This preview shows pages 1–2. Sign up to view the full content.
Homework 7 Solution
1. [3pt] We derived
a
0
and
a
k
from the following equation in the class. Find
b
k
by using
the cosine and sine laws and the facts that
∫
T
0
cos(
αt
)
dt
=
∫
T
0
sin(
βt
)
dt
= 0. Note that
w
0
= 2
π/T
.
f
(
t
) =
a
0
+
∞
∑
k
=1
[
a
k
cos(
kw
0
t
) +
b
k
sin(
kw
0
t
)]
Hint: Begin by multiplying ‘
sin(
mw
0
t
)
’ to both sides of the equation and integrate over
(0, T).
Q1:
Multiply sin(
mw
0
t
) to both sides of the equation and integrate over(0, T):
left
=
±
T
0
f
(
t
) sin(
mw
0
t
)
dt
right
=
±
T
0
a
0
sin(
mw
0
t
)
dt
+
∞
∑
k
=1
[
a
k
±
T
0
cos(
kw
0
t
) sin(
mw
0
t
)
dt
+
b
k
±
T
0
sin(
kw
0
t
) sin(
mw
0
)
dt
]
Considering the right side
The ﬁrst term:
±
T
0
a
0
sin(
mw
0
t
)
dt
= 0
The second term:
a
k
±
T
0
cos(
kw
0
t
) sin(
mw
0
t
)
dt
=
1
2
[
a
k
±
T
0
sin(
kw
0
t
+
nw
0
t
)+
a
k
±
T
0
sin(
mw
0
t
−
kw
0
t
)
dt
] = 0
The third term:
for any
k
̸
=
m
b
k
±
T
0
sin(
kw
0
t
) sin(
mw
0
t
)
dt
=
1
2
[
b
k
±
T
0
cos(
kw
0
t
−
nw
0
t
)
−
b
k
±
T
0
cos(
mw
0
t
+
kw
0
t
)
dt
] = 0
for
k
=
m
b
k
±
T
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.
This note was uploaded on 11/14/2011 for the course EAME 250 taught by Professor Lee during the Spring '11 term at Case Western.
 Spring '11
 LEE

Click to edit the document details