PHY4604
R. D. Field
Department of Physics
Chapter1_4.doc
University of Florida
Re(
Ψ
)
A
Ψ
= Ae
i(kx
ω
t)
Im(
Ψ
)
φ
= kx
ω
t
t
Im(
Ψ
) = Asin(kx
ω
t)
A
A
Crest
Trough
A
Ψ
(0,t) = Ae
i
ω
t
Distance r
φ
= kr
ω
t
x = 0
φ
=
ω
t
A
x = r
Ψ
(r,t) = Ae
i(
kr

ω
t)
Representing Waves as Complex Numbers
We can use complex numbers to represent traveling waves.
If we let
)
(
t
kx
i
Ae
ω
−
=
Ψ
then
)
cos(
)
Re(
t
kx
A
ω
−
=
Ψ
is a traveling plane wave with wave number
k = 2
π
/
λ
, “angular” frequency
ω
= 2
π
f
, and amplitude
A
.
The intensity,
I
, is proportional to
A
2
.
∗
ΨΨ
=
Ψ
=
A
2
2
Ψ
=
∝
A
I
PhaseShift Due to a Path Length Difference
Consider two traveling wave that are in phase at their source, but
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 Spring '07
 FIELDS
 mechanics, Complex Numbers, path length difference, maximal destructive interference, maximal constructive interference

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