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The angle through which an optical element changes a light ray's propagation direction, with the chosen branch and convention specified. For the one-internal-reflection branch through a spherical drop, entry refraction, internal specular reflection, and exit refraction give $\delta =\pi +2\alpha -4\beta$, with $\sin \alpha =n\sin \beta$. The corresponding primary rainbow radius about the antisolar direction is $\pi -\delta$.

Wave dispersion means that different frequencies propagate with different phase velocities, so a wave packet generally changes shape. Equivalently the dispersion relation is not proportional to $k$ over the relevant band. Three natural examples are the colour separation in a rainbow, where the refractive index of water depends on wavelength; deep-water surface gravity waves, for which ${\omega }^2=gk$ and long waves have larger phase velocity; and capillary waves, whose short-wave limit is ${\omega }^2=(\gamma /\rho )k^3$ and whose shorter waves have larger phase velocity. Here $g$ is gravitational acceleration, $\gamma$ is surface tension, and $\rho$ is fluid mass density, with positive $k$ and negligible viscosity.

A distinct application is an optical prism spectrometer: the wavelength-dependent refractive index produces different refraction angles, separating spectral components for measurement. This exploits wave dispersion deliberately.

Regard the refraction angle as a function of the angle of incidence, $\beta =\arcsin (\sin \alpha /n)$. Implicit differentiation of Snell's law gives ${\beta }^{\prime }=\cos \alpha /(n\cos \beta )$. Differentiating the deviation angle gives ${\delta }^{\prime }=2-4{\beta }^{\prime }$. An interior stationary point therefore satisfiesThe stationary deviation of a primary rainbow ray is a minimum: differentiating $\cos \alpha =n\cos \beta {\beta }^{\prime }$ again and using Snell's law gives ${\beta }^{\prime \prime }=\tan \beta (({\beta }^{\prime }{)}^2-1)$. At the stationary point, ${\beta }^{\prime }=1/2$, and hence ${\delta }^{\prime \prime }=-4{\beta }^{\prime \prime }=3\tan \beta >0$ for a positive interior refraction angle. This minimum is responsible for the concentration of emerging light rays near the primary rainbow direction.

With the Sun behind the observer and raindrops ahead, the one-reflection branch sends many light rays toward directions close to the minimum deviation angle. Nearby values of the angle of incidence produce nearly the same outgoing direction there, forming a wave caustic. For water the resulting cone is approximately $42^{\circ }$ from the antisolar direction. The observer sees the intersecting drops as a circular rainbow, or the part of that circle visible above the ground.

Normal dispersion makes the refractive index larger for violet than for red light. At the minimum, increasing the refractive index increases the deviation angle, and therefore decreases the viewing angle $\pi -{\delta }_{\min }$. Red is on the outer edge of the primary rainbow, violet on the inner edge. The internal specular reflection is partial: with $n=4/3$, the relevant $\beta \simeq 40.2^{\circ }$ is below the critical angle $\arcsin (1/n)\simeq 48.6^{\circ }$. Diffraction and the finite angular size of the Sun smooth the geometrical concentration; the physical intensity does not actually become infinite.