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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

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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