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18Superposition and Standing Waves
CHAPTER OUTLINE
18.1 Superposition and Interference18.2 Standing Waves18.3 Standing Waves in a String Fixed
at Both Ends18.4 Resonance18.5 Standing Waves in Air Columns18.6 Standing Waves in Rod and
Membranes18.7 Beats: Interference in Time18.8 Nonsinusoidal Wave Patterns
ANSWERS TO QUESTIONS
Q18.1 No. Waves with all waveforms interfere. Waves with other wave shapes are also trains of disturbance that add together when waves from different sources move through the same medium at the same time.
*Q18.2 (i) If the end is fi xed, there is inversion of the pulse upon refl ection. Thus, when they meet, they cancel and the amplitude is zero. Answer (d).
(ii) If the end is free, there is no inversion on refl ection. When they meet, the amplitude is 2 2 0 1 0 2A = ( ) =. . mm . Answer (b).
*Q18.3 In the starting situation, the waves interfere constructively. When the sliding section is moved out by 0.1 m, the wave going through it has an extra path length of 0.2 m = λ �4, to show partial interference. When the slide has come out 0.2 m from the starting confi guration, the extra path length is 0.4 m = λ �2, for destructive interference. Another 0.1 m and we are at r
2 − r
1 = 3λ �4
for partial interference as before. At last, another equal step of sliding and one wave travels one wavelength farther to interfere constructively. The ranking is then d > a = c > b.
Q18.4 No. The total energy of the pair of waves remains the same. Energy missing from zones of destructive interference appears in zones of constructive interference.
*Q18.5 Answer (c). The two waves must have slightly different amplitudes at P because of their different distances, so they cannot cancel each other exactly.
Q18.6 Damping, and non–linear effects in the vibration turn the energy of vibration into internal energy.
*Q18.7 The strings have different linear densities and are stretched to different tensions, so they carry string waves with different speeds and vibrate with different fundamental frequencies. They are all equally long, so the string waves have equal wavelengths. They all radiate sound into air, where the sound moves with the same speed for different sound wavelengths. The answer is (b) and (e).
*Q18.8 The fundamental frequency is described by fL1 2
= v, where v =
⎛⎝⎜
⎞⎠⎟
T
μ
1 2
(i) If L is doubled, then f L11� − will be reduced by a factor
1
2. Answer (f ).
(ii) If μ is doubled, then f11 2� μ− will be reduced by a factor
1
2. Answer (e).
(iii) If T is doubled, then f T1 � will increase by a factor of 2. Answer (c).
473
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*Q18.9 Answer (d). The energy has not disappeared, but is still carried by the wave pulses. Each par-ticle of the string still has kinetic energy. This is similar to the motion of a simple pendulum. The pendulum does not stop at its equilibrium position during oscillation—likewise the particles of the string do not stop at the equilibrium position of the string when these two waves superimpose.
*Q18.10 The resultant amplitude is greater than either individual amplitude, wherever the two waves are nearly enough in phase that 2Acos(φ �2) is greater than A. This condition is satisfi ed whenever the absolute value of the phase difference φ between the two waves is less than 120°. Answer (d).
Q18.11 What is needed is a tuning fork—or other pure-tone generator—of the desired frequency. Strike the tuning fork and pluck the corresponding string on the piano at the same time. If they are pre-cisely in tune, you will hear a single pitch with no amplitude modulation. If the two pitches are a bit off, you will hear beats. As they vibrate, retune the piano string until the beat frequency goes to zero.
*Q18.12 The bow string is pulled away from equilibrium and released, similar to the way that a guitar string is pulled and released when it is plucked. Thus, standing waves will be excited in the bow string. If the arrow leaves from the exact center of the string, then a series of odd harmonics will be excited. Even harmonies will not be excited because they have a node at the point where the string exhibits its maximum displacement. Answer (c).
*Q18.13 (a) The tuning fork hits the paper repetitively to make a sound like a buzzer, and the paper effi ciently moves the surrounding air. The tuning fork will vibrate audibly for a shorter time.
(b) Instead of just radiating sound very softly into the surrounding air, the tuning fork makes the chalkboard vibrate. With its large area this stiff sounding board radiates sound into the air with higher power. So it drains away the fork’s energy of vibration faster and the fork stops vibrating sooner.
(c) The tuning fork in resonance makes the column of air vibrate, especially at the antinode of displacement at the top of the tube. Its area is larger than that of the fork tines, so it radiates louder sound into the environment. The tuning fork will not vibrate for so long.
(d) The tuning fork ordinarily pushes air to the right on one side and simultaneously pushes air to the left a couple of centimeters away, on the far side of its other time. Its net disturbance for sound radiation is small. The slot in the cardboard admits the ‘back wave’ from the far side of the fork and keeps much of it from interfering destructively with the sound radiated by the tine in front. Thus the sound radiated in front of the screen can become noticeably louder. The fork will vibrate for a shorter time.
All four of these processes exemplify conservation of energy, as the energy of vibration of the fork is transferred faster into energy of vibration of the air. The reduction in the time of audible fork vibration is easy to observe in case (a), but may be challenging to observe in the other cases.
Q18.14 Walking makes the person’s hand vibrate a little. If the frequency of this motion is equal to the natural frequency of coffee sloshing from side to side in the cup, then a large–amplitude vibra-tion of the coffee will build up in resonance. To get off resonance and back to the normal case of a small-amplitude disturbance producing a small–amplitude result, the person can walk faster, walk slower, or get a larger or smaller cup. Alternatively, even at resonance he can reduce the amplitude by adding damping, as by stirring high–fi ber quick–cooking oatmeal into the hot coffee. You do not need a cover on your cup.
474 Chapter 18
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*Q18.15 The tape will reduce the frequency of the fork, leaving the string frequency unchanged. If the bit of tape is small, the fork must have started with a frequency 4 Hz below that of the string, to end up with a frequency 5 Hz below that of the string. The string frequency is 262 + 4 = 266 Hz, answer (d).
Q18.16 Beats. The propellers are rotating at slightly different frequencies.
SOLUTIONS TO PROBLEMS
Section 18.1 Superposition and Interference
P18.1 y y y x t x= + = −( ) + −1 2 3 00 4 00 1 60 4 00 5 0 2. cos . . . sin . .000t( ) evaluated at the given x values.
