For solutions to the 3-Dimensional wave equation, usecomplex notation
)(01ˆ
trkieEaE
)(02ˆ trkieBaB
frequencyangular
vector waveˆˆ
constantscomplex are ,
ectorposition vCartesian ),,(
rsunit vecto are ˆ,ˆ
00
21
nkk
BE
zyxr
aa
where
Before we go further, let’s review complex numbers
Argand Diagram
imaginary
real
y
x
x+iy
Complex number
zr
iyxz Complex conjugate iyxz
1ix = real part of zy = imaginary part of z
In polar coordinates sin cos ryrx
so )sin(cos iriyxz
The Euler Formula
sincos iei
implies sincos irrrez i
r = magnitude of zθ = phase angle of z
Re(z) = real part of z = rcosθIm(z) = imaginary part of z = r sinθ
Since exponentials are so easy to integrate and differentiate,it is convenient to describe waves as
)(),( tkziAetz Where A is a real constant
To get the physically meaningful quantity, which must be a real number,one solves the wave equation and then takes the REAL part of thesolution.
This is OK, since the wave equation is linear, so that the real part ofΨ and its imaginary part are each separately solutions.
So for example you can write
ctorcomplex ve a is where
Re
0Re
00
)(0
imaginaryal
tkzi
EiEE
eEE
then
€
rE = Re
r E 0
real + ir E 0
imaginary( ) cos(kz −ωt) − isin(kz −ωt)( )[ ]
=r E 0
real cos(kz −ωt) +r E 0
imaginary sin(kz −ωt)
vector vector
Solution to the wave equations:
)(01ˆ
trkieEaE
)(02ˆ trkieBaB
scalarscomplex are , 00 BE
nkk
vector wave
k
n
The waves travel in direction or
surfaces of constant phase travel with time in direction k
)(01ˆ
trkieEaE
)(02ˆ trkieBaB
These must also satisfy Maxwell’s Equations
z
E
y
E
x
E E zyx
€
∇⋅ ˆ a 1E0ei(
r k ⋅
r r −ωt )
[ ] = 0
Recall the definition of the divergence:
0E
)(01)ˆ( trki
xxx iekEa
x
E
)(01)ˆ( trki
xx eEaE
0E
0ˆ 01 Eaki
So
“Can show” the other equations are
0ˆ 01 Eaki
0ˆ 02 Baki
)(01ˆ
trkieEaE
)(02ˆ trkieBaB
0201 ˆˆ Bac
iEaki
0102 ˆˆ Ea
c
iBaki
ktransverse
BaEa
akak
n,propagatio ofdirection the to are
) is(that ˆ and ) is(that ˆboth
0ˆ and 0ˆ Since
21
21
0ˆ 01 Eaki
0ˆ 02 Baki
)(01ˆ
trkieEaE
)(02ˆ trkieBaB
0201 ˆˆ Bac
iEaki
0102 ˆˆ Ea
c
iBaki
00 Bkc
E
00 Ekc
B
Also
0
2
0 Ekc
E
222 kc
Require k>0 and ω>0 ω = c k
Hence, E0=B0
0ˆ 01 Eaki
0ˆ 02 Baki
)(01ˆ
trkieEaE
)(02ˆ trkieBaB
0201 ˆˆ Bac
iEaki
€
ir k × ˆ a 2B0 = −
iω
cˆ a 1E0
othereach lar toperpendicu bemust
) is(that ˆ and ) is(that ˆ
ˆ opposite is 0ˆ and
ˆdirection in is 0ˆ Since
21
12
21
BaEa
aak
aak
othereach lar toperpendicu always
and n,propagatio ofdirection theto
larperpendicu always are and
SO
BE
Qualitative Picture: For “one” wave with one λ
In real situations, one wants to consider the superposition of many waveslike this – and the more general case where the direction of E(and hence B) is random as the wave propagates.
Phase Velocity v. Group Velocity
The speed at which the sine moves is the phase velocity
ckphase
v
The group velocity is
kg
v
This is usually discussed when you have several waves superimposed,which make a modulated wave: the modulation envelope travels with the group velocity
In a dispersive medium ω=ω(k) so ynecessaril vkg
However, in a vacuum, vgroup= c
The Radiation Spectrum
The spectrum depends on the time variation of the electric field (or, equivalently, the magnetic field)
It is impossible to know what the spectrum is, if the electric field is only specified at a single instant of time. One needs to record the electric field for some sufficiently long time.
The spectrum (energy as a function of frequency) is related to the E-field (as a function of frequency) through the Poynting Vector.
The E-field (as a function of frequency) is related to the E-field (as a function of time) through the Fourier Transform
dtetEE ti
)(
2
1)(
deEtE ti)(2
1)(
Likewise,
ω = angular frequency
Fourier Transforms
dxexFf ix 2)()(
A function’s Fourier Transform is a specification of the amplitudes and phases of sinusoidals, which, when added together, reproduce the function
Given a function F(x)The Fourier Transform of F(x) is f(σ)
dxefxF ix 2)()(
see Bracewell’s book:FT and Its Applications
The inverse transform is
note change in sign
Not all functions have Fourier Transforms.
F.T. sometimes called the “power spectrum” e.g. search for periods in a variable star
Visualizing the F.T.
)()()( xiFxFxF IR
xixeix sincos
dxxxFdxxxFi
dxxxFidxxxFf
IR
IR
)2sin()()2sin()(
)2cos()()2cos()()(
Suppose you have a complex function:
Recall Euler’s formula:
FT(F(x)) =
Notes:When F(x) is real (FI=0) the fourier transform f(σ) can still be complex.
For fixed σ, these integrals involve multiplying F by a sine (or cosine) with period 1/σ and summing the area underneath the result.Changing the frequency of the sines and cosines and repeating the process gives f(σ) at a second value of σ, and so on.
Some examples
22
for 1
2 and
2for 0)(
x
xxxB
(1) F.T. of box function
€
b(σ ) = B(x)e2πixσ dx−∞
+∞
∫
= e2πixσ dx−ω
2
ω2
∫
=e2πixσ
2πixσ
⎡
⎣ ⎢
⎤
⎦ ⎥
−ω2
ω2
=1
2πixσe2πi(ω
2 )σ − e−2πi(ω2 )σ
[ ]
= ωsin(πωσ )
πωσ= ω sinc(πωσ )
“Ringing” -- sharp discontinuity ripples in spectrum
When ω is large, the F.T. is narrow: first zero at
1
other zeros at
1
(2) Gaussian
F.T. of gaussian is a gaussian with narrower width
22 /1)(
xexG
222
)( egFT
Dispersion of G(x) β
Dispersion of g(σ)
1
1
1
2
12
21
)(
)()(
:function delta theof
ix
ix
ix
e
dxxxe
dxexxf
FT
Amplitude of F.T. of delta function = 1 (constant with sigma)Phase = 2πxiσ linear function of sigma
(4) )()()( 11 xxxxxF
)2cos(2
)(
1
22 11
x
eefFT ixix
So, cosine with wavelength1
1
x
transforms to delta functions at +/ x1
x
0-x1 +x1