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16
Wave Motion
CHAPTER OUTLINE
16.1
Propagation of a Disturbance
16.2
The Traveling Wave Model
16.3
The Speed of Waves on Strings
16.5
Rate of Energy Transfer by Sinusoidal
Waves on Strings
16.6
The Linear Wave Equation
ANSWERS TO QUESTIONS
Q16.1
As the pulse moves down the string, the particles of the
string itself move side to side. Since the medium—here,
the string—moves perpendicular to the direction of wave
propagation, the wave is transverse by deF
nition.
*Q16.2
±rom
v
=
T
μ
,
we must increase the tension by a factor
of 4 to make
v
double. Answer (b).
*Q16.3
(i)
Look at the coefF
cients of the sine and cosine functions: 2, 4, 6, 8, 8, 7. The ranking is d
=
e > f > c > b > a.
(ii)
Look at the coefF
cients of
x
. Each is the wave number, 2
π
/
λ
,
so the smallest
k
goes with the
largest wavelength. The ranking is d > a
=
b
=
c > e > f.
(iii)
Look at the coefF
cients of
t
. The absolute value of each is the angular frequency
ω
=
2
f
. The
ranking is f > e > a
=
b
=
c
=
d.
(iv)
Period is the reciprocal of frequency, so the ranking is the reverse of that in part iii: d
=
c
=
b
=
a > e > f.
(v)
±rom
v
=
f
λ
=
/
k
, we compute the absolute value of the ratio of the coefF
cient of
t
to the
coefF
cient of
x
in each case. ±rom a to f respectively the numerical speeds are 5, 5, 5, 7.5,
5, 4. The ranking is d > a
=
b
=
c
=
e > f.
Q16.4
To use a slinky to create a longitudinal wave, pull a few coils back and release. ±or a transverse
wave, jostle the end coil side to side.
*Q16.5
Answer (b). Wave speed is inversely proportional to the square root of linear density.
Q16.6
Yes, among other things it depends on. The particle speed is described by
v
y
,max
v
==
=
ωπ
Af
A
A
2
2
.
Here
v
is the speed of the wave.
Q16.7
Each element of the rope must support the weight of the rope below it. The tension increases with
height. (It increases linearly, if the rope does not stretch.) Then the wave speed
v
=
T
increases
with height.
*Q16.8
(i)
Answer (a). Higher tension makes wave speed higher.
(ii)
Answer (b).
Greater linear density makes the wave move more slowly.
383
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Chapter 16
*Q16.9
(i)
Answer a. As the wave passes from the massive string to the less massive string, the wave
speed will increase according to
v
=
T
μ
.
(ii)
Answer c. The frequency will remain unchanged. However often crests come up to the
boundary they leave the boundary.
(iii)
Answer a. Since
v
=
f
λ
,
the wavelength must increase.
Q16.10
Since the frequency is 3 cycles per second, the period is
1
3
second = 333 ms.
Q16.11
Let
Δ
tt t
sp
=−
represent the difference in arrival times of the two waves at a station at distance
dt
t
ss
pp
==
vv
from the hypocenter. Then
⎛
⎝
⎜
⎞
⎠
⎟
−
Δ
11
1
. Knowing the distance from the
F
rst station places the hypocenter on a sphere around it. A measurement from a second sta
tion limits it to another sphere, which intersects with the F
rst in a circle. Data from a third
noncollinear station will generally limit the possibilities to a point.
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This note was uploaded on 01/28/2011 for the course PHYS 011 taught by Professor Nianlin during the Fall '08 term at HKUST.
 Fall '08
 NianLin
 Physics, Energy

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