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6.003
Homework
1
Due
at
the
beginning
of
recitation
on
Wednesday,
February
10,
2010
.
Problems
1.
Independent
and
Dependent
Variables
Assume
that
the
height
of
a
water
wave
is
given
by
g
(
x
−
vt
) where
x
is
distance,
v
is
velocity,
and
t
is
time.
Assume
that
the
height
of
the
wave
is
a
sinusoidal
function
of
distance
at
each
instant
of
time.
Also
assume
that
the
positive
peaks
have
a
height
of
1 meter
(relative
to
the
average
water
level)
and
that
they
occur
at
integer
multiples
of
2 meters
when
the
time
t
= 3 seconds.
a.
Determine
an
expression
for
the
height
of
the
wave
h
(
x,t
) as
a
function
of
distance
x
and
time
t
if
the
wave
is
traveling
in
the
positive
x
direction
at
5 meters/second.
What
is
the
function
g
( ) for
this
case?
·
b.
Determine
an
expression
for
the
height
of
the
wave
h
(
x,t
) as
a
function
of
distance
x
and
time
t
if
the
wave
is
traveling
in
the
negative
x
direction
at
4 meters/second.
What
is
the
function
g
( ) for
this
case?
·
c.
Determine
the
speed
of
the
wave
if
successive
positive
peaks
at
x
= 1
.
3 meters
are
separated
by
0
.
75 seconds.
2.
Even
and
Odd
The
even
and
odd
parts
of
a
signal
x
[
n
] are
deﬁned
by
the
following:
x
e
[
−
n
] =
x
e
[
n
] (i.e.,
x
e
is
an
even
function
of
n
)
•
x
o
[
−
n
] =
−
x
o
[
n
] (i.e.,
x
o
is
an
odd
function
of
n
)
•
x
[
n
] =
x
e
[
n
] +
[
n
]
•
x
o
Let
x
represent
the
signal
whose
samples
are
given
by
x
[
n
] =
±
(
2
1
)
n
n
≥
0
.
0
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 Spring '11
 DennisM.Freeman

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