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Rainbow maths! A. French. March 2004 Rain cloud Parallel rays of incident sunlight Sun Rainbow Horizon Observer Close up of rainbow showing deflection of incident rays of sunlight by internal reflection within a spherical raindrop of radius a Incident ray a a a Spherical raindrop Deflected ray Figure 1. Schematic of rainbow seen by ground based observer. The rainbow is formed by the deflection and focus of incident sunlight by spherical raindrops into a narrow range of elevation . Assumptions: (1) All raindrops are perfectly spherical. Since water molecules are polar, electric fields can distort the spherical shape of rain droplets so this assumption may become invalid during thunderstorms. (2) Assume raindrops have identical liquid content (water) whose refractive index variation with optical frequency follows a defined semi-empirical model. This implies the raindrops are fairly uniform in temperature, pressure and impurity content. We will be using a model of ‘pure water’ to compute the refractive index, so we neglect the latter. (3) The refractive index of the ambient air is unity. (4) Assume a ray model of sunlight propagation through the raincloud . i.e. don’t include any interference effects from the wave nature of light. For this assumption to be valid the wavelength of light within the raindrop a . Note wavelength of light varies between 0.4 and 0.8 microns. Colour Red Orange Yellow Green Blue Violet Table 1. Wavelength in vacuo /nm 780-622 622-597 597-577 577-492 492-455 455-390 Approximate refractive index 1.330 1.341 Primary colours related to ranges of optical wavelength (measured in vacuo). Hypothesis: Descartes Theory of the Rainbow Assert a rainbow is formed by deflection and focus of incident sunlight by spherical raindrops into a narrow range of elevation . The deflection occurs via internal reflection of incident sunlight and the focusing effect results from the observation that the total angle of deflection passes through an extremum as the angle between incidence and raindrop surface normal is varied. The large multitude of raindrops within a raincloud allows us to assume that all possible angles of are explored within the ensemble. If a mathematical model can determine ( ) then the postulate of ‘extremum deviation’ implies the first derivative of must pass through zero. When d is small, this can mean varies little for large deviations d in . Hence a wide range of sunlight rays (each with a different value) will result in similar angles. This is another way of saying sunlight is focused in a narrow range of elevation angle. A mathematical model of the deflection of light by spherical raindrops. We can compute an expression for ( ) using the geometrical construction at the bottom of figure 1. The construction was created noting all surface normals on a sphere point radially outwards and thus emanate from the center of the sphere. Thus the reflected ray paths bound two joined isosceles triangles. Also assumed is the Law of Reflection, that is the angle of incidence of light upon a surface equals the angle of reflection from that surface. (Angles measured from the ray directions to the surface normal). Decomposing the total deflection into (A) surface refraction at incidence, (B) internal reflection & (C) surface refraction at ray emergence: 2 A Incident ray B (1) C A a a B a C Spherical raindrop Deflected ray Hence: 4 2 (2) Let the refractive index of water within the raindrop be n and let n vary with optical frequency f by some known relation n( f ) . This is described in equation (?) below. Snell’s law relates angle to in terms of n . sin sin n (3) Therefore: sin 2 n ( ) 4 sin 1 (4) sin 2 n ( ) 4 sin1 Angle of rainbow Gradient of From (2) we can differentiate to yield: d d 4 2 d d (5) Implicit differentiation of Snell’s law (3) gives: cos n cos d d cos d d n cos (6) Using the trigonometric identity sin 2 cos 2 1 and (3) we can write: sin 2 cos 1 2 n (7) d cos d n 2 sin 2 (8) d 4 cos 2 d n 2 sin 2 (9) Hence: This gives: The minimum of is found using the angle min which solves the equation 4 cos min 2 n 2 sin 2 min i.e. d 0. d (10) 4 1 sin 2 min 2 n 2 sin 2 min 41 sin 2 min n 2 sin 2 min 3 sin 2 min 4 n 2 min sin 1 Therefore: 4 n2 3 (11) The minimum value of is thus 4 sin 1 since from (3) 4 n2 3n 2 2 2 sin 1 4 n 3 (12) sin . n sin 1 Hence one expects to see rings of focused light (a separate ring for each frequency since n varies with frequency) and elevations . Note the land based observer shown in Figure 1 will only see the upper arc. A flying observer, experiencing illumination above and below will see both upper and lower arcs and therefore see a circular rainbow. Rain cloud Incident sunlight Sun Rainbow Incident sunlight Refractive index of water (for optical frequencies) Hecht [1] gives a formula for the frequency variation of refractive index of a transparent medium. Using data from Kay & Layby [2] we can deduce the following semi-empirical relationship between the refractive index of water and (optical) frequencies of light. n 2 1 2 f 15 10 Hz 2 (13) where (n*n-1)^-2 1.7587 and 0.3612 . 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 y = -0.3612x + 1.7587 R2 = 0.9876 0.0 0.5 1.0 Refractive index of water Linear (Refractive index of water) 1.5 (frequency/10^15 Hz)^2 2.0 2.5 The positive roots of (13) were used to compute the graph below for water at 20 degrees celsius. In summary, (12) combined with (13) can be used to give the elevation of each colour component of the rainbow. Note that the + and - solutions of (12) yield the same numerical result. So a rainbow is observed at a mean angle of about 41.7o with an angular width of about 1.6o. Moving outwards in angle, colours go from violet (higher frequency, smaller wavelength) to red (lower frequency, larger wavelength) Rain cloud 1.6o Light from rain cloud Rainbow Horizon 41.7o Final notes A full wave-theoretic description is necessary to explain all the properties of the rainbow, but Descartes' simple model is sufficient for the majority of observations. Rare phenomenon such as 'multiple rainbows' and 'fog bows' result from (respectively) multiple internal reflections (i.e. more than just one) and interference effects when the wavelength of light is similar to the raindrop size. Computation of the light intensity of the rainbow requires a wave-theoretic approach. References: [1] [2] HECHT, E. Optics. 3rd Edition, 1998. Addison, Wesley, Longman Inc. LAY & LAYBY. pp95. A. French. March 2004.