proportional to the (second) spatial derivative of the wave function (recall the wave equa
tion!), we may write the condition on the acceleration as
∂ψ
/
∂
x

x
=0

=
∂ψ
/
∂
x

x
=0
+
. (The
second spatial derivative being continuous guarantees that the first derivative is continuous
and di
ff
erentiable.)
We choose to represent the waves in complex representation.
That is, a harmonic wave
traveling to the right would be given by
e
i
(
ω
t

kx
)
and wave traveling to the left would be
e
i
(
ω
t
+
kx
)
. What is the meaning of a complex valued wave? Recall that we may superpose
two solutions of the wave equation, and the result would still be a solution, because of the
superposition principle and the linearity of the wave equation. Therefore, the complex valued
wave function is a superposition of cosine and sine waves:
e
i
(
ω
t
∓
kx
)
= cos(
ω
t
∓
kx
) +
i
sin(
ω
t
∓
kx
)
.
Because of the superposition principle, we may be interested only in the cosine wave, say,
work with the full complex wave, but take only the real part for the cosine wave. This way,
we may benefit form all the niceties of complex analysis, yet retain the “realness” of the
physical field.
We now write the incident wave as
ψ
I
=
A
I
e
i
(
ω
t

k
1
x
)
, the reflected wave as
ψ
R
=
A
R
e
i
(
ω
t
+
k
1
x
)
,
and the transmitted wave as
ψ
T
=
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 Spring '09
 LIORBURKO
 Physics, Acceleration

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