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Wave Equations
8/20/2019 08 Wave Equations
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UCF Wave Equations (I)
likewise
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UCF Wave Equations (II)
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UCF Scalar Wave Equation
equationescalar wav1D 01
havewe,0and 0oronly,zondependsthatAssume
equation.wave3DaisThis.1
where
01
havewe,ororeitherrepresentLet
equation.escalar wavsatisfyshould field electricof componenteachThen
s,coordinaterrectangulaIn
0:equationector waveConsider v
2
2
22
2
2
2
2222
2
22
t u
v zu
y
u
x
uu
v
uv
u
u
t
p
p
p
z y x
z z y y x
EEE
EEE aaax
E
E E
z
0 ,0
y x
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UCF General Solution to 1D Scalar Wave Equation
solution.aalsois)/( provealsocanweLikewise,
,11
,
1
.1and 1
havewe,Let
:solutionais)(Prove
).()(),(:issolutiongeneralthe
01
:equationescalar wav1DFor
2
2
2
2
2
2
22
2
2
2
22
2
p
p p
p
p p
p
p p
p
v zt g
u
t
u
t
uu
v z
u
v z
u
u
t
u
t
uu
v z
u
z
u
t v zv
zt
v
zt f
v zt g
v zt f t zu
t
u
v z
u
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UCF Physical Understanding of General Solution
)(Plot pv
zt f
z
)( pv
z f
)(
)(
p
p
p
vt v z f
v
zt f
t v z p10 .itywith velocdirectioning propagatinwaveaforstands :)( p
p
v zv
zt f
“backward wave”, or “negative wave”, or “reflected wave”
.itywith velocdirectioning propagatinwaveaforstands :)( p p
v zv
zt g
“forward wave”, or “positive wave”, or “incident wave”Likewise:
lightof yvelocitm/s10997.2 111 8
00 c
cv
r r r r
p
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UCF Sinusoidal Case
)cos()cos(
])(cos[])(cos[
),(),(),(
B A
B
p
A
p
kzt Bkzt A
v zt B
v zt A
t zut zut zu
frequency.spatialorrwavenumbeisfrequency,angulariswhere pv
k
In general, we can use plane waves to expand any arbitrary wave which
is the Fourier Transform in both time and spatial domain.
8/20/2019 08 Wave Equations
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UCF Period and Wavelength
A
-A
)cos(),(Plot kzt At zu
k k T T T
2
,2 :lengthwave
2
,2 : period
z0
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UCF Electromagnetic Spectrum
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UCF Traveling Wave and Standing Wave
)cos()cos(
),(),(),(
B A kzt Bkzt A
t zut zut zu
wave.travelingais),(wavetotalthe
0,isorif Orwave.travelingais),(or),(Each
t zu
B At zut zu
wave.standingais),(wavetotal,If t zu B A
wave.standing-travelingacalled is),(Otherwise, t zu
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UCF Standing Wave (1)
A2
2
A2
0 2/ 2/
When , 0 and 0, ( , ) [cos( ) cos( )]
2Asin( )sin( )
A B B u z t A t kz t kz
t kz
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UCF Standing Wave (2)
z
))sin(Asin(2),(For kzt t zu
0 2/ 2/
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UCF
Phasor Form Representation
)Re()Re
)cos()cos(
),(),(),(
t j jkz jt j jkz j
B A
ee Beee(Ae
kzt Bkzt A
t zut zut zu
B A
jkz jkz j
jkz jkz j
eU e BeU
eU e AeU
B
A
0
0
(propagating in +z direction)
(propagating in -z direction)
z
0 ,0
y x