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Propagation of waves

Friday October 18, 2002

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Propagation of waves in 3D

Imagine a disturbane that results in waves propagating equally in all directionsE.g. sound wave source in air or water, light

source in a dielectric medium etc.. The generalization of the wave equation to 3-

dimensions is straight forward if the medium is homogeneous

Let = amplitude of disturbance (could be amplitude of E-field also)

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Propagation of waves in 3D

01

2

2

22

2

2

2

2

2

tvxyx

01

2

2

22

tv

depends on x, y and z such that it satisfies the wave equationdepends on x, y and z such that it satisfies the wave equation

zz

yy

xx

ˆˆˆ

or,or,

where in cartesian co-ordinates,where in cartesian co-ordinates,

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1. Special Case: Plane Waves along x

Suppose (x, y, z, t)=(x, t) (depends only on x)

Then = f(kx-ωt) + g(kx+ωt) Then for a given position xo, has the same

value for all y, z at any time to. i.e. the disturbance has the same value in the

y-z plane that intersects the x-axis at xo. This is a surface of constant phase

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Plane waves along x

Planes perpendicular to the x-axis are wave fronts – by definitionPlanes perpendicular to the x-axis are wave fronts – by definition

xkˆ

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2. Plane waves along an arbitrary direction (n) of propagation

Now will be constant in plane perpendicular to n – if wave is plane

For all points P’ in plane

dnr ˆ OO

zz

xx

yy

ddP’P’

r

n̂PP

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2. Plane waves along an arbitrary direction (n) of propagation

tnrkf

tkdf

ˆ

For all points P’ in plane For all points P’ in plane

or, for the disturbance at Por, for the disturbance at P

tkdf

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2. Plane waves along an arbitrary direction (n) of propagation

OO

zz

xx

yy

ddP’P’

r

n̂PP

If wave is plane, If wave is plane, must be the must be thesame everywhere in plane same everywhere in plane to n to n

This plane is defined byThis plane is defined by

constdOPnnr

or

nOPr

ˆˆ

,

0ˆ

is equation of a plane is equation of a plane to n, to n,a distance d from the origina distance d from the origin

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2. Plane waves along an arbitrary direction (n) of propagation

trkf

trnkf

tnrkf

ˆ

is the equation of a plane wave propagating in k-directionis the equation of a plane wave propagating in k-direction

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3. Spherical Waves Assume has spherical symmetry about

origin (where source is located) In spherical polar co-ordinates

tr ,

2

2

2222

22

sin

1sin

sin

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rrr

rrr

θθ

φφxx

yy

zz

rr

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3. Spherical Waves

Given spherical symmetry, depends only on r, not φ or θ Consequently, the wave equation can be written,

012

,

011

2

2

22

2

2

2

22

2

tvrrr

or

tvrr

rr

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3. Spherical WavesNow note that,Now note that,

2

2

22

2

2

2

2

2

2

2

2

2

2

1

2

2

t

r

vr

r

tv

r

rrrr

rr

r

rr

rr

r

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3. Spherical Waves

0

12

2

22

2

t

r

vr

r But,But,

is just the wave equation, whose solution is,is just the wave equation, whose solution is,

tkrfr

tkrgtkrfr

1

i.e. amplitude decreases as 1/ r !! i.e. amplitude decreases as 1/ r !! Wave fronts are spheresWave fronts are spheres

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4. Cylindrical Waves (e.g. line source)

tkA

cos

The corresponding expression is,The corresponding expression is,

for a cylindrical wave traveling along positive for a cylindrical wave traveling along positive

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Electromagnetic waves

Consider propagation in a homogeneous medium (no absorption) characterized by a dielectric constant

o

oo = permittivity of free space = permittivity of free space

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Electromagnetic wavesMaxwell’s equations are, in a region of no free charges,Maxwell’s equations are, in a region of no free charges,

t

E

t

EjB

t

BE

B

E

oo

4

0

04 Gauss’ law – electric fieldGauss’ law – electric fieldfrom a charge distributionfrom a charge distribution

No magnetic monopolesNo magnetic monopoles

Electromagnetic inductionElectromagnetic induction(time varying magnetic field(time varying magnetic fieldproducing an electric field)producing an electric field)

Magnetic fields being inducedMagnetic fields being inducedBy currents and a time-varyingBy currents and a time-varyingelectric fieldselectric fields

µµoo = permeability of free space (medium is diamagnetic) = permeability of free space (medium is diamagnetic)

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Electromagnetic waves

2

2

22

t

EB

t

EEEE

o

02

22

t

EE o

or,or,

For the electric field E,For the electric field E,

i.e. wave equation with vi.e. wave equation with v22 = 1/ = 1/µµoo

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Electromagnetic waves

02

22

t

BB o

Similarly for the magnetic fieldSimilarly for the magnetic field

i.e. wave equation with vi.e. wave equation with v22 = 1/ = 1/µµoo

In free space, In free space, = = oo = = oo ( ( = 1) = 1)

oo

c

1 c = 3.0 X 10c = 3.0 X 1088 m/s m/s

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Electromagnetic waves

In a dielectric medium, In a dielectric medium, = n = n2 2 and and = = oo = n = n2 2 oo

n

c

nv

ooo

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Electromagnetic waves: Phase relations

02

22

t

EE o

02

22

t

BB o

The solutions to the wave equations,The solutions to the wave equations,

can be plane waves,can be plane waves,

ti

o

tio

eBB

eEErk

rk