(a) x = 1.00, t = 1.00 y = ( ) + +( ) = −3 00 2 40 4 00 3 00 1 6. cos . . sin . .rad rad 55 cm
(b) x = 1.00, t = 0.500 y = +( ) + +( ) = −3 00 3 20 4 00 4 00 6. cos . . sin . .rad rad 002 cm
(c) x = 0.500, t = 0 y = +( ) + +( ) =3 00 2 00 4 00 2 50 1 1. cos . . sin . .rad rad 55 cm
P18.2
P18.3 (a) y f x t1 = −( )v , so wave 1 travels in the +x direction
y f x t2 = +( )v , so wave 2 travels in the −x direction
(b) To cancel, y y1 2 0+ = : 5
3 4 2
5
3 4 6 22 2x t x t−( ) += +
+ −( ) +
3 4 3 4 6
3 4 3 4 6
2 2x t x t
x t x t
−( ) = + −( )− = ± + −( )
from the positive root, 8 6t = t = 0 750. s
(at t = 0 750. s, the waves cancel everywhere)
(c) from the negative root, 6 6x = x = 1 00. m
(at x = 1.00 m, the waves cancel always)
Superposition and Standing Waves 475
FIG. P18.2
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P18.4 Suppose the waves are sinusoidal.
The sum is 4 00 4 00 90 0. sin . sin .cm cm( ) −( ) + ( ) − +kx t kx tω ω °(( )
2 4 00 45 0 45 0. sin . cos .cm( ) − +( )kx tω ° °
So the amplitude is 8 00 45 0 5 66. cos . .cm cm( ) =° .
P18.5 The resultant wave function has the form
y A kx t= ⎛⎝
⎞⎠ − +⎛
⎝⎞⎠2
2 20 cos sinφ ω φ
(a) A A= ⎛⎝
⎞⎠ = ( ) −⎡
⎣⎢⎤⎦⎥=2
22 5 00
4
29 240 cos . cos .
φ πm
(b) f = = =ωπ
ππ2
1 200
2600 Hz
P18.6 (a) Δx = + − = − =9 00 4 00 3 00 13 3 00 0 606. . . . . m
The wavelength is λ = = =vf
343
3001 14
m s
Hzm.
Thus, Δx
λ= =
0 606
1 140 530
.
..
of a wave,
or Δφ π= ( ) =2 0 530 3 33. . rad
(b) For destructive interference, we want Δ Δx
fx
λ= =0 500.
v
where Δx is a constant in this set up. fx
= =( )
=v
2
343
2 0 606283
Δ .Hz
P18.7 We suppose the man’s ears are at the same level as the lower speaker. Sound from the upper
speaker is delayed by traveling the extra distance L d L2 2+ − .
He hears a minimum when this is 2 1
2
n −( )λ with n = 1 2 3, , , …
Then,
L d Ln
f2 2 1 2+ − =
−( )v
L dn
fL
L dn
fL
n
2 2
2 22 2
22
1 2
1 2 2 1 2
+ =−( ) +
+ =−( ) + +
−(
v
v ))
=− −( )
−( ) =
v
vv
L
f
Ld n f
n fn
2 2 2 21 2
2 1 21 2 3, , ,…
This will give us the answer to (b). The path difference starts from nearly zero when the man is very far away and increases to d when L = 0. The number of minima he hears is the greatest
integer solution to dn
f≥
−( )1 2 v
n = greatest integer ≤ +df
v1
2
continued on next page
476 Chapter 18
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(a) df
v+ =
( )( ) + =1
2
4 00 200
330
1
22 92
..
m s
m s
He hears two minima.
(b) With n = 1,
L
d f
f=
− ( )( ) =
( ) − ( )2 2 2 2 2 21 2
2 1 2
4 00 330 4 2vv
. m m s 000
330 200
9 28
2s
m s s
m
( )( )
=L .
With n = 2,
Ld f
f=
− ( )( ) =
2 2 2 23 2
2 3 21 99
vv
. m
P18.8 Suppose the man’s ears are at the same level as the lower speaker. Sound from the upper speaker
is delayed by traveling the extra distance Δr L d L= + −2 2 .
He hears a minimum when Δr n= −( )⎛⎝⎜
⎞⎠⎟2 1
2
λ with n = 1 2 3, , ,…
Then,
L d L nf
2 2 1
2+ − = −⎛
⎝⎞⎠⎛⎝⎜
⎞⎠⎟
v
L d nf
L
L d nf
2 2
2 22
1
2
1
2
+ = −⎛⎝
⎞⎠⎛⎝⎜
⎞⎠⎟+
+ = −⎛⎝
⎞⎠
⎛⎝
v
v⎜⎜
⎞⎠⎟
+ −⎛⎝
⎞⎠⎛⎝⎜
⎞⎠⎟
+2
221
2n
fL L
v
d nf
nf
L22 2
1
22
1
2− −⎛⎝
⎞⎠
⎛⎝⎜
⎞⎠⎟
= −⎛⎝
⎞⎠⎛⎝⎜
⎞⎠⎟
v v (1)
Equation 1 gives the distances from the lower speaker at which the man will hear a minimum. The path difference Δr starts from nearly zero when the man is very far away and increases to d when L = 0.
(a) The number of minima he hears is the greatest integer value for which L ≥ 0. This is the
same as the greatest integer solution to d nf
≥ −⎛⎝
⎞⎠⎛⎝⎜
⎞⎠⎟
1
2
v, or
number of minima heard greatest intege= nmax = rr ≤ df
v⎛⎝⎜
⎞⎠⎟ +
1
2
(b) From equation 1, the distances at which minima occur are given by
Ld n f
n fnn =
− −( ) ( )−( )( ) =
2 2 21 2
2 1 21 2
vv
where , ,…,, maxn
Superposition and Standing Waves 477
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P18.9 (a) φ1 20 0 5 00 32 0 2 00= ( )( ) − ( )(. . . .rad cm cm rad s s)) = 36 0. rad
φ1 25 0 5 00 40 0 2 00= ( )( ) − ( )(. . . .rad cm cm rad s s)) =
= = =
45 0
9 00 516 156
.
.
rad
radiansΔφ ° °
(b) Δφ = − − −[ ] = − +20 0 32 0 25 0 40 0 5 00 8 00. . . . . .x t x t x t
At t = 2.00 s, the requirement is
Δφ π= − + ( ) = +( )5 00 8 00 2 00 2 1. . .x n for any integer n.
