The Physics and Formation of Rainbows

AUnlike a bridge or a tower, a rainbow possesses no fixed physical location in the landscape. If a person moves towards the apparent position of the colourful arc, it recedes at an identical pace, maintaining a constant distance from the viewer. This occurs because a rainbow is not a solid entity anchored in space, but an optical illusion generated by light interacting with millions of suspended water droplets. For an observer to witness the phenomenon, two basic conditions must be met: the sun must be situated behind them at a relatively low elevation, and airborne moisture, such as rain or mist, must lie ahead. Consequently, every individual observer views their own unique visual arc, formed by a distinct collection of raindrops that happen to redirect light directly into their eyes.

BThe emergence of this circular arc relies on the path of sunlight inside individual droplets. When parallel rays of white sunlight strike a spherical drop of water, they bend as they pass from the less dense medium of air into the denser water. Most of this light travels straight through and exits the other side, but a proportion reflects off the inner rear boundary of the droplet. As this reflected beam exits the front surface, it bends once more into the surrounding air. Crucially, the geometry of spherical refraction dictates that the redirected rays do not scatter evenly in all directions. Instead, they become intensely concentrated along a specific exit trajectory, emerging at an angle of roughly forty-two degrees relative to the incoming solar rays.

CWhile geometric concentration explains the brightness of the bow, it does not account for the vivid separation of colours. Sunlight is composed of a continuous blend of different wavelengths, each of which reacts slightly differently when entering a medium of higher density. Shorter wavelengths, corresponding to blue and violet light, experience a greater degree of deceleration and consequently bend more sharply than longer wavelengths such as red. When the light reflects internally and exits the droplet, this slight angular deviation is amplified. As a result, red light emerges at an angle of approximately forty-two degrees, while violet light exits at roughly forty degrees. Because these separated wavelengths reach the observer's eye from slightly different heights in the sky, a distinct spectrum of concentric colour bands is perceived.

DObservers frequently notice a second, fainter rainbow arching above the primary one. This secondary feature is produced when sunlight undergoes two distinct reflections within each water droplet before escaping back into the atmosphere. Because some energy is lost during each reflection, the secondary bow is noticeably dimmer than the primary arc. Furthermore, the additional reflection inverts the spatial arrangement of the colours, causing red to appear along the inner rim and violet along the outer edge. The space separating these two bows is known to scientists as Alexander's band. In this intermediate zone, no light is directed towards the observer from single or double internal reflections, making this strip of sky appear distinctly darker than the surrounding areas.

EAlthough all raindrops follow the same optical laws, the visual characteristics of a rainbow vary substantially depending on the physical dimensions of the water droplets involved. Large raindrops, often exceeding a millimetre in diameter, produce exceptionally bright and narrow bands with sharply defined colours. Conversely, when droplets are microscopic, as in sea spray or mountain fog, the principles of geometric optics are modified by wave diffraction. In such conditions, the distinct colour bands spread out and overlap, often creating a completely white arc known as a fogbow. In other instances, wave interference inside small drops produces faint, pastel-tinted bands along the inner edge of the primary rainbow, a phenomenon referred to as supernumerary arcs.

FThe systematic explanation of these optical processes developed over many centuries through the contributions of diverse natural philosophers. In the fourteenth century, the Dominican scholar Theodoric of Freiberg made a major breakthrough by conducting laboratory experiments with spherical glass flasks filled with water, allowing him to trace the internal path of light rays through individual simulated droplets. Later, in the seventeenth century, René Descartes applied rigorous mathematics to determine the exact angles at which light concentrates after internal reflection, successfully calculating the geometric basis of both the primary and secondary bows. Isaac Newton subsequently refined this model by demonstrating that white light is a composite of different colours, thereby explaining the chromatic dispersion that Descartes had not fully resolved.

GUnder unusual atmospheric and environmental circumstances, the standard circular display can transform into far rarer optical configurations. For instance, when sunlight reflects off a completely calm body of water before striking raindrops, it produces an upward-curving reflection rainbow that intersects the standard arc. Other variations occur at night, where bright moonlight replaces sunlight to generate a faint, pale apparition known as a moonbow. Furthermore, strong updrafts can flatten falling raindrops into non-spherical shapes, splitting the single primary arc into two distinct, branching bows that share a single base. These complex forms demonstrate that the interplay between light and atmospheric moisture can generate patterns far more intricate than the traditional single arc.