The Mathematics of the Rainbow

Each internal reflection turns the beam by 180°-2β, two refractions (in and out) turn it by 2(α-β).


For k internal reflections the total deflexion δ is:

δ = 2(α-β) + k(180°-2β) = 2α - 2(k+1)β + k 180°    (1)

To find the angle α of incidence at extremal deflexion we differentiate the equation (1) for α and solve for zero:
dbda         (2)

Using Snell's law: sin(β) = sin(α)/n we have β = arcsin[sin(α)/n]. Differentiating β for α:

diffentiate     (3)
Equating and squaring (2) and (3) we get the angles α and β at extremal deflexion:


folgt   sos alpha     or      sinus

sinus beta

Inserting into equation (1) for extremal deflexion:


For one reflexion (primary rainbow):
deflexion angle delta

Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet regi

Web Links
The Work of G.G. Stokes in Evaluating the Airy Rainbow Integral and its Ramifications Today (Anne B. O’Donnell)

From Alexander of Aphrodisias to Young and Airy (J.D. Jackson)

The Mathematics of Rainbows (American Mathematical Society)

Zwei Bemerkungen zu Airy's Theorie des Regensbogens (W. Wirtinger)
The International Association for the Properties of Water and Steam
Dietze, Gerhard: Einführung in die Optik der Atmosphäre; Akadem. Verl.-Ges. Geest & Portig, Leipzig 1957.

Pernter, J. M., Exner, F. M.: Meteorologische Optik; Salzwasser Verlag, Paderborn, 2012; 978-3-86444-515-6
Nachdruck der 2. Auflage 1922

Pernter, J. M., Exner, F. M.: Meteorologische Optik; Wilhelm Braumüller, Wien und Leipzig, 2. Aufl. 1922

Vollmer, Michael: Lichtspiele in der Luft, Atmosphärische Optik für Einsteiger; Spektrum Akademischer Verlag,
978-3-8274-3092-2  (Softcover).

2016 J. Giesen

updated: 2016, Apr 28