For x < 3.20, − 5.00x + 16.0 is positive, so we have
− + = +( )
= − +( )5 00 16 0 2 1
3 202 1
5 00
. .
..
x n
xn
ππ
, or
The smallest positive value of x occurs for n = 2 and is
x = − +( )= − =3 20
4 1
5 003 20 0 058 4.
.. .
π π cm
*P18.10 (a) First we calculate the wavelength: λ = = =vf
344
21 516 0
m s
Hzm
..
Then we note that the path difference equals 9 00 1 001
2. .m m− = λ
Point A is one-half wavelength farther from one speaker than from the other. The waves from the two sources interfere destructively, so the receiver records a minimum in sound intensity.
(b) We choose the origin at the midpoint between the speakers. If the receiver is located at point (x, y), then we must solve:
x y x y+( ) + − −( ) + =5 00 5 001
22 2 2 2. . λ
Then, x y x y+( ) + = −( ) + +5 00 5 001
22 2 2 2. . λ
Square both sides and simplify to get: 20 04
5 002
2 2. .x x y− = −( ) +λ λ
Upon squaring again, this reduces to: 400 10 016 0
5 002 24
2 2 2 2x x x y− + = −( )..
.λ λ λ λ+
Substituting λ = 16 0. m, and reducing, 9 00 16 0 1442 2. .x y− =
or x y2 2
16 0 9 001
. .− =
The point should move along the hyperbola 9x2 − 16y2 = 144.
(c) Yes. Far from the origin the equation might as well be 9x2 − 16y2 = 0, so the point can move along the straight line through the origin with slope 0.75 or the straight line through the origin with slope −0.75.
478 Chapter 18
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Section 18.2 Standing Waves
P18.11 y x t A kx= ( ) ( ) ( ) =1 50 0 400 200 2 0. sin . cos sin cosm ωtt
Therefore, k = =20 400
πλ
. rad m λ π= =2
0 40015 7
..
rad mm
and ω π= 2 f so f = = =ωπ π2
200
231 8
rad s
radHz.
The speed of waves in the medium is v = = = = =λ λπ
π ωf f
k22
200
0 400500
rad s
rad mm s
.
P18.12 From y A kx t= 2 0 sin cosω we fi nd
∂∂
=y
xA k kx t2 0 cos cosω
∂∂
= −y
tA kx t2 0ω ωsin sin
∂∂
= −2
2 022
y
xA k kx tsin cosω
∂∂
= −2
2 022
y
tA kx tω ωsin cos
Substitution into the wave equation gives − = ⎛⎝
⎞⎠ −( )2
120
22 0
2A k kx t A kx tsin cos sin cosω ω ωv
This is satisfi ed, provided that v = ωk
. But this is true, because v = = =λ λπ
π ωf f
k22
P18.13 The facing speakers produce a standing wave in the space between them, with the spacing between nodes being
dfNN
m s
sm= = = ( ) =−
λ2 2
343
2 8000 2141
v.
If the speakers vibrate in phase, the point halfway between them is an antinode of pressure at a distance from either speaker of
1 25
0 625.
.m
2=
Then there is a node at 0 6250 214
20 518.
..− = m
a node at 0 518 0 214 0 303. . .m m m− =
a node at 0 303 0 214 0 089 1. . .m m m− =
a node at 0 518 0 214 0 732. . .m m m+ =
a node at 0 732 0 214 0 947. . .m m m+ =
and a node at 0 947 0 214 1 16. . .m m m+ = from either speaker.
Superposition and Standing Waves 479
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*P18.14 (a)
−4
4
02 4 6
x
t = 0y
−4
4
02 4 6
x
t = 20 msy
−4
4
02 4 6
x
t = 5 msy
−4
4
02 4 6
x
t = 10 msy
−4
4
02 4 6
x
t = 15 msy
(b) In any one picture, the distance from one positive-going zero crossing to the next is λ = 4 m.
(c) f = 50 Hz. The oscillation at any point starts to repeat after a period of 20 ms, and f = 1�T.
(d) 4 m. By comparison with the wave function y = (2A sin kx)cos ω t, we identify k = π �2, and
then compute λ = 2π �k.
(e) 50 Hz. By comparison with the wave function y = (2A sin kx)cos ω t, we identify ω = 2πf =100π.
P18.15 y x t1 3 00 0 600= +( )[ ]. sin .π cm; y x t2 3 00 0 600= −( )[ ]. sin .π cm
y y y x t x= + = ( ) ( ) + ( )1 2 3 00 0 600 3 00. sin cos . . sinπ π π ccos .
. sin cos .
0 600
6 00 0 60
ππ
t
y x
( )[ ]= ( ) ( )
cm
cm 00πt( )
(a) We can take cos .0 600 1πt( ) = to get the maximum y.
At x = 0 250. cm, ymax . sin . .= ( ) ( ) =6 00 0 250 4 24cm cmπ
(b) At x = 0 500. cm, ymax . sin . .= ( ) ( ) =6 00 0 500 6 00cm cmπ
(c) Now take cos .0 600 1πt( ) = − to get ymax:
At x = 1 50. cm, ymax . sin . .= ( ) ( ) −( ) =6 00 1 50 1 6 00cm cmπ
480 Chapter 18
continued on next page
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(d) The antinodes occur when xn= λ4
n =( )1 3 5, , ,…
But k = =2πλ
π so λ = 2 00. cm
and x1 40 500= =λ. cm as in (b)
x2
3
41 50= =λ. cm as in (c)
x3
5
42 50= =λ. cm
*P18.16 (a) The resultant wave is y A kx t= +⎛⎝
⎞⎠ −⎛
⎝⎞⎠2
2 2sin cos
φ ω φ
The oscillation of the sin(kx + φ �2) factor means that this wave shows alternating nodes and antinodes. It is a standing wave.
The nodes are located at kx n+ =φ π2
so xn
k k= −π φ
2
which means that each node is shifted φ2k
to the left by the phase difference between the traveling waves.
(b) The separation of nodes is Δx nk k
n
k k= +( ) −⎡⎣⎢
⎤⎦⎥− −⎡⎣⎢
⎤⎦⎥
12 2
π φ π φΔx
k= =π λ
2
The nodes are still separated by half a wavelength.
(c) As noted in part (a), the nodes are all shifted by the distance φ�2k to the left.
