracted depends
on the direction of the initial polarisation relative to the
surface. Rotate a pair of sunglasses while looking at a reflection
in glass to show this. Rainbows are therefore polarised (rotate
sunglasses while viewing one). Polarisation does not invalidate
the above qualitative explanation but it does affect the quantitative description of rainbows and fogbows.
||The simulations are
for a 0.5° diameter
source, sun sized, and droplets all of one size. In nature, far
fewer supernumeraries (if any) are visible because there is a distribution
of drop sizes and larger drops are flattened to some extent.