A sinusoidal wave traveling to the right can be expressed with a wave function 16.5 where A is the amplitude, l is the wavelength, and v is the wave speed.. y 1x, t2 f 1x ; vt2 The spee
Trang 1In addition to kinetic energy, there is potential energy associated with each ment of the string due to its displacement from the equilibrium position and therestoring forces from neighboring elements A similar analysis to that above for
ele-the total potential energy Ulin one wavelength gives exactly the same result:
The total energy in one wavelength of the wave is the sum of the potential andkinetic energies:
(16.20)
As the wave moves along the string, this amount of energy passes by a given point
on the string during a time interval of one period of the oscillation Therefore,the power , or rate of energy transfer TMWassociated with the mechanical wave, is
sinu-Quick Quiz 16.5 Which of the following, taken by itself, would be most effective
in increasing the rate at which energy is transferred by a wave traveling along astring? (a) reducing the linear mass density of the string by one half (b) doublingthe wavelength of the wave (c) doubling the tension in the string (d) doublingthe amplitude of the wave
Conceptualize Consider Active Figure 16.10 again and notice that the vibrating blade supplies energy to the string
at a certain rate This energy then propagates to the right along the string
Categorize We evaluate quantities from equations developed in the chapter, so we categorize this example as a
sub-stitution problem
Power Supplied to a Vibrating String
Evaluate the wave speed on the string from
Equa-tion 16.18:
vBT
mB 80.0 N5.00 2 kg>m 40.0 m>s
Evaluate the angular frequency v of the
sinu-soidal waves on the string from Equation 16.9:
v 2pf 2p160.0 Hz2 377 s1
Use these values and A 2m in
Equa-tion 16.21 to evaluate the power:
Trang 216.6 The Linear Wave Equation
In Section 16.1, we introduced the concept of the wave function to represent
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
Furthermore, the linear wave equation is basic to many forms of wave motion In
this section, we derive this equation as applied to waves on strings
Suppose a traveling wave is propagating along a string that is under a tension T.
Let’s consider one small string element of length x (Fig 16.19) The ends of the
element make small angles uA and uB with the x axis The net force acting on the
element in the vertical direction is
Because the angles are small, we can use the small-angle approximation sin u
tan u to express the net force as
(16.22)
Imagine undergoing an infinitesimal displacement outward from the end of the
rope element in Figure 16.19 along the blue line representing the force This
displacement has infinitesimal x and y components and can be represented by the
vector The tangent of the angle with respect to the x axis for this
dis-placement is dy/dx Because we evaluate this tangent at a particular instant of
time, we must express it in partial form as y/x Substituting for the tangents in
a F y T sin u B T sin u A T1sin uB sin uA2
Section 16.6 The Linear Wave Equation 465
What If? What if the string is to transfer energy at a rate of 1 000 W? What must be the required amplitude if allother parameters remain the same?
Answer Let us set up a ratio of the new and old power, reflecting only a change in the amplitude:
Solving for the new amplitude gives
2mv2
A2oldv A
2 new
A2 old
Trang 3The right side of Equation 16.25 can be expressed in a different form if we notethat the partial derivative of any function is defined as
Associating f (x x) with (y/x)B and f (x) with ( y/x)A, we see that, in thelimit x S 0, Equation 16.25 becomes
(16.26)
This expression is the linear wave equation as it applies to waves on a string.The linear wave equation (Eq 16.26) is often written in the form
(16.27)
Equation 16.27 applies in general to various types of traveling waves For waves on
strings, y represents the vertical position of elements of the string For sound waves, y corresponds to longitudinal position of elements of air from equilibrium
or variations in either the pressure or the density of the gas through which the
sound waves are propagating In the case of electromagnetic waves, y corresponds
to electric or magnetic field components
We have shown that the sinusoidal wave function (Eq 16.10) is one solution ofthe linear wave equation (Eq 16.27) Although we do not prove it here, the linear
wave equation is satisfied by any wave function having the form y f(x vt)
Fur-thermore, we have seen that the linear wave equation is a direct consequence ofNewton’s second law applied to any element of a string carrying a traveling wave
466 Chapter 16 Wave Motion
Linear wave equation for a
A one-dimensional sinusoidal wave is one for which
the positions of the elements of the medium vary
sinu-soidally A sinusoidal wave traveling to the right can be
expressed with a wave function
(16.5)
where A is the amplitude, l is the wavelength, and v is
the wave speed.
y 1x, t2 A sin c2pl 1x vt2 d
The angular wave number k and angular frequency v
of a wave are defined as follows:
A transverse wave is one in which the elements of the medium move in a direction perpendicular to the direction of propagation A longitudinal wave is one in which the elements of the medium move in a direction parallel to the
direction of propagation
(continued)
Trang 4Questions 467
CO N C E P T S A N D P R I N C I P L E S
Any one-dimensional wave traveling with a speed v in the x direction can be
represented by a wave function of the form
(16.1, 16.2)
where the positive sign applies to a wave traveling in the negative x direction
and the negative sign applies to a wave traveling in the positive x direction.
