lengths. For each length, the time interval for 50 oscillations
is measured with a stopwatch. For lengths of
1.000 m, 0.750 m, and 0.500 m, total time intervals of
99.8 s, 86.6 s, and 71.1 s are measu
energy is lost due to air resistance, (a) show that the ensuing
motion is periodic and (b) determine the period of
the motion. (c) Is the motion simple harmonic? Explain.
Section 15.2 The Particle in
Angular frequency _
Wave function for a _
sinusoidal wave
Speed of a sinusoidal wave
General expression for a _
sinusoidal wave
_
y (cm)
40.0 cm
15.0 cm
x (cm)
Figure 16.9 (Example 16.2) A sinusoidal
depends only on the relative speed
of source and observer. As you listen
to an approaching source, you
will detect increasing intensity but
constant frequency. As the source
passes, you will hear the
the sinusoidal wave is traveling can be written
(16.4)
where the constant A represents the wave amplitude and the constant l is the
wavelength. Notice that the vertical position of an element of the m
(a)
Undisturbed gas
Figure 17.1 Motion of a longitudinal
pulse through a compressible gas.
The compression (darker region) is
produced by the moving piston.
17.2 Periodic Sound Waves
One can produce a
(b) What If? Write the expression for y as a function of x
and t for the wave in part (a) assuming that y(x, 0) _ 0 at
the point x _ 10.0 cm.
16. A sinusoidal wave traveling in the _x direction (to th
2rv
I _ 12
rv 1vsmax2
I_
_
A
_
El
T
_
2
12
1rA2v2s2
maxl
T
_ 12
1rA2v2s2
max a
l
T
b _ 12
rAvv2s2
max
El _ Kl _ Ul _
1rA2v2s2
maxl
Kl _ 14
1rA2v2s2
maxl
12
478 Chapter 17 Sound Waves
Because the sine
at time t _ 0 as in Figure 16.5a. Consequently, an element of the string at x at this
time has the same y position as an element located at x _ vt had at time t _ 0:
In general, then, we can represent
investigated. Finally, we explore digital reproduction of sound, focusing in particular
on sound systems used in modern motion pictures.
17.1 Speed of Sound Waves
Let us describe pictorially the motio
L
P
y
Pivot
My=0
Figure P15.47
48. An object of mass m1 _ 9.00 kg is in equilibrium, connected
to a light spring of constant k _ 100 N/m that is
fastened to a wall as shown in Figure P15.48a. A second
Ear plugs are recommended whenever sound levels exceed 90 dB. Recent evidence
suggests that noise pollution may be a contributing factor to high blood
pressure, anxiety, and nervousness. Table 17.2 gi
analog recording, any small imperfection in the record surface or the recording
equipment can cause a distortion of the waveform. For example, clipping all peaks
of a waveform by 10% has a major effec
waves, but a homogeneous fluid can transport only
longitudinal waves. Why?
15. In an earthquake both S (transverse) and P (longitudinal)
waves propagate from the focus of the earthquake.
The focus is
tension in the wire? (The density of copper is 8.92 g/cm 3.)
25. An astronaut on the Moon wishes to measure the local
value of the free-fall acceleration by timing pulses traveling
down a wire that ha
strength of a sound. Let us now extend our discussion from Example 17.4 concerning
the psychological measurement of the strength of a sound.
Of course, we dont have instruments in our bodies that can
(c) the maximum speed of the object. Where does this
maximum speed occur? (d) Find the maximum acceleration
of the object. Where does it occur? (e) Find the total
energy of the oscillating system. Fin
16
Wave Motion
449
ferred from the point at which the pebble is dropped to the position of the object.
This feature is central to wave motion: energy is transferred over a distance, but
matter is not.
unit time interval. The frequency of a sinusoidal wave is related to the period by
the expression
454 Chapter 16 Wave Motion
Answer One new feature in this expression is the plus sign in the denominat
b
(a) What are the speed and direction of travel of the
wave? (b) What is the vertical position of an element of
the string at t _ 0, x _ 0.100 m? (c) What are the wavelength
and frequency of the wave
correct? (a) The speed of the wave changes. (b) The frequency of the wave
changes. (c) The maximum transverse speed of an element of the medium
changes. (d) Statements (a) through (c) are all true. (e
x2 _ 1
Write the
2
1x _ 3.02
Write the
2
1x _ 6.02
wave function expression at t _ 1.0 s: y 1x, 1.0 2 _
_1
wave function expression at t _ 2.0 s: y 1x, 2.0 2 _
2
2
_1
16.2 The Traveling Wave Model
In
convenient to choose a different inertial reference frame that moves along with
the pulse with the same speed as the pulse so that the pulse is at rest within the
frame. This change of reference frame
of propagation is called a transverse wave.
Compare this wave with another type of pulse, one moving down a long,
stretched spring as shown in Figure 16.3. The left end of the spring is pushed
briefly
according to their amplitudes from the largest to
the smallest. If two waves have the same amplitude, show
them as having equal rank.
(a) y _ 2 sin (3x _ 15t _ 2) (b) y _ 4 sin (3x _ 15t)
(c) y _ 6 co
and (d) its velocity at the end of this time interval.
9. A piston in a gasoline engine is in simple harmonic
motion. Taking the extremes of its position relative to its
center point as _5.00 cm, find
its maximum negative position. It then repeats these two
motions in the reverse direction to return to its original
position and complete one cycle.
15.2 (f). The object is in the region x _ 0, so the
pulse
ACTIVE FIGURE 16.13
The reflection of a traveling pulse at
the fixed end of a stretched string.
The reflected pulse is inverted, but its
shape is otherwise unchanged.
Sign in at www.thomsonedu.c
If the wave at t _ 0 is as described in Active Figure 16.10b, the wave function
can be written as
y _ A sin 1kx _ vt 2
Section 16.2 The Traveling Wave Model 457
Write the wave function: y _ A sin a kx