Section 18.3 Standing Waves in a String Fixed at Both Ends
P18.17 L = 30 0. m; μ = × −9 00 10 3. kg m; T = 20 0. N; fL1 2
= v
where v =⎛⎝⎜
⎞⎠⎟
=T
μ
1 2
47 1. m s
so f147 1
60 00 786= =.
.. Hz f f2 12 1 57= = . Hz
f f3 13 2 36= = . Hz f f4 14 3 14= = . Hz
P18.18 The tension in the string is T = ( )( ) =4 9 8 39 2kg m s N2. .
Its linear density is μ = = × = ×−
−m
L
8 101 6 10
33kg
5 mkg m.
and the wave speed on the string is v = =×
=−
T
μ39 2
10156 53
..
N
1.6 kg mm s
In its fundamental mode of vibration, we have λ = = ( ) =2 2 5 10L m m
Thus, f = = =vλ
156 5
1015 7
..
m s
mHz
Superposition and Standing Waves 481
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P18.19 (a) Let n be the number of nodes in the standing wave resulting from the 25.0-kg mass. Then
n + 1 is the number of nodes for the standing wave resulting from the 16.0-kg mass. For
standing waves, λ = 2L
n, and the frequency is f = v
λ
Thus, fn
L
Tn=2 μ
and also fn
L
Tn= + +1
21
μ
Thus, n
n
T
T
g
gn
n
+ = =( )( ) =
+
1 25 0
16 0
5
41
.
.
kg
kg
Therefore, 4n + 4 = 5n, or n = 4
Then, f =( )
( )( )=4
2 2 00
25 0 9 80
0 002 00.
. .
.m
kg m s
kg m
2
3350 Hz
(b) The largest mass will correspond to a standing wave of 1 loop
(n = 1) so 3501
2 2 00
9 80
0 002 00Hz
m
m s
kg m
2
=( )
( ).
.
.
m
yielding m = 400 kg
P18.20 For the whole string vibrating, dNN = =0 642
. mλ
; λ = 1 28. m
The speed of a pulse on the string is
v = = =fs
λ 3301
1 28 422. m m s
(a) When the string is stopped at the fret,
dNN = =2
30 64
2. m
λ ; λ = 0 853. m
f = = =v
λ422
0 853495
m s
mHz
.
(b) The light touch at a point one third of the way along the string damps out vibration in the two lowest vibration states of the string as a whole. The whole string vibrates
in its third resonance possibility: 3 0 64 32
dNN = =. mλ
λ = 0 427. m
f = = =v
λ422
0 4279
m s
m90 Hz
.
P18.21 dNN = 0 700. m = λ �2
λ
λ
=
= = =×( ) ( )−
1 40
3081 20 10 0 7003
.
. .
m
m sfT
v
(a) T = 163 N
(b) With one-third the distance between nodes, the frequency
is f3 3 220 660= ⋅ Hz = Hz
482 Chapter 18
FIG. P18.20(a)
FIG. P18.20(b)
FIG. P18.21
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P18.22 λGGf
= ( ) =2 0 350. mv
; λA AA
Lf
= =2v
L L Lf
fL L
f
fG A GG
AG G
G
A
− = −⎛⎝⎜
⎞⎠⎟
= −⎛⎝⎜
⎞⎠⎟=1 0 350. mm m( ) −⎛
⎝⎞⎠ =1
392
4400 038 2.
Thus, L LA G= − = − =0 038 2 0 350 0 038 2 0 312. . . .m m m m, or the fi nger should be placed
31 2. cm from the bridge .
Lf f
TA
A A
= =v2
1
2 μ; dL
dT
f TAA
=4 μ
; dL
L
dT
TA
A
= 1
2
dT
T
dL
LA
A
= =−( ) =2 2
0 600
3 823 84
.
.. %
cm
35.0 cm
P18.23 In the fundamental mode, the string above the rod has only two nodes, at A and B, with an anti-node halfway between A and B. Thus,
λ
θ2= =AB
L
cos or λ
θ= 2L
cos
Since the fundamental frequency is f, the wave speed inthis segment of string is
v = =λθ
fLf2
cos
Also,
v = = =T T
m AB
TL
mμ θcos
where T is the tension in this part of the string. Thus,
2Lf TL
mcos cosθ θ= or
4 2 2
2
L f TL
mcos cosθ θ=
and the mass of string above the rod is:
mT
Lf= cosθ
4 2 [1]
Now, consider the tension in the string. The light rod would rotate about point P if the string exerted any vertical force on it. Therefore, recalling Newton’s third law, the rod must exert only a horizontal force on the string. Consider a free-body diagram of the string segment in contact with the end of the rod.
F T Mg TMg
y∑ = − = ⇒ =sinsin
θθ
0
Then, from Equation [1], the mass of string above the rod is
mMg
Lf
Mg
Lf= ⎛⎝
⎞⎠ =
sin
cos
tanθθ
θ4 42 2
Superposition and Standing Waves 483
L
M
A
Bq
T
F
Mg
q
FIG. P18.23
13794_18_ch18_p473-496.indd 48313794_18_ch18_p473-496.indd 483 1/3/07 8:15:04 PM1/3/07 8:15:04 PM
P18.24 Let m = ρV represent the mass of the copper cylinder. The original tension in the wire is T1 = mg =
ρVg. The water exerts a buoyant force ρwater
Vg
2⎛⎝
⎞⎠ on the cylinder, to reduce the tension to
T VgV
g Vg2 2 2= − ⎛
⎝⎞⎠ = −⎛
⎝⎞⎠ρ ρ ρ ρ
waterwater
The speed of a wave on the string changes from T1
μ to T2
μ. The frequency changes from
fT
11 1 1= =vλ μ λ
to fT
22 1=μ λ
where we assume λ = 2L is constant.
Then
f
f
T
T2
1
2
1
2 8 92 1 00 2
8 92= =
−= −ρ ρ
ρwater . .
.
f2 3008 42
8 92291= =Hz Hz
.
.
P18.25 Comparing y = (0.002 m) sin ((π rad�m)x) cos ((100 π rad�s)t)
with y A kx t= 2 sin cosω
we fi nd k = = −2 1πλ
π m , λ = 2 00. m, and ω π π= = −2 100 1f s : f = 50 0. Hz
(a) Then the distance between adjacent nodes is dNN m= =λ2
1 00.
and on the string are L
dNN
m
mloops= =3 00
1 003
.
.
For the speed we have v = = ( ) =−f λ 50 2 1001s m m s
(b) In the simplest standing wave vibration, d bNN m= =3 00
2.
