>With respect to rainbows they form every time white light passes though
>airborne water droplets.
Actually this is not correct. If it was, we would see a rainbow every
time we looked up in the direction of the sun when it is hidden behind
clouds. We don't.
First of all 'water droplets' have nothing to do with rainbow formation.
The kind of piece of water necessary to make a rainbow is the *drop* --
not droplet. Droplets are much smaller than drops since typical droplets
as in clouds tend to have sizes on the order of microns, whereas drops
tend to have sizes on the order of millimeters. Drops exist when rain is
falling. Droplets exist when clouds are present. In order for for a
rainbow to exist the sunlight must interact with round transparent
objects much larger than the range of wavelengths contained in the light
in order for the geometric (i.e. ray) optics limit to hold, and so that
diffraction effects become negligible. Water droplets are so small that
their size is close enough to that of the light's wavelengths that
diffraction effects are significant, ray optics doesn't accurately
describe the situation properly, and the interaction between the droplets
and the light is dominated by a scattering process called Mie scattering.
In this process all the wavelengths of the sunlight are effectively
scattered widely from each droplet over all possible angles relative to
the direction of the incident sunlight. The result of all this Mie
scattering is that the batch of droplets illuminated by a white light
source looks *white*, i.e. like a cloud (not like a rainbow).
The conditions necessary for a rainbow are: 1. A batch of spherical (or
nearly spherical) scattering centers made of a substance that is
transparent to white light but whose index of refraction is both
significantly larger than 1 and is dispersive over the visible wavelength
band (water does a good job meeting this requirement) exists and the
typical intersphere distance is relatively large compared to the sphere
diameters, and the spheres are distributed approximately randomly
throughout the batch. 2. The size of the spheres must be so large
compared to the wavelengths in the incident light that the laws of
geometric optics hold (i.e. a ray description is valid and those rays
obey the equal angles of incidence and reflection law for reflection from
the spherical surfaces, and obey Snell's law for refraction across the
interface between the interior of the sphere and the outside air). 3. The
white light source must be directionally localized as an effectively
point source of light. 4. The observer's point of view must be such that
the observer looks at the batch of spheres from approximately the same
side as the direction of the incident light but the viewing direction to
the batch of spheres must satisfy a constraint where the line connecting
the illumination source with the part of the batch of spheres that the
observer is observing (when viewing a section of the bow) must make a
correct angle (i.e. about 42 degrees for rainbows from water spheres)
with the line connecting the same observed part of the batch of spheres
to the observer. 5. The angle subtended by the whole illuminated batch
of spheres at the location of the observer must be large enough so that a
significant fraction of the arc of the circular bow can be seen for it to
appear as a bow, 6. The background behind the batch of spheres must be
dark enough.
If these conditions are not all met a bow is not visible. These
conditions *can* be met when it rains (on relatively rare occasions), and
when a spray of drops is thrown into the air from various sources such as
garden hoses, waterfalls, sea spray, geysers, fountains, etc. Drops of
water tend to automatically satisfy condition 1. because of the optical
properties of water and because surface tension tends to shape the drops
into the necessary spherical shape. The drops can't be too large in size
either since then the force of air resistence on their underside as they
fall through the air causes them to have a flattened bottom side thus
destroying the necessary spherical shape. Droplets (as in clouds) don't
work because they are too small to satisfy condition 2.. Condition 3. is
met when the sun is shining and not met when the sky is overcast. This
creates a problem for observing *rain*bows because there is a strong
tendency towards being overcast when it rains. Also, as a consequence of
condition 4. the sun can't be any higher in the sky than 42 degrees above
the horizon in order for at least some of the rainbow to appear above the
horizon. Since the rain in the air tends to be seen above the horizon
this tends to limit the times of rainbow viewing (for real rain) to the
hours just after sunrise or just before sunset in order for a significant
fraction of the rainbow circle to appear above the horizon so it can be
visible. If it is possible to view the raindrops as well below the
horizon (as when they are viewed from above from an airplane, then it may
be possible to see the whole rainbow as a complete circle centered on the
direction directly opposite the sun.
> I have seen them in sea spray, water falls, and in
>condensing water round hot springs. ....
Yes, they can be visible with drops from these sources, but in all cases
the rainbow is visible because of round *drops* in the air, not droplets.
David Bowman
dbowman@georgetowncollege.edu