The shape of the wave at any instant in time (a snapshot of the wave) is
obtained by holding t constant.
y 1x, t2 f 1x ; vt2
The speed of a wave traveling
on a taut string of mass per
unit length m and tension T is
(16.18)
vBTm
A wave is totally or partially reflected
when it reaches the end of the medium in
which it propagates or when it reaches a
boundary where its speed changes
discon-tinuously If a wave traveling on a string
meets a fixed end, the wave is reflected
and inverted If the wave reaches a free
end, it is reflected but not inverted
The power transmitted by a sinusoidal wave on a stretched string is
(16.21)
1
2mv2
A2v
Wave functions are solutions to a differential equation called the
linear wave equation:
v
l
Questions
denotes answer available in Student Solutions Manual/Study Guide; O denotes objective question
1. Why is a pulse on a string considered to be transverse?
2. How would you create a longitudinal wave in a stretched
spring? Would it be possible to create a transverse wave in
a spring?
3 O (i) Rank the waves represented by the following
func-tions 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 (4x 20t) (f) y 7 sin (6x 24t)
(ii) Rank the same waves according to their wavelengths
from largest to smallest (iii) Rank the same waves ing to their frequencies from largest to smallest (iv) Rank
accord-the same waves according to accord-their periods from largest to
smallest (v) Rank the same waves according to their
speeds from largest to smallest.
4 OIf the string does not stretch, by what factor would you have to multiply the tension in a taut string so as to double the wave speed? (a) 8 (b) 4 (c) 2 (d) 0.5 (e) You could not change the speed by a predictable factor by changing the tension.
Trang 55 O When all the strings on a guitar are stretched to the
same tension, will the speed of a wave along the most
massive bass string be (a) faster, (b) slower, or (c) the
same as the speed of a wave on the lighter strings?
Alter-natively, (d) is the speed on the bass string not necessarily
any of these answers?
6 O If you stretch a rubber hose and pluck it, you can
observe a pulse traveling up and down the hose (i) What
happens to the speed of the pulse if you stretch the hose
more tightly? (a) It increases (b) It decreases (c) It is
constant (d) It changes unpredictably (ii) What happens
to the speed if you fill the hose with water? Choose from
the same possibilities.
7. When a pulse travels on a taut string, does it always invert
upon reflection? Explain.
8. Does the vertical speed of a segment of a horizontal taut
string, through which a wave is traveling, depend on the
wave speed?
9 O(a) Can a wave on a string move with a wave speed that
is greater than the maximum transverse speed v y, maxof an
element of the string? (b) Can the wave speed be much
greater than the maximum element speed? (c) Can the
wave speed be equal to the maximum element speed?
(d) Can the wave speed be less than v y, max?
10. If you shake one end of a taut rope steadily three times
each second, what would be the period of the sinusoidal
wave set up in the rope?
11. If a long rope is hung from a ceiling and waves are sent
up the rope from its lower end, they do not ascend with
constant speed Explain.
468 Chapter 16 Wave Motion
12 O A source vibrating at constant frequency generates a sinusoidal wave on a string under constant tension If the power delivered to the string is doubled, by what factor does the amplitude change? (a) 4 (b) 2 (c) (d) 1 (e) 0.707 (f) cannot be predicted
13 OIf one end of a heavy rope is attached to one end of a light rope, a wave can move from the heavy rope into
the lighter one (i) What happens to the speed of the
wave? (a) It increases (b) It decreases (c) It is
con-stant (d) It changes unpredictably (ii) What happens
to the frequency? Choose from the same possibilities.
(iii)What happens to the wavelength? Choose from the same possibilities.
14. A solid can transport both longitudinal waves and verse waves, but a homogeneous fluid can transport only longitudinal waves Why?
trans-15. In an earthquake both S (transverse) and P nal) waves propagate from the focus of the earthquake The focus is in the ground below the epicenter on the surface Assume the waves move in straight lines through uniform material The S waves travel through the Earth more slowly than the P waves (at about 5 km/s versus
(longitudi-8 km/s) By detecting the time of arrival of the waves, how can one determine the distance to the focus of the earthquake? How many detection stations are necessary to locate the focus unambiguously?
16. In mechanics, massless strings are often assumed Why is that not a good assumption when discussing waves on strings?
12
2 = intermediate; 3 = challenging; = SSM/SG; = ThomsonNOW; = symbolic reasoning; = qualitative reasoning
Problems
The Problems from this chapter may be assigned online in WebAssign.
Sign in at www.thomsonedu.com and go to ThomsonNOW to assess your understanding of this chapter’s topics
with additional quizzing and conceptual questions.