λ, λb = 6 00. m
and
fba
b
= = =vλ
100
6 0016 7
m s
mHz
..
(c) In v00=
T
μ, if the tension increases to T Tc = 9 0 and the string does not stretch, the speed
increases to
v vc
T T= = = = ( ) =9
3 3 3 100 3000 00μ μ
m s m s
Then
λcc
af= = =−
v 300
506 001
m s
sm. d c
NN m= =λ2
3 00.
and one loop fi ts onto the string.
484 Chapter 18
13794_18_ch18_p473-496.indd 48413794_18_ch18_p473-496.indd 484 1/3/07 8:15:05 PM1/3/07 8:15:05 PM
Section 18.4 Resonance
P18.26 The wave speed is v = = ( )( ) =gd 9 80 36 1 18 8. . .m s m m s2
The bay has one end open and one closed. Its simplest resonance is with a node of horizontal veloc-ity, which is also an antinode of vertical displacement, at the head of the bay and an antinode of velocity, which is a node of displacement, at the mouth. The vibration of the water in the bay is like that in one half of the pond shown in Figure P18.27.
Then, dNA m= × =210 104
3 λ
and λ = ×840 103 m
Therefore, the period is Tf
= = = × = × =1 840 104 47 10 12
34λ
vm
18.8 m ss h 24. min
The natural frequency of the water sloshing in the bay agrees precisely with that of lunar excita-tion, so we identify the extra-high tides as amplifi ed by resonance.
P18.27 (a) The wave speed is v = =9 153 66
..
m
2.50 sm s
(b) From the fi gure, there are antinodes at both ends of the pond, so the distance between adjacent antinodes
is dAA m= =λ2
9 15.
and the wavelength is λ = 18 3. m
The frequency is then f = = =vλ
3 66
18 30 200
.
..
m s
mHz
We have assumed the wave speed is the same for all wavelengths.
P18.28 The distance between adjacent nodes is one-quarter of the circumference.
d dNN AA
cmcm= = = =λ
2
20 0
45 00
..
so
λ = 10 0. cm
and
f = = = =vλ
900
0 1009 000 9 00
m s
mHz kHz
..
The singer must match this frequency quite precisely for some interval of time to feed enough energy into the glass to crack it.
Superposition and Standing Waves 485
13794_18_ch18_p473-496.indd 48513794_18_ch18_p473-496.indd 485 1/3/07 8:15:06 PM1/3/07 8:15:06 PM
Section 18.5 Standing Waves in Air Columns
P18.29 (a) For the fundamental mode in a closed pipe, λ = 4L, as inthe diagram.
But v = f λ, therefore Lf
= v4
So,
L = ( ) =−
343
4 2400 3571
m s
sm.
(b) For an open pipe, λ = 2L, as in the diagram.
So,
Lf
= = ( ) =−
v2
343
2 2400 7151
m s
sm.
P18.30 dAA m= 0 320. ; λ = 0 640. m
(a) f = =vλ
531 Hz
(b) λ = =v� f 0 085 0. m; dAA mm= 42 5.
P18.31 The wavelength is λ = = =vf
3431 31
m s
261.6 sm.
so the length of the open pipe vibrating in its simplest (A-N-A) mode is
dA to A m= =1
20 656λ .
A closed pipe has (N-A) for its simplest resonance,
(N-A-N-A) for the second,
and (N-A-N-A-N-A) for the third.
Here, the pipe length is 55
4
5
41 31 1 64dN to A m m= = ( ) =λ
. .
P18.32 The air in the auditory canal, about 3 cm long, can vibrate with a node at the closed end and antinode at the open end,
with dN to A cm= =34
λ
so λ = 0 12. m
and f = = ≈vλ
343
0 123
m s
mkHz
.
A small-amplitude external excitation at this frequency can, over time, feed energy into a larger-amplitude resonance vibration of the air in the canal, making it audible.
486 Chapter 18
L
NA
AA N
l/4
l/2
FIG. P18.29
FIG. P18.31
13794_18_ch18_p473-496.indd 48613794_18_ch18_p473-496.indd 486 1/3/07 8:15:07 PM1/3/07 8:15:07 PM
P18.33 For a closed box, the resonant frequencies will have nodes at both sides, so the permitted
wavelengths will be L n n= =( )1
21 2 3λ , , , , … .
i.e.,
Ln n
f= =λ
2 2
v and f
L=
nv2
Therefore, with L = 0.860 m and ′ =L 2 10. m, the resonant frequencies are f nn = ( )206 Hz for
L = 0.860 m for each n from 1 to 9 and ′ = ( )f nn 84 5. Hz for ′ =L 2 10. m for each n from 2 to 23.
P18.34 The wavelength of sound is λ = vf
The distance between water levels at resonance is df
= v2
∴ = =Rt r dr
fπ π2
2
2
v
and tr
Rf=
π 2
2
v
P18.35 For both open and closed pipes, resonant frequencies are equally spaced as numbers. The set of resonant frequencies then must be 650 Hz, 550 Hz, 450 Hz, 350 Hz, 250 Hz, 150 Hz, 50 Hz.
These are odd-integer multipliers of the fundamental frequency of 50 0. Hz . Then the pipe
length is dfNA = = = ( ) =λ
4 4
340
4 501
v m s
s.70 m .
*P18.36 (a) The open ends of the tunnel are antinodes, so dAA
= 2000 m �n with n = 1, 2, 3, … . Then λ = 2d
AA = 4000 m �n. And f = v�λ = (343 m �s)�(4000 m �n) =
0.0858 Hz, with 1, 2, 3,n n= …
(b) It is a good rule. Any car horn would produce several or many of the closely-spaced reso-nance frequencies of the air in the tunnel, so it would be greatly amplifi ed. Other drivers might be frightened directly into dangerous behavior, or might blow their horns also.
P18.37 For resonance in a narrow tube open at one end,
f nL
=v
4 n =( )1 3 5, , ,…
(a) Assuming n = 1 and n = 3,
3844 0 228
=( )
v.
and 3843
4 0 683=
( )v
.
In either case, v = 350 m s .
(b) For the next resonance n = 5, and
Lf
= =( )( ) =−
5
4
5 350
4 3841 141
v m s
sm.
