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15.82:
a)
,
cos
sin
sin
cos
SW
W
S
ωt
kx
ω
ωA
ωt,
kx
kA
t
y
x
y

=
=
∂
∂
∂
∂
and so the instantaneous
power is
).
2
sin(
)
2
sin(
4
1
)
cos
)(sin
cos
(sin
SW
SW
2
2
ωt
kx
ωk
FA
ωt
ωt
kx
kx
ωk
FA
P
=
=
b) The average value of
P
is proportional to the average value of
),
2
sin(
t
ϖ
and the
average of the sine function is zero;
.
0
av
=
P
c) The waveform is the solid line, and the
power is the dashed line. At time
0
and
0
,
2
=
=
=
P
y
ω
π
t
and the graphs coincide. d)
When the standing wave is at its maximum displacement at all points, all of the energy is
potential, and is concentrated at the places where the slope is steepest (the nodes). When
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