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Wikipedia :   Rainbow   |   CIE chromaticity diagram (1931)   |   Fraunhofer lines

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(2005-09-29)   Dispersion Relation
The celerity of a wave as a function of its frequency.

In a given propagation medium, the  dispersion relation, is whatever gives the celerity of a wave in terms of either its frequency  (n )  or its wavelength  (l).

The simplest dispersion relation is that of a nondispersive medium, for which the celerity (u) is constant.  For example, the celerity of electromagnetic waves in a vacuum is equal to Einstein's constant  (u = c).

One common way to specify the  dispersion relation  is by giving the  pulsatance  w = 2pn  as a function of the  wave number  k = 2p/l

w   =   w(k)

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(2005-09-29)   Group Speed
The speed at which a wave may carry information.

A wave where a single frequency is present is unable to carry any information. 

v   =   dw / dk   =   -l²  dn / dl

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(2008-01-24)   Rayleigh scattering   (Tyndall effect, 1859)
(2007-07-24)   What makes the sky blue and sunsets red?
(2007-07-13)   Why do we perceive the Sun as yellow?

In 1859, John Tyndall (1820-1893) observed that small particles suspended in a fluid scatter bluish light  (short wavelength)  more strongly than reddish light  (long wavelength).  This scattering of light by tiny particles is known either as the  Tyndall effect  or  (more commonly)  Rayleigh scattering.  The intensity of the effect varies inversely as the  fourth power  of the wavelength involved.

One crude way to explain the main part of effect is to consider that an incoming electromagnetic wave produces induced dipoles which radiate energy away at the same frequency as the driving wave.

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(2008-01-24)   Index of refraction of water
Different colors travel at different speeds in water.

For visible light in water, the index of refraction  (n)  goes from  1.331  for red light  to about  1.343  for violet light.  More precise data is tabulated below.

Absolute Index of Refraction of Water  (n)

n  (20°C)	l  (vacuum)	Fraunhofer Line
1.3312	656.281 nm	C   ( Ha )	Red
	627.661 nm	a   ( O2 )	Orange
1.3330	589.3 nm	D   ( Na )	Yellow
	527.039 nm	E   ( Fe )	Green
1.3372	486.134 nm	F   ( Hb )	Blue
1.3404	434.047 nm	G'   ( Hg )	Indigo
1.3435	396.847 nm	H   ( Ca+ )	Violet

Data gleaned for the relative index of water with respect to either air or vacuum:

• Sodium light  (yellow, 589.3 nm)  in water at  t °C  (accuracy 0.00002):
  n_vacuum   =   1.33401 - 10^-7 (66 t + 26.2 t² - 0.1817 t³ + 0.000755 t⁴ )

Index of Refraction of Water

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(2008-01-24)   Reflection by a raindrop
Several types of reflections are possible.

Let  n  be the index of refraction of the water inside a spherical raindrop  (relative  to the surrounding air).  The dominant mode of reflection is pictured at right.

Elementary geometry gives the angle  q  between the incident and emergent rays as a function of the angles of incidence  (i)  and refraction  (r)  which the rays make with the [centripetal] normal lines at each of the three relevant diopters:

q   =   4 r  -  2 i

As  i  increases  (starting from 0)   so does  q,  until a maximum is reached where the relation  2 dr = di  makes  dq  vanish.  At that point, Snell's law and its derivative translate into two simultaneous equations:

n  sin r   =     sin i
  n  cos r   =   2  cos i

Putting   sin i  =  x ,  we have   sin r  =  x/n.  Squaring the second relation gives:

n ² ( 1 - x ² / n ² )   =   4 ( 1 - x ² )

Therefore,   x ²  =  (4-n ² ) / 3 .   Using   cos 2i  =  1-2x ²   we obtain:

i   =   ½ arccos (2n²/3 - 5/3)

Similarly,  cos 2r  =  1-2x² / n²   gives   r   =   ½ arccos (5/3 - 8/3n² ) .  So:

qmax   =   2 arccos (5/3 - 8/3n2 )  -  arccos (2n2/3 - 5/3)

With  n = 1.3312  (red light in water at 20°C)  we obtain  q_max = 42.34°.  On the other hand,  n = 1.3435  (violet light)  yields  q_max = 40.58°. 

For graphics, we used  q = 42.4°,  i = 59.4°,  r = 40.4°  (n = 1.3308).  As  i  is near the Brewster angle of 53.08°, strong polarization occurs.

What the main relection mode produces is the familiar sight of a beautiful  42°  rainbow (the primary rainbow) around the direction  opposite  to the Sun, as explained in the next article.

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(2008-01-27)   Primary and Secondary Rainbows
The spectacular show put on by water droplets.

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(2008-01-27)   The  22° Halo
From ice crystals in high-altitude cirrus clouds.

Under the same conditions, an halo also exists around the Sun but it's much harder to detect because of the blinding effect of direct sunlight.