Superposition and Standing Waves 487
22.8 cm
68.3 cm
f = 384 Hz
warmair
FIG. P18.37
13794_18_ch18_p473-496.indd 48713794_18_ch18_p473-496.indd 487 1/3/07 8:15:08 PM1/3/07 8:15:08 PM
P18.38 The length corresponding to the fundamental satisfi es fL
= v4
: Lf1 4
34
4 5120 167= =
( ) =v
. m.
Since L > 20.0 cm, the next two modes will be observed, corresponding to
fL
= 3
4 2
v and f
L= 5
4 3
v
or
Lf2
3
40 502= =v. m and L
f3
5
40 837= =v. m
*P18.39 Call L the depth of the well and v the speed of sound.
Then for some integer n L n nf
n= −( ) = −( ) =
−( )( )2 1
42 1
4
2 1 343
4 51 51
1
λ v m s
. ss−( )1
and for the next resonance L n nf
n= +( ) −[ ] = +( ) =
+( )( )2 1 1
42 1
4
2 1 343
4 62
2
λ v m s
00 0 1. s−( ) Thus,
2 1 343
4 51 5
2 1 343
4 601
n n−( )( )( ) =
+( )( )−
m s
s
m s
. ..0 1s−( ) and we require an integer solution to
2 1
60 0
2 1
51 5
n n+ = −. .
The equation gives n = =111 5
176 56
.. , so the best fi tting integer is n = 7.
Then the results L =( ) −[ ]( )
( ) =−
2 7 1 343
4 51 521 61
m s
sm
..
and L =( ) +[ ]( )
( ) =−
2 7 1 343
4 60 021 41
m s
sm
..
suggest that we can say
the depth of the well is (21.5 ± 0.1) m. The data suggest 0.6-Hz uncertainty in the frequency measurements, which is only a little more than 1%.
P18.40 (a) For the fundamental mode of an open tube,
Lf
= = = ( ) =−
λ2 2
343
2 8800 1951
v m s
sm.
(b) v = + −( )=331 1
5 00
273328m s m s
.
We ignore the thermal expansion of the metal.
fL
= = =( ) =
v vλ 2
328
2 0 195841
m s
mHz
.
The fl ute is fl at by a semitone.
488 Chapter 18
13794_18_ch18_p473-496.indd 48813794_18_ch18_p473-496.indd 488 1/3/07 8:15:09 PM1/3/07 8:15:09 PM
Section 18.6 Standing Waves in Rod and Membranes
P18.41 (a) fL
= =( )( ) =
v2
5 100
2 1 601 59
.. kHz
(b) Since it is held in the center, there must be a node in the center as well as antinodes at the
ends. The even harmonics have an antinode at the center so only the odd harmonics are present.
(c) fL
= ′ =( )( ) =
v2
3 560
2 1 601 11
.. kHz
P18.42 When the rod is clamped at one-quarter of its length, the vibration pattern reads ANANA and the rod length is L d= 2 AA = λ .
Therefore,
Lf
= = =v 5 100
4 4001 16
m s
Hzm.
Section 18.7 Beats: Interference in Time
P18.43 f T� �v fnew Hz= =110540
600104 4.
Δ f = −110 104 4 5 64� �s s = beats s. .
P18.44 (a) The string could be tuned to either 521 Hz or 525 Hz from this evidence.
(b) Tightening the string raises the wave speed and frequency. If the frequency were originally 521 Hz, the beats would slow down.
Instead, the frequency must have started at 525 Hz to become
526 Hz .
(c) From fT
L L
T= = =vλ
μμ2
1
2
f
f
T
T2
1
2
1
= and Tf
fT T T2
2
1
2
1
2
1
5230 989=
⎛⎝⎜
⎞⎠⎟
= ⎛⎝
⎞⎠ =Hz
526 Hz. 11
The fractional change that should be made in the tension is then
fractional change = − = − = =T T
T1 2
1
1 0 989 0 011 4 1 14. . . % lower
The tension should be reduced by 1.14% .
P18.45 For an echo ′ =+( )−( )f f s
s
v vv v
the beat frequency is f f fb = ′ − .
Solving for fb , gives f fbs
s
=( )−( )2v
v v when approaching wall.
(a) fb = ( ) ( )−( ) =256
2 1 33
343 1 331 99
.
.. Hz beat frequency
(b) When he is moving away from the wall, vs changes sign. Solving for sv gives
vv
sb
b
f
f f=
−= ( )( )( )( ) − =
2
5 343
2 256 53 38. m s
Superposition and Standing Waves 489
13794_18_ch18_p473-496.indd 48913794_18_ch18_p473-496.indd 489 1/3/07 8:15:10 PM1/3/07 8:15:10 PM
P18.46 Using the 4 and 22
3-foot pipes produces actual frequencies of 131 Hz and 196 Hz and a com-
bination tone at 196 131 65 4−( ) =Hz Hz. , so this pair supplies the so-called missing fundamental. The 4 and 2-foot pipes produce a combination tone 262 131 131−( ) =Hz Hz, so this does not work.
The 22
32and -foot pipes produce a combination tone at 262 196 65 4−( ) =Hz Hz. , so this works.
Also, 4 22
32, , and -foot pipes all playing together produce the 65.4-Hz combination tone.
Section 18.8 Nonsinusoidal Wave Patterns
P18.47 We list the frequencies of the harmonics of each note in Hz:
HarmonicNote 1 2 3 4 5
A 440.00 880.00 1 320.0 1 760.0 2 200.0C# 554.37 1 108.7 1 663.1 2 217.5 2 771.9E 659.26 1 318.5 1 977.8 2 637.0 3 296.3
The second harmonic of E is close the the thhird harmonic of A, and the fourth harmonicc of C# is
close to the fifth harmonic of A..
P18.48 We evaluate
s = + + +100 157 2 62 9 3 105 4sin sin . sin sinθ θ θ θ
+ 51 9. siin . sin . sin5 29 5 6 25 3 7θ θ θ+ +
where s represents particle displacement in nanometers and θ represents the phase of the wave in radians. As θ advances by 2π , time advances by (1�523) s. Here is the result:
Additional Problems
*P18.49 (a) The yo-yo’s downward speed is dL �dt = 0 + (0.8 m �s2)(1.2 s) = 0.960 m �s. The instanta-neous wavelength of the fundamental string wave is given by d
NN = λ �2 = L so λ = 2L and
dλ �dt = 2 dL �dt = 2(0.96 m �s) = 1.92 m �s.
