question: If water scatters light in all directions, why do rainbows appear in a stationary position?
comment: A water droplet is like a glass ball of water. While light may scatter in all directions, only light that enter in a specific angle can bounce twice off the inside edges of the sphere and return in a direction close to that which it came from. That light is separated into the rainbow by the transition from air to water so slightly different angles mean slightly different colors. The rainbow forms an arc, really a circle, because all the light originates from one direction (usually the sun) and every point where the angle is that particular angle from the source back to your eyes makes a virtual circle in space in front of you.
passages: Droplets (or spheres) composed of materials with different refractive indices than plain water produce rainbows with different radius angles. Since salt water has a higher refractive index, a sea spray bow doesn't perfectly align with the ordinary rainbow, if seen at the same spot. Tiny plastic or glass marbles may be used in road marking as a reflectors to enhance its visibility by drivers at night. Due to a much higher refractive index, rainbows observed on such marbles have a noticeably smaller radius. One can easily reproduce such phenomena by sprinkling liquids of different refractive indices in the air, as illustrated in the photo.
   BULLET::::- the positions of the primary and secondary rainbows
  
  BULLET::::- the path of sunlight within a drop: light beams are refracted when entering the atmospheric droplets, then reflected inside the droplets and finally refracted again when leaving them.
  
  BULLET::::- the formation of the rainbow: he explains the role of the individual drops in creating the rainbow
  
  BULLET::::- the phenomenon of color reversal in the secondary rainbow
   Section::::Geographic Distribution.
   When sunlight encounters a raindrop, part of the light is reflected and the rest enters the raindrop. The light is refracted at the surface of the raindrop. When this light hits the back of the raindrop, some of it is reflected off the back. When the internally reflected light reaches the surface again, once more some is internally reflected and some is refracted as it exits the drop. (The light that reflects off the drop, exits from the back, or continues to bounce around inside the drop after the second encounter with the surface, is not relevant to the formation of the primary rainbow.) The overall effect is that part of the incoming light is reflected back over the range of 0° to 42°, with the most intense light at 42°. This angle is independent of the size of the drop, but does depend on its refractive index. Seawater has a higher refractive index than rain water, so the radius of a "rainbow" in sea spray is smaller than a true rainbow. This is visible to the naked eye by a misalignment of these bows.
   A rainbow is not located at a specific distance from the observer, but comes from an optical illusion caused by any water droplets viewed from a certain angle relative to a light source. Thus, a rainbow is not an object and cannot be physically approached. Indeed, it is impossible for an observer to see a rainbow from water droplets at any angle other than the customary one of 42 degrees from the direction opposite the light source. Even if an observer sees another observer who seems "under" or "at the end of" a rainbow, the second observer will see a different rainbow—farther off—at the same angle as seen by the first observer.
   In addition to the common primary and secondary rainbows, it is also possible for rainbows of higher orders to form. The order of a rainbow is determined by the number of light reflections inside the water droplets that create it: One reflection results in the first-order or "primary" rainbow; two reflections create the second-order or "secondary" rainbow. More internal reflections cause bows of higher orders—theoretically unto infinity. As more and more light is lost with each internal reflection, however, each subsequent bow becomes progressively dimmer and therefore increasingly harder to spot. An additional challenge in observing the third-order (or "tertiary") and fourth-order ("quaternary") rainbows is their location in the direction of the sun (about 40° and 45° from the sun, respectively), causing them to become drowned in its glare.
   In theory, every rainbow is a circle, but from the ground, usually only its upper half can be seen. Since the rainbow's centre is diametrically opposed to the sun's position in the sky, more of the circle comes into view as the sun approaches the horizon, meaning that the largest section of the circle normally seen is about 50% during sunset or sunrise. Viewing the rainbow's lower half requires the presence of water droplets "below" the observer's horizon, as well as sunlight that is able to reach them. These requirements are not usually met when the viewer is at ground level, either because droplets are absent in the required position, or because the sunlight is obstructed by the landscape behind the observer. From a high viewpoint such as a high building or an aircraft, however, the requirements can be met and the full-circle rainbow can be seen. Like a partial rainbow, the circular rainbow can have a secondary bow or supernumerary bows as well. It is possible to produce the full circle when standing on the ground, for example by spraying a water mist from a garden hose while facing away from the sun.
   Rainbows are the result of a combination of internal reflection and dispersive refraction of light in raindrops. A single reflection off the backs of an array of raindrops produces a rainbow with an angular size on the sky that ranges from 40° to 42° with red on the outside. Double rainbows are produced by two internal reflections with angular size of 50.5° to 54° with violet on the outside. Because rainbows are seen with the sun 180° away from the centre of the rainbow, rainbows are more prominent the closer the sun is to the horizon.
  
