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 measured for 50 oscillations.
(a) Determine the period of mo
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 denominator rather than
the minus sign. The new
expression repre
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? (d) What is the maximum
transverse speed of an elemen
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) None of statements (a)
through (c) is true.
16.3 The
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 this section, we introduce an important wave function w
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 is permitted because Newtons laws are
valid in either
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 to the right and then pulled briefly to the left. This
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 cos (3x _ 15t _ 2) (d) y _ 8 sin (2x _ 15t)
(e) y _ 8 cos
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 the maximum velocity and
acceleration of the piston wh
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 position is
negative. Because the object is moving bac
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.com and
go to ThomsonNOW to adjust the linear
mass densi
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 _ vt _
p
2
b _ A cos 1kx _ vt 2
(B) Determine the phas
one end. The planks other end is supported by a spring
of force constant k (Fig. P15.57). The moment of inertia
of the plank about the pivot is . The plank is displaced
by a small angle u from its horizontal equilibrium
position and released. (a) Show tha
waves traveling on a string. All wave functions y(x, t) represent solutions of an
equation called the linear wave equation. This equation gives a complete description
of the wave motion, and from it one can derive an expression for the wave speed.
Further
cord in this stretched position? (c) Find the speed of a
transverse wave in the cord if the block is held in this lowest
position.
50. Review problem. A block of mass M hangs from a rubber
cord. The block is supported so that the cord is not
stretched. Th
mounted at its center on a frictionless axle. The assembly
is rotated through a small angle u from its equilibrium
position and released. (a) Show that the speed of the center
of the small disk as it passes through the equilibrium
position is
(b) Show tha
function
where x and y are in meters and t is in seconds. The mass
per unit length of this string is 12.0 g/m. Determine
(a) the speed of the wave, (b) the wavelength, (c) the frequency,
and (d) the power transmitted to the wave.
38. The wave function for
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.
16.1 Propagation of a Disturbance
The introduction to
(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. Find (f) the speed and
(g) the acceleration of the object
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 Simple Harmonic Motion
2. In an engine, a piston oscill
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
wave of wavelength l _
40.0 cm and amplitude A _ 15.0 c
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 frequency
suddenly drop to a new constant
value and 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 medium is the
same whenever x is increased by an integra
(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 one-dimensional periodic sound wave in a long, narrow
(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 the
left) has an amplitude of 20.0 cm, a wavelength of
35
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 function has a maximum value of 1,
identify the maximum
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 the transverse position y for all positions and
times,
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 motion of a one-dimensional longitudinal pulse
moving throug
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
object, m2 _ 7.00 kg, is slowly pushed up against m1,