1, 2 3 denotes straightforward, intermediate, challenging; denotes full solution available in Student Solutions Manual/Study Guide ; denotes coached solution with hints available at www.thomsonedu.com; denotes developing symbolic reasoning;
denotes asking for qualitative reasoning; denotes computer useful in solving problem
Section 16.1 Propagation of a Disturbance
1. At t 0, a transverse pulse in a wire is described by the
function
where x and y are in meters Write the function y(x, t)
that describes this pulse if it is traveling in the positive x
direction with a speed of 4.50 m/s.
2. Ocean waves with a crest-to-crest distance of 10.0 m can
be described by the wave function
where v 1.20 m/s (a) Sketch y(x, t) at t 0 (b) Sketch
y(x, t) at t 2.00 s Compare this graph with that for part
(a) and explain similarities and differences What has the
wave done between picture (a) and picture (b)?
3. Two points A and B on the surface of the Earth are at the
same longitude and 60.0° apart in latitude Suppose an
y 1x, t2 10.800 m2 sin 30.628 1x vt2 4
x2 3
earthquake at point A creates a P wave that reaches point
B by traveling straight through the body of the Earth at a
constant speed of 7.80 km/s The earthquake also ates a Rayleigh wave that travels along the surface of the Earth at 4.50 km/s (a) Which of these two seismic waves
radi-arrives at B first? (b) What is the time difference between the arrivals of these two waves at B ? Take the radius of the
Earth to be 6 370 km.
4. A seismographic station receives S and P waves from an earthquake, 17.3 s apart Assume the waves have traveled over the same path at speeds of 4.50 km/s and 7.80 km/s Find the distance from the seismograph to the hypocen- ter of the earthquake.
Section 16.2 The Traveling Wave Model
5. The wave function for a traveling wave on a taut string
is (in SI units)
y 1x, t2 10.350 m2 sin a 10pt 3px p
4 b
Trang 6(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
wave-length and frequency of the wave? (d) What is the
maxi-mum transverse speed of an element of the string?
6. A certain uniform string is held under constant
ten-sion (a) Draw a side-view snapshot of a sinusoidal wave
on a string as shown in diagrams in the text (b)
Immedi-ately below diagram (a), draw the same wave at a moment
later by one quarter of the period of the wave (c) Then,
draw a wave with an amplitude 1.5 times larger than the
wave in diagram (a) (d) Next, draw a wave differing from
the one in your diagram (a) just by having a wavelength
1.5 times larger (e) Finally, draw a wave differing from
that in diagram (a) just by having a frequency 1.5 times
larger.
7. A sinusoidal wave is traveling along a rope The oscillator
that generates the wave completes 40.0 vibrations in 30.0 s.
Also, a given maximum travels 425 cm along the rope in
10.0 s What is the wavelength of the wave?
8. For a certain transverse wave, the distance between two
successive crests is 1.20 m, and eight crests pass a given
point along the direction of travel every 12.0 s Calculate
the wave speed.
9. A wave is described by y (2.00 cm) sin (kx vt), where
k 2.11 rad/m, v 3.62 rad/s, x is in meters, and t is in
seconds Determine the amplitude, wavelength,
fre-quency, and speed of the wave.
10. When a particular wire is vibrating with a frequency of
4.00 Hz, a transverse wave of wavelength 60.0 cm is
pro-duced Determine the speed of waves along the wire.
11. The string shown in Active Figure 16.10 is driven at a
fre-quency of 5.00 Hz The amplitude of the motion is
12.0 cm, and the wave speed is 20.0 m/s Furthermore,
the wave is such that y 0 at x 0 and t 0 Determine
(a) the angular frequency and (b) wave number for this
wave (c) Write an expression for the wave function
Cal-culate (d) the maximum transverse speed and (e) the
maximum transverse acceleration of a point on the string.
12. Consider the sinusoidal wave of Example 16.2 with the
wave function
At a certain instant, let point A be at the origin and point
B be the first point along the x axis where the wave is
60.0° out of phase with A What is the coordinate of B ?
13. A sinusoidal wave is described by the wave function
where x and y are in meters and t is in seconds
Deter-mine for this wave the (a) amplitude, (b) angular
fre-quency, (c) angular wave number, (d) wavelength, (e) wave
speed, and (f) direction of motion.
14. (a) Plot y versus t at x 0 for a sinusoidal wave of the
form y (15.0 cm) cos (0.157x 50.3t), where x and y
are in centimeters and t is in seconds (b) Determine the
period of vibration from this plot State how your result
compares with the value found in Example 16.2.
15. (a) Write the expression for y as a function of x and t
for a sinusoidal wave traveling along a rope in the
(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 35.0 cm, and a frequency of 12.0 Hz The transverse posi-
tion of an element of the medium at t 0, x 0 is y
3.00 cm, and the element has a positive velocity here.
