Wave Packets+Solution of EM Wave Eq
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Transcript of Wave Packets+Solution of EM Wave Eq
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Superposition of Waves & Wave packets
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y(t) = Sin t 0 < t < 200
Superposition of waves and wave packet formation
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y(t) = [Sin t + Sin (1.08 t)]/2 0 < t < 200
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y(t) = [Sin t + Sin(1.04 t) + Sin (1.08 t)]/3
0 < t < 400
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y(t) = [Sin t + Sin(1.02 t) + Sin (1.04 t)
+ Sin(1.06 t) + Sin (1.08 t)]/5
0 < t < 400
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y(t) = [Sin t + Sin(1.01 t) + Sin (1.02 t)
+ Sin(1.03 t) + Sin (1.04 t) + Sin (1.05 t)
+ Sin (1.06 t) + Sin (1.07 t)+ Sin (1.08 t)]/9
0 < t < 800
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y(t) = [Sin t + Sin(1.01 t) + Sin (1.02 t)
+ Sin(1.03 t) + Sin (1.04 t) + Sin (1.05 t)
+ Sin (1.06 t) + Sin (1.07 t)+ Sin (1.08 t)]/9
0 < t < 400
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Wave Equations: EM Waves
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Electromagnetic waves
for E field
for B field
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In general,electromagnetic waves
2
2
2
2 1tc
Where represents E or Bor their components
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#A plane wavesatisfies waveequation in Cartesian coordinates
#Aspherical wavesatisfies wave
equation inspherical polarcoordinates
#
A cylindrical wavesatisfies waveequation in cylindrical coordinates
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Solution of 3D wave equation
In Cartesian coordinates
2
2
22
2
2
2
2
22
1 tczyx
Separation of variables
)()()()(),,,( tTzZyYxXtzyx
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Substituting forwe obtain
2
2
22
2
2
2
2
211111
t
T
Tcz
Z
Zy
Y
Yx
X
XVariables are separated out
Each variable-term independentAnd must be a constant
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So we may write
2
2
22
2
2
2
2
22
2
2
1;1
;1;1
t
T
Tk
z
Z
Z
ky
Y
Yk
x
X
X
z
yx
where we use222222
kkkkc zyx
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Solutions are then
tizik
yikxik
etTezZ
eyYexX
z
yx
)(;)(
;)(;)(
Total Solution is
)()()()(),,,( tTzZyYxXtzyx
)]([ zkykxkti zyxAe
].[ rktiAe
plane wave
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Traveling 3D plane wave
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spherical waves
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spherical waves
2
2
222
2
222
2
2 1sin
cossin12
rrrrrr
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rr
rrrrr
2
22
22 12
2
2
2
2
2
11
tcrr
rr
Alternatively
The wave equation becomes
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2
1
r
u
r
u
rr
Put r
rur
)()(
Then ur
urr
r
2
ru
rur
ru
ur
urrr
rr
2
2
2 Hence
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2
22
2
11
r
u
rrr
rr
Therefore
Wave equation transforms to
2
2
22
2 111tu
rcru
r
2
2
22
2 1tu
cru
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)()(),( tTrRtru
22
2
22
2
111 ktT
TcrR
R
kcetTerR tiikr
)(;)(
Which follows that
Separation of variables
)()(
krtieru
Solutions are
Total solution is
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)(1)(krti
er
r
)()( 11)(krtikrti
e
r
e
r
r
outgoingwaves
incomingwaves
Final form of solution
General solution
spherical wave
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Cylindrical waves
2
2
2
2
22
2
2 11
zrrrr
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with angular and azimuthal symmetry, theLaplacian simplifies and the wave equation
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The solutions are Bessel functions.For large r, they are approximated as
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A plane wave satisfies one-dimensionalwave equation in Cartesian coordinates
The position vector must remainperpendicular to the given plane
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The wave then satisfies the generalizationof the one-dimension wave equation
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Plane EM waves in vacuum
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Wave vector k is perpendicular to E
Wave vectorkis perpendicular toB
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B is perpendicular to E
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B, kandE make a right handed
Cartesian co-ordinate system
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Plane EM waves in vacuum