1 The Subtlety of Rainbows Dr Andrew French A. French. 2011. Page 1
Dec 16, 2015
1
The Subtlety of RainbowsDr Andrew French
A. French. 2011. Page 1
Summary• When I worked in the radar systems simulation & modelling
group of BAE Systems we often constructed mathematical models of physical processes (such as reflection of microwaves off aeroplanes, rain, sea…) in order to run accurate computer simulations which allowed us to minimise the amount of (expensive) tests using real radar equipment.
• The example here of a mathematical model of a rainbow aims to help us to understand how simple principles of physics can be used to predict observable quantities such as the shape, angular width and order of colours.
• All models are built upon assumptions - essentially what set of scientific laws can we apply to the situation.
A. French. 2011. Page 2
Descartes theory of the rainbow
‘des cartes postal’ !A. French. 2011. Page 3
Descartes theory of the rainbow - assumptions
• Light is internally reflected off the interior of spherical raindrops.• The wavelength of light is much smaller than the dimensions of the
raindrop, so interference effects can be ignored.• The mathematical relation between the angle that light is bent by the
raindrops and the angle of incidence to the raindrop, has an ‘extremum’.* This results in the focussing of light of particular wavelengths into particular angles.
Incident light
Sphericalraindrop
Light from rain cloud
Maximum
0 degrees 90 degrees
*max or min.A. French. 2011. Page 4
Schematic of a rainbow
‘Anti solar direction’
Sun
Parallel rays of incident sunlight
Rain cloud
Rainbow
Horizon
Observer
See next page for close up!
A. French. 2011. Page 5
Internal reflection of sunlight by a spherical raindrop
Close up of rainbow showing deflection of incident rays of sunlight by internal reflection within a spherical raindrop of radius a
a
a
a
Spherical raindrop
Incident ray
Deflected ray
A. French. 2011. Page 6
a
a
Spherical raindrop
Incident ray
Deflected ray
A
CBA 2
Internal reflection of sunlight by a spherical raindrop
a
A. French. 2011. Page 7
a
a
Spherical raindrop
Incident ray
Deflected ray
A
CBA 2
B
Internal reflection of sunlight by a spherical raindrop
a
A. French. 2011. Page 8
a
a
Spherical raindrop
Incident ray
Deflected ray
A
CBA 2
B
C
Internal reflection of sunlight by a spherical raindrop
a
A. French. 2011. Page 9
a
a
Spherical raindrop
Incident ray
Deflected ray
A
CBA 2
B
C 24
Internal reflection of sunlight by a spherical raindrop
a
A. French. 2011. Page 10
air
water
Light ray
Air-water interfaceNormal to interface
Snell’s Law of refraction
A. French. 2011. Page 11
Snell’s Law of refraction
n sin
sin
Refractive index
air
water
Light ray
Air-water interfaceNormal to interface
A. French. 2011. Page 12
Refractive index
n sin
sin
Refractive index is the ratio of the speed of light in a medium to that of the speed of light in a vacuum
medium
vacuum
cc
n
For air n~1For water n~1.4
Note n often varies with thefrequency of light
A. French. 2011. Page 13
Snell’s law again!n
sinsin
n = 1,
n >1,
n
So for light entering a medium of higher refractive index, light is always bent towards the normal to the interface
A. French. 2011. Page 14
Formula for rain cloud light elevation
24
n sin
sin
2sin
sin4)( 1
n
Incident light
Sphericalraindrop
Light from rain cloud
Maximum
0 degrees 90 degrees
A. French. 2011. Page 15
Visible light and refractiveindices
2sin
sin4)( 1
n
Colour Wavelength in vacuo /nm Approximate refractive index Red 780-622 1.330 Orange 622-597 Yellow 597-577 Green 577-492 Blue 492-455 Violet 455-390 1.341
Rememberfc
FrequencyWavelength
Speed
nm=10-9 metres
In water
A. French. 2011. Page 16
Frequency (or wavelength) variation of refractive index n of water
A. French. 2011. Page 17
Minimum deviation and light focussing 2
sinsin4)( 1
n
Graph produced using MATLAB
Gradient of
Angle of rainbow
A. French. 2011. Page 18
Rainbow structure! 2sin
sin4)( 1
n
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)
A. French. 2011. Page 19
Rainbow structure!
Rain cloud
RainbowHorizon
41.7o
Light from rain cloud
1.6o
A. French. 2011. Page 20
See a circular rainbow when flying since rain cloud is illuminated above and below aircraft
Sun
Incident sunlight
Incident sunlight Rainbow
Rain cloud
A. French. 2011. Page 21