Chapter 16
Questions: 1, 3, 7, 10, 11, 13, 14, 17, 18, 21, 23
Problems: 1, 9, 13, 15, 23, 35, 39, 43, 47, 49, 50, 52
Questions
1. An oscillation does not propagate, while a wave does.
3. A light wave satis…es, c = f ; while a sound wave satis…es vs = f :
Since c >> vs ; the frequency of the same wavelength of light is much greater
than the frequency of a sound wave.
7. A wave pulse has a speed and an amplitude with the obvious definitions. However a pulse is a sum of wave with di¤erent frequencies and
periods. Hence it does not have a well de…ned frequency or period.
11. A wave traveling in the positive x direction has the functional form
f (kx !t) : Note that k plays the same role spatially that ! does in the
time domain.
13. In the absence of tension there is no restoring force that is required
for the wave to propagate.
14. Power is proportional to the amplitude squared. Thus if the amplitude is reduced by 1/2, the power is reduced by 1/4.
17. No! The energy is spread out over a larger area, A = 4 r2 :
18. For a large amplitude wave on a string, the string is stretched more
for a small wavelength wave as compared to a wave with a larger wavelength.
This will increase the tension for the small wavelength as compared to the
large wavelength. Hence the wave speed will increase as the wavelength
decreases.
21. The nodes and antinodes depend on the relative distances to the two
sources. This doesn’t change in time.
23. The beat frequency is f = (102 100) =2kHz = 1kHz. Since this is
in the range of human hearing you will hear a tone at this frequency.
Problems
1. The time interval is the period, where
v = =T ! T = =v = 18=5:3 = 3: 40 sec
9. The frequency of the wave is f = 6=60Hz = :1Hz. The wavelength of
this wave is
= v=f = 10v = 22m.
The minimum distance between the boats is =2 = 11m.
1
13. An expression for a wave propagating in the negative x direction is
y = A cos
2
(x + vt + xo ) = A cos (kx + !t + kxo ) ;
where k = 2 = and ! = vk = 2 v= :
(a) Since = 16cm and v = 35cm=s, ! = 2
35=16 ' 13:74s 1 ;
1
(b) and k = 2 =16 ' :393cm .
(c) For this wave A = 2:5 and the arbitrary phase, 2 xo = = 0: Hence
y = 2:5 cos (:393x + 13:74t)
15. Given a water wave y = :25 sin (:52x 2:3t) = :25 sin 2 (x=
f t) ;
(a) The amplitude yo = :25m = 25cm,
(b) the frequency f = 2:3=2 = 0:366 Hz,
(c) the wavelength = 2 =:52 = 12: 08m,
(d) the velocity v = 2:3=:52 = 4: 423 m=s.
p
23. (a) The velocity of a transverse wave on a string is v = F= : Hence
F = v 2 = :17
6:72 = 7:6N
35. The average power propagating along the slinky is
q _
_
1 2 2
! A v ! A = 2P = ! 2 v
P =
2
s
2 1:1
A =
= :35m = 35cm
(:6=10) (2
1:8)2 2:3
39. The intensity is
I=
P
=
4 r2
4
50
= :0123W=m2 = 12:3mW=m2
(18)2
43. (a) The intensity at the laser
I=
P
=
A
5
(5
10 3
= 6:37kW=m2
10 4 )2
(b) at the wall
I=
P
5 10 3
=
= 4; 9W=m2
A
(:018)2
2
47. (a) For the …rst pulse the amplitude is found when x = t; and
A1 = 2cm: The amplitude for the second pulse occurs when x = 5 t; and
A2 = 2cm.
(b) At t = 0 the peak of the …rst pulse occurs at x = 0 and it is moving
in the positive x direction. At t = 0 the pulse of the second pulse occurs at
x = 5cm and it is moving in the negative x direction.
(c) At t = 2:5 sec the peak of both pulses are located at x = 2:5cm at
which point they cancel.
49. The beat frequency is
f = fh
fl = 10= min
Hence the period is once every 6 sec.
50. The two wave forms are
y1 = A cos (kx
!t) = A Re ei(kx
!t)
!t + ) = A Re ei(kx
and y2 = A cos (kx
Summing the two wave forms we …nd:
y = y1 + y2 = A Re ei(kx
!t)
+ ei(kx
!t+ )
y = A Re ei(kx !t) 1 + ei
= A Re ei(kx
y = 2A cos =2 cos (kx !t + =2) :
!t+ =2)
ei
=2
+e
This is a simple harmonic wave with an amplitude
A0 = 2A cos =2:
52. The distance from the top speaker to the …rst nodal line is
q
r1 = 3:52 + (:83 :75)2 = 3:5009m
The distance from the bottom speaker to the …rst nodal line is
q
r2 = 3:52 + (:83 + :75)2 = 3:8401m
A node occurs when
r = =2: Hence
= 2 (3:8401
3:5009) = :6784m
The speed of the sound waves is
vs = f = 500
:6784 = 339m=s
3
i =2
!t+ )