  Section::::See also.
   The Irish leprechaun's secret hiding place for his pot of gold is usually said to be at the end of the rainbow. This place is impossible to reach, because the rainbow is an optical effect which depends on the location of the viewer. When walking towards the end of a rainbow, it will appear to "move" further away (two people who simultaneously observe a rainbow at different locations will disagree about where a rainbow is).
   In certain circumstances, one or several narrow, faintly coloured bands can be seen bordering the violet edge of a rainbow; i.e., inside the primary bow or, much more rarely, outside the secondary. These extra bands are called "supernumerary rainbows" or "supernumerary bands"; together with the rainbow itself the phenomenon is also known as a "stacker rainbow". The supernumerary bows are slightly detached from the main bow, become successively fainter along with their distance from it, and have pastel colours (consisting mainly of pink, purple and green hues) rather than the usual spectrum pattern. The effect becomes apparent when water droplets are involved that have a diameter of about 1 mm or less; the smaller the droplets are, the broader the supernumerary bands become, and the less saturated their colours. Due to their origin in small droplets, supernumerary bands tend to be particularly prominent in fogbows.
   Rainbows can be caused by many forms of airborne water. These include not only rain, but also mist, spray, and airborne dew.
  
  Section::::Visibility.
   Usually the Wave wears off on a half way to the equator. If not, special energy-absorbing machines called "charybdis" (after mythological Charybdis from Odyssey) are used to lower the potential energy of the Wave thus wearing it off. It is also known that the Wave can disappear itself any moment for no apparent reason.
   In Book I of "Naturales Quaestiones" (c. 65 AD), the Roman philosopher Seneca the Younger discusses various theories of the formation of rainbows extensively, including those of Aristotle. He notices that rainbows appear always opposite to the sun, that they appear in water sprayed by a rower, in the water spat by a fuller on clothes stretched on pegs or by water sprayed through a small hole in a burst pipe. He even speaks of rainbows produced by small rods (virgulae) of glass, anticipating Newton's experiences with prisms. He takes into account two theories: one, that the rainbow is produced by the sun reflecting in each water drop, the other, that it is produced by the sun reflected in a cloud shaped like a concave mirror; he favours the latter. He also discusses other phenomena related to rainbows: the mysterious "virgae" (rods), halos and parhelia.
   Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement. In other places, the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre.
   A very similar experiment consists in using a cylindrical glass vessel filled with water or a solid transparent cylinder and illuminated either parallel to the circular base (i.e. light rays remaining at a fixed height while they transit the cylinder) or under an angle to the base. Under these latter conditions the rainbow angles change relative to the natural phenomenon since the effective index of refraction of water changes (Bravais' index of refraction for inclined rays applies).
  
  Other experiments use small liquid drops, see text above.
  
