1b24 waves optics and acoustics lecture 5 of 33 (ucl)

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Sinusoidal Waves A single-frequency wave may be written as fxt , ( ) A cos k x . ϖ t . ( ) . where ϖ 2 π . ν k 2 π . λ 0 1 2 3 4 5 1 0 1 f 0 t , ( ) t λ 0 1 2 3 4 5 1 0 1 fx 0 , ( ) x

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1B24 Waves Optics and Acoustics Lecture 5 of 33 (UCL), astronomy, astrophysics, cosmology, general relativity, quantum mechanics, physics, university degree, lecture notes, physical sciences

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Page 1: 1B24 Waves Optics and Acoustics Lecture 5 of 33 (UCL)

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Sinusoidal Waves

A single-frequency wave may be written as

f x t,( ) A cos k x. ω t.( ).

where

ω2 π.

νk2 π.

λ

0 1 2 3 4 51

0

1

f 0 t,( )

0 1 2 3 4 51

0

1

f x 0,( )

x

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Page 4: 1B24 Waves Optics and Acoustics Lecture 5 of 33 (UCL)

Refractive indexAbsorptionLimiting value = 1

Typical variation of refractive index

Frequency

n

Infrared Visible Ultraviolet X-ray

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Page 6: 1B24 Waves Optics and Acoustics Lecture 5 of 33 (UCL)

Two Superposed Waves

y ωω 1 ωω 2, k 1, k 2, x, t, a cos ωω 1 t. k 1 x.. a cos ωω 2 t. k 2 x..

Take the angular frequencies to be 10 ππ. and 11.5 ππ. , wavevectors to be 10 ππ. and 12 ππ.

0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2

1

1

2

time t=0

0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2

1

1

2

time t=0.15

The carrier has edged slightly ahead of the envelope.

Groupv.mcd 1 04/02/99 12:29/{OTs3:<287$����lkH�¡~��CE0�U8:<�§7�2e�¥03Ue7�_BU8OP=N7dCG4�a37-U�x§U8R<0´C OP=Ns�I8R<7d>�4H2e28OP7-2 4H=<_�7�=JaH7�zP0H�¥7blPhase velocity : 10 ππ. 11.5 ππ.

10 ππ. 12 ππ.0.977=

Group velocity: 11.5 1012 10

0.75=

We expect the envelope to lag behind

0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2

1

1

2

The carrier wavelength is 2 ππ.

11 ππ.0.182=

By t 0.4 the envelope has travelled to x 0.3 but the carrier has moved to x 0.4 , i.e. the carrier is ahead by half a wavelength.

Groupv.mcd 2 04/02/99 12:30

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