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Homework Assignment #2 Solutions  1
ECE 473/TAM 413
Homework Assignment #2 Solutions
Note to graduate students taking the course for 4 hours of graduate credit (NB: only
graduate students can receive 4 hours of credit)
: For the additional 1 hour of credit, you are
required to write a paper (typically about 10 pages, double spaced) that discusses in some detail
any topic on acoustics for which the fundamentals of engineering acoustics are explicitly
described
. The paper needs to be based on 5 peerreviewed publications. The paper will be due
Monday, December 5, 2008. However, I must approve the topic and peerreviewed publications.
For the approval process, prepare a onepage outline (including 5 peerreviewed citations) for
submission
Monday, October 6, 2008
.
1. The tension in a string is provided by hanging a 3kg mass at one end; the other end is rigidly
fixed. The length of the string is 2.5 m and its mass is 50 g. Determine the phase speed of
waves on the string.
Tension in the string is T = F = mg = (3 kg)(9.81 m/s
2
) = 29.4 N
Mass per unit length is
!
L
=
m
L
=
0.05 kg
2.5 m
=
0.02 kg /m
Phase speed is
c
=
T
!
L
=
29.4 N
0.02 kg / m
=
38.3m /s
2. Represent the calculations to the correct number of significant figures:
(a) 32.4 x 41.4 = 1,340
(b) 32.1 x 41.43 = 1,340
3. Problem 2.3.1 in Kinsler et al. (show all work). Put your final answers in the form of a wave
equation.
(a) Assume the linear density varies with position, that is,
!
L
=
!
L
x
( )
.
The tension force on each element is given by:
df
y
=
!
!
x
T
!
y
!
x
"
#
$
%
&
’
dx
=
T
!
2
y
!
x
2
dx
(see Eq. 2.3.4)
Using Newton’s 2nd Law:
f
y
=
ma
=
m
!
2
y
!
t
2
, but the mass for an element is:
dm
=
!
L
x
( )
dx
for
some element at x, giving
df
y
=
dm
( )
a
=
!
L
x
( )
"
2
y
"
t
2
dx
.
Equating forces
T
!
2
y
!
x
2
dx
=
"
L
x
( )
!
2
y
!
t
2
dx
and rearranging yields:
!
2
y
!
t
2
=
T
"
L
x
( )
!
2
y
!
x
2
.
(b) Assume the string hangs vertically supported only at the upper end. Therefore, the tension is
given by the force of gravity and the mass of the string hanging below the position x for a dx
element at x. Thus, the tension at x is given by:
T
=
m x
( )
g
=
!
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 Fall '08
 OBRIAN

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