  Section::::Culture.
   The colour pattern of a rainbow is different from a spectrum, and the colours are less saturated. There is spectral smearing in a rainbow owing to the fact that for any particular wavelength, there is a distribution of exit angles, rather than a single unvarying angle. In addition, a rainbow is a blurred version of the bow obtained from a point source, because the disk diameter of the sun (0.5°) cannot be neglected compared to the width of a rainbow (2°). Further red of the first supplementary rainbow overlaps the violet of the primary rainbow, so rather than the final colour being a variant of spectral violet, it is actually a purple. The number of colour bands of a rainbow may therefore be different from the number of bands in a spectrum, especially if the droplets are particularly large or small. Therefore, the number of colours of a rainbow is variable. If, however, the word "rainbow" is used inaccurately to mean "spectrum", it is the number of main colours in the spectrum.
   Section::::Background.
   A rainbow does not exist at one particular location. Many rainbows exist; however, only one can be seen depending on the particular observer's viewpoint as droplets of light illuminated by the sun. All raindrops refract and reflect the sunlight in the same way, but only the light from some raindrops reaches the observer's eye. This light is what constitutes the rainbow for that observer. The whole system composed by the sun's rays, the observer's head, and the (spherical) water drops has an axial symmetry around the axis through the observer's head and parallel to the sun's rays. The rainbow is curved because the set of all the raindrops that have the right angle between the observer, the drop, and the sun, lie on a cone pointing at the sun with the observer at the tip. The base of the cone forms a circle at an angle of 40–42° to the line between the observer's head and their shadow but 50% or more of the circle is below the horizon, unless the observer is sufficiently far above the earth's surface to see it all, for example in an aeroplane (see above). Alternatively, an observer with the right vantage point may see the full circle in a fountain or waterfall spray.
   Section::::Variations.:Reflected rainbow, reflection rainbow.
  
  When a rainbow appears above a body of water, two complementary mirror bows may be seen below and above the horizon, originating from different light paths. Their names are slightly different.
  
  A reflected rainbow may appear in the water surface below the horizon. The sunlight is first deflected by the raindrops, and then reflected off the body of water, before reaching the observer. The reflected rainbow is frequently visible, at least partially, even in small puddles.
   BULLET::::- Rainbows. These result from a combination of internal reflection and dispersive refraction of light in raindrops. Because rainbows are seen on the opposite side of the sky from the sun, rainbows are more visible the closer the sun is to the horizon. For example, if the sun is overhead, any possible rainbow appears near an observer's feet, making it hard to see, and involves very few raindrops between the observer's eyes and the ground, making any rainbow very sparse.
  
  Other phenomena that are remarkable because they are forms of visual illusions include:
  
  BULLET::::- Crepuscular rays,
  
  BULLET::::- Anticrepuscular rays, and
   Section::::Explanation.:Mathematical derivation.
  
  It is possible to determine the perceived angle which the rainbow subtends as follows.
  
  Given a spherical raindrop, and defining the perceived angle of the rainbow as , and the angle of the internal reflection as , then the angle of incidence of the sun's rays with respect to the drop's surface normal is . Since the angle of refraction is , Snell's law gives us
  
  where is the refractive index of water. Solving for , we get
   A reflection rainbow may be produced where sunlight reflects off a body of water before reaching the raindrops), if the water body is large, quiet over its entire surface, and close to the rain curtain. The reflection rainbow appears above the horizon. It intersects the normal rainbow at the horizon, and its arc reaches higher in the sky, with its centre as high above the horizon as the normal rainbow's centre is below it. Due to the combination of requirements, a reflection rainbow is rarely visible.
   Judith Bauer Stamper's 1987 Find Your Fate book "Jem and The Holograms #3: The Secret of Rainbow Island ", involves the reader in saving Rainbow Island, a tropical island that frequently has rainbows.
  
  American legend retelling "The Rough Face Girl" involves the heroines seeing an invisible Chief, who wears a rainbow as a sash.
  
  Stephen King's 1985 book 'It" features a scene where Ben Hanscom sees a spray bow while fighting with Henry Bowers in The Barrens, and comments on finding gold at the end of it.
   If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves of that disturbance, after reflecting off the walls, converge simultaneously to a single point: the "second focus". This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.
   Section::::Variations.:Fogbow.
  