Ph12a Solution Set 3
October 21, 2009
2.30
(a)
Choosing the origin of time at the center of Δ
t
, the sine terms all vanish
(see eqn (52) on p. 66, with
F
(
t
) an even function).
(b)
B
0
=
1
T
1
R
T
1
/
2

T
1
/
2
f
(
t
)
dt
=
Δ
t
T
1
B
n
=
2
T
1
R
T
1
/
2

T
1
/
2
f
(
t
)cos
2
πnt
T
1
=
2
T
1
R
Δ
t/
2

Δ
t/
2
cos
2
πnt
T
1
=
2
πn
sin
πn
Δ
t
T
1
(c)
For Δ
t << T
1
, one gets approximately
B
n
=
2
πn
πn
Δ
t
T
1
=
2Δ
t
T
1
for small
n
,
which is independent of
n
.
(d)
The formula for
B
n
derived in (b) can be expressed in the form
B
n
= 2
ν
1
Δ
t
sin(
nπν
1
Δ
t
)
nπν
1
Δ
t
Pretending for a moment that
n
is a continuous variable, we can produce
the plot shown in Figure 1. Precisely which points on this plot correspond
to integer values of
n
, depends on the value of
ν
1
Δ
t
, so we can’t mark
those points on the plot. However, we can note that there must be many
of those points within a unit interval of the
x
axis of the plot, because
ν
1
Δ
t
= Δ
t/T
1
<<
1.
(e)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 DUDKO
 mechanics, Trigraph, 1 m, Ford Ka, 1 2 2 2 2 M, 2 1 2 2 M

Click to edit the document details