(b) For the second harmonic, the wavelength is equal to the length of the string. Then the rate of
change of wavelength is equal to dL �dt = 0.960 m/s, half as much as for the f irst harmonic .
(c) A yo-yo of different mass will hold the string under different tension to make each string wave vibrate with a different frequency, but the geometrical argument given in parts (a) and
(b) still applies to the wavelength. The answers are unchanged : dλ1�dt = 1.92 m �s and
dλ2 �dt = 0.960 m �s.
490 Chapter 18
FIG. P18.48
13794_18_ch18_p473-496.indd 49013794_18_ch18_p473-496.indd 490 1/3/07 8:15:12 PM1/3/07 8:15:12 PM
*P18.50 (a) Use the Doppler formula
′ =±( )
( )f fs
v vv v
0
∓ With ′=f1 frequency of the speaker in front of student and
′ =f2 frequency of the speaker behind the student.
′= ( ) +( )−( ) =f1 456
343 1 50
343 0458Hz
m s m s
m s
.HHz
Hzm s m s
m s′ = ( ) −( )
+( ) =f2 456343 1 50
343 04
.554 Hz
Therefore, f f fb = ′− ′ =1 2 3 99. Hz
(b) The waves broadcast by both speakers have λ = = =vf
343
4560 752
m s
sm. . The standing
wave between them has dAA = =λ2
0 376. m. The student walks from one maximum to the
next in time Δt = =0 376
1 500 251
.
..
m
m ss, so the frequency at which she hears maxima is
fT
= =13 99. Hz
(c) The answers are identical. The models are equally valid. We may think of the interference of the two waves as interference in space or in time, linked to space by the steady motion of the student.
P18.51 f = 87 0. Hz
speed of sound in air: va = 340 m s
(a) λb = � v = = ( )( )−f bλ 87 0 0 4001. .s m
v = 34 8. m s
(b) λλ
a
a a
L
f
==
⎫⎬⎭
4
v L
fa= = ( ) =−
v4
340
4 87 00 9771
m s
sm
..
Superposition and Standing Waves 491
FIG. P18.51
13794_18_ch18_p473-496.indd 49113794_18_ch18_p473-496.indd 491 1/3/07 8:15:13 PM1/3/07 8:15:13 PM
P18.52
dNA
dNA
dNA
dNA
dNA
IIIII I
A
A
AA
A
A
N
N
N
N
NN
FIG. P18.52
(a) μ = × = ×−
−5 5 106 40 10
33.
.kg
0.86 mkg m
v = = ⋅×
=−
T
μ1 30
6 40 1014 33
.
..
kg m s
kg mm s
2
(b) In state I, dNA m= =0 8604
.λ
(c) λ = 3 44. m f = = =vλ
14 3
3 444 14
.
..
m s
mHz
In state II, dNA m m= ( ) =1
30 86 0 287. .
λ = ( ) =4 0 287 1 15. .m m f = = =vλ
14 3
1 1512 4
.
..
m s
mHz
In state III, dNA m m= ( ) =1
50 86 0 172. .
f = =( ) =
vλ
14 3
4 0 17220 7
.
..
m s
mHz
P18.53 If the train is moving away from station, its frequency is depressed:
′ = − =f 180 2 00 178. Hz: 178 180343
343=
− −( )v
Solving for v gives v = ( )( )2 00 343
178
.
Therefore, v = 3 85. m s away from station
If it is moving toward the station, the frequency is enhanced:
′ = + =f 180 2 00 182. Hz: 182 180343
343=
− v
Solving for v gives 42 00 343
182= ( )( ).
Therefore, v = 3 77. m s toward the station
P18.54 v = ( )( )×
=−
48 0 2 00
4 80 101413
. .
.m s
dNN m= 1 00. ; λ = 2 00. m; f = =vλ
70 7. Hz
λaa
f= = =v 343
70 74 85
m s
Hzm
..
492 Chapter 18
13794_18_ch18_p473-496.indd 49213794_18_ch18_p473-496.indd 492 1/3/07 8:15:14 PM1/3/07 8:15:14 PM
P18.55 (a) Since the fi rst node is at the weld, the wavelength in the thin wire is 2L or 80.0 cm. The frequency and tension are the same in both sections, so
fL
T= =( ) ×
=−
1
2
1
2 0 400
4 60
2 00 1059 93μ .
.
.. Hz
(b) As the thick wire is twice the diameter, the linear density is 4 times that of the thin wire. ′ =μ 8 00. g m
so ′ =′
Lf
T1
2 μ
′ =
( )( )⎡⎣⎢
⎤⎦⎥ × −L
1
2 59 9
4 60
8 00 10 3.
.
. = 20 0. cm half the length of the thin wire.
*P18.56 The wavelength stays constant at 0.96 m while the wavespeed rises according tov = (T�μ)1/2 = [(15 + 2.86t)�0.0016]1/2 = [9375 + 1786t]1/2 so the frequency rises asf = v�λ = [9375 + 1786t]1/2�0.96 = [10 173 + 1938t]1/2 The number of cycles is f dt ineach incremental bit of time, or altogether
10173 1938
1
193810173 19381 2
0
3 5 1+( ) = +( )∫ t dt t/. //. 2
0
3 51938 dt∫
/ .3 2
0
31
1938
10173 1938
3 2
t=
+[ ]�
55 3 2 3 216954 10173
2906407=
( ) − ( )=
/ /
cycles
P18.57 (a) fn
L
T=2 μ
so ′ =′
= =f
f
L
L
L
L2
1
2
The frequency should be halved to get the same number of antinodes for twice the
length.
(b) ′ =
′n
n
T
T so ′ =
′⎛⎝
⎞⎠ =
+⎡⎣⎢
⎤⎦⎥
T
T
n
n
n
n
2 2
1
The tension must be ′ =+
⎡⎣⎢
⎤⎦⎥
Tn
nT
1
2
(c) ′ = ′′
′f
f
n L
nL
T
T so ′ = ′ ′
′⎛⎝⎜
⎞⎠⎟
T
T
nf L
n fL
2
′ =⋅
⎛⎝
⎞⎠
T
T
3
2 2
2
′ =T
T
9
16 to get twice as many antinodes.
Superposition and Standing Waves 493
13794_18_ch18_p473-496.indd 49313794_18_ch18_p473-496.indd 493 1/5/07 6:56:33 PM1/5/07 6:56:33 PM
P18.58 (a) For the block:
F T Mgx∑ = − =sin .30 0 0°
so T Mg Mg= =sin .30 01
2°
(b) The length of the section of string parallel to the incline ish
hsin .30 0
2°= . The total length of the string is then 3h .
(c) The mass per unit length of the string is μ = m
h3
(d) The speed of waves in the string is v = = ⎛⎝
⎞⎠⎛⎝
⎞⎠ =T Mg h
m
Mgh
mμ 2
3 3
2
(e) In the fundamental mode, the segment of length h vibrates as one loop. The distance
between adjacent nodes is then d hNN = =λ2
, so the wavelength is λ = 2h.
The frequency is fh
Mgh
m
Mg
mh= = =vλ
1
2
3
2
3
8
(g) When the vertical segment of string vibrates with 2 loops (i.e., 3 nodes), then h = ⎛⎝⎜
⎞⎠⎟2
2
λ
and the wavelength is λ = h
(f ) The period of the standing wave of 3 nodes (or two loops) is
Tf
hm
Mgh
mh
Mg= = = =1 2
3
2
3
λv
(h) f f f fMg
mhb = − = ×( ) = ×( )− −1 02 2 00 10 2 00 103
82 2. . .
P18.59 We look for a solution of the form
5 00 2 00 10 0 10 0 2 00 10 0. sin . . . cos . .x t x t−( ) + −( )
A= ssin . .
sin . . cos
2 00 10 0
2 00 10 0
x t
A x t
− +( )= −( )
φφ A+ ccos . . sin2 00 10 0x t−( ) φ
This will be true if both 5 00. cos= A φ and10 0. sin= A φ,
requiring 5 00 10 02 2 2. .( ) + ( ) = A
A = 11 2. and φ = 63 4. °
The resultant wave 11 2 2 00 10 0 63 4. sin . . .x t− +( )° is sinusoidal.
494 Chapter 18
FIG. P18.58
13794_18_ch18_p473-496.indd 49413794_18_ch18_p473-496.indd 494 1/3/07 8:15:16 PM1/3/07 8:15:16 PM
P18.60 dAA m= = × −λ2
7 05 10 3. is the distance between antinodes.
Then λ = × −14 1 10 3. m
and f = = ××
= ×−
vλ
3 70 102 62 10
35.
.m s
14.1 10 mHz3
The crystal can be tuned to vibrate at 218 Hz, so that binary counters canderive from it a signal at precisely 1 Hz.
P18.61 (a) Let θ represent the angle each slanted rope makes with the vertical.
In the diagram, observe that:
sin.θ = =1 00 2
3
m
1.50 m or θ = 41 8. °
Considering the mass,
Fy∑ = 0: 2T mgcosθ =
or T =( )( )
=12 0 9 80
2 41 878 9
. .
cos ..
kg m sN
2
°
(b) The speed of transverse waves in the string is v = = =T
μ78 9
281. N
0.001 00 kg mm s
For the standing wave pattern shown (3 loops), d = 3
2λ
or λ = ( )=2 2 00
31 33
..
mm
Thus, the required frequency is f = = =vλ
281
1 33211
m s
mHz
.
ANSWERS TO EVEN PROBLEMS
P18.2 see the solution
P18.4 5.66 cm
P18.6 (a) 3.33 rad (b) 283 Hz
P18.8 (a) The number is the greatest integer ≤ ⎛⎝
⎞⎠ +d
f
v1
2 (b) L
d n f
n fn =− −( ) ( )
−( )( )2 2 2
1 2
2 1 2
vv
where
n n= 1 2, , , max…
P18.10 (a) Point A is one half wavelength farther from one speaker than from the other. The waves it receives interfere destructively. (b) Along the hyperbola 9x2 − 16y2 = 144. (c) Yes; along the straight line through the origin with slope 0.75 or the straight line through the origin with slope −0.75.
P18.12 see the solution
P18.14 (a) see the solution (b) 4 m is the distance between crests. (c) 50 Hz. The oscillation at any point starts to repeat after a period of 20 ms, and f = 1�T. (d) 4 m. By comparison with equation 18.3,k = π �2, and λ = 2π �k. (e) 50 Hz. By comparison with equation 18.3, ω = 2πf = 100π.
Superposition and Standing Waves 495
FIG. P18.61
FIG. P18.60
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P18.16 (a) Yes. The resultant wave contains points of no motion. (b) and (c) The nodes are still separated by λ �2. They are all shifted by the distance φ � 2k to the left.
P18.18 15.7 Hz
P18.20 (a) 495 Hz (b) 990 Hz
P18.22 31.2 cm from the bridge; 3.84%
P18.24 291 Hz
P18.26 The natural frequency of the water sloshing in the bay agrees precisely with that of lunar excita-tion, so we identify the extra-high tides as amplifi ed by resonance.
P18.28 9.00 kHz
P18.30 (a) 531 Hz (b) 42.5 mm
P18.32 3 kHz; a small-amplitude external excitation at this frequency can, over times, feed energy into a larger-amplitude resonance vibration of the air in the canal, making it audible.
P18.34 Δtr
Rf=π 2
2
v
P18.36 (a) 0.0858 n Hz, with n = 1, 2, 3, … (b) It is a good rule. Any car horn would produce several or many of the closely-spaced resonance frequencies of the air in the tunnel, so it would be greatly amplifi ed. Other drivers might be frightened directly into dangerous behavior, or might blow their horns also.
P18.38 0.502 m; 0.837 m
P18.40 (a) 0.195 m (b) 841 m
P18.42 1.16 m
P18.44 (a) 521 Hz or 525 Hz (b) 526 Hz (c) reduce by 1.14%
P18.46 4-foot and 22
3-foot; 2
2
32and -foot; and all three together
P18.48 see the solution
P18.50 (a) 3.99 beats�s (b) 3.99 beats�s (c) The answers are identical. The models are equally valid. We may think of the interference of the two waves as interference in space or in time, linked to space by the steady motion of the student.
P18.52 (a) 14.3 m �s (b) 86.0 cm, 28.7 cm, 17.2 cm (c) 4.14 Hz, 12.4 Hz, 20.7 Hz
P18.54 4.85 m
P18.56 407 cycles
P18.58 (a)1
2Mg (b) 3h (c) m
h3 (d)
3
2
Mgh
m (e)
3
8
Mg
mh (f )
2
3
mh
Mg (g) h (h) 2 00 10
3
82. ×( )− Mg
mh
P18.60 262 kHz
496 Chapter 18
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