(a) Sketch the wave at t 0 (b) Find the angular wave number, period, angular frequency, and wave speed of the wave (c) Write an expression for the wave function
y(x, t ).
17. A transverse wave on a string is described by the wave function
(a) Determine the transverse speed and acceleration of
the string at t 0.200 s for the point on the string
located at x 1.60 m (b) What are the wavelength, period, and speed of propagation of this wave?
18. A transverse sinusoidal wave on a string has a period T
25.0 ms and travels in the negative x direction with a speed
of 30.0 m/s At t 0, an element of the string at x 0 has
a transverse position of 2.00 cm and is traveling downward with a speed of 2.00 m/s (a) What is the amplitude of the wave? (b) What is the initial phase angle? (c) What is the maximum transverse speed of an element of the string? (d) Write the wave function for the wave.
19. A sinusoidal wave of wavelength 2.00 m and amplitude 0.100 m travels on a string with a speed of 1.00 m/s to the right Initially, the left end of the string is at the origin Find (a) the frequency and angular frequency, (b) the angular wave number, and (c) the wave function for this wave Determine the equation of motion for (d) the left
end of the string and (e) the point on the string at x 1.50 m to the right of the left end (f) What is the maxi- mum speed of any point on the string?
20. A wave on a string is described by the wave function y
(0.100 m) sin (0.50x 20t) (a) Show that an element of the string at x 2.00 m executes harmonic motion (b) Determine the frequency of oscillation of this particu- lar point.
Section 16.3 The Speed of Waves on Strings
21. A telephone cord is 4.00 m long The cord has a mass of 0.200 kg A transverse pulse is produced by plucking one end of the taut cord The pulse makes four trips down and back along the cord in 0.800 s What is the tension in the cord?
22. A transverse traveling wave on a taut wire has an tude of 0.200 mm and a frequency of 500 Hz It travels with a speed of 196 m/s (a) Write an equation in SI units
ampli-of the form y A sin (kx vt) for this wave (b) The
mass per unit length of this wire is 4.10 g/m Find the tension in the wire.
23. A piano string having a mass per unit length equal to 5.00 3kg/m is under a tension of 1 350 N Find the speed with which a wave travels on this string.
y 10.120 m2 sin ap
8 x 4ptb
2 = intermediate; 3 = challenging; = SSM/SG; = ThomsonNOW; = symbolic reasoning; = qualitative reasoning
Trang 724. Transverse pulses travel with a speed of 200 m/s along a
taut copper wire whose diameter is 1.50 mm What is the
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
travel-ing down a wire that has an object of large mass
sus-pended from it Assume a wire has a mass of 4.00 g and a
length of 1.60 m and assume a 3.00-kg object is
sus-pended from it A pulse requires 36.1 ms to traverse the
length of the wire Calculate gMoonfrom these data (You
may ignore the mass of the wire when calculating the
ten-sion in it.)
26. A simple pendulum consists of a ball of mass M hanging
from a uniform string of mass m and length L, with
m M Let T represent the period of oscillations for
the pendulum Determine the speed of a transverse wave
in the string when the pendulum hangs at rest.
27. Transverse waves travel with a speed of 20.0 m/s in a
string under a tension of 6.00 N What tension is required
for a wave speed of 30.0 m/s in the same string?
28 Review problem. A light string with a mass per unit
length of 8.00 g/m has its ends tied to two walls separated
by a distance equal to three-fourths the length of the
string (Fig P16.28) An object of mass m is suspended
from the center of the string, putting a tension in the
string (a) Find an expression for the transverse wave
speed in the string as a function of the mass of the
hang-ing object (b) What should be the mass of the object
sus-pended from the string if the wave speed is to be 60.0 m/s?
470 Chapter 16 Wave Motion
Section 16.5 Rate of Energy Transfer by Sinusoidal Waves
on Strings
32. A taut rope has a mass of 0.180 kg and a length of 3.60 m What power must be supplied to the rope so as to gener- ate sinusoidal waves having an amplitude of 0.100 m and
a wavelength of 0.500 m and traveling with a speed of 30.0 m/s?
33. A two-dimensional water wave spreads in circular ripples.
Show that the amplitude A at a distance r from the initial
disturbance is proportional to Suggestion: Consider
the energy carried by one outward-moving ripple.
34. Transverse waves are being generated on a rope under constant tension By what factor is the required power increased or decreased if (a) the length of the rope is doubled and the angular frequency remains constant, (b) the amplitude is doubled and the angular frequency
is halved, (c) both the wavelength and the amplitude are doubled, and (d) both the length of the rope and the wavelength are halved?
35. Sinusoidal waves 5.00 cm in amplitude are to be
trans-mitted along a string that has a linear mass density of 4.00 2 kg/m The source can deliver a maximum power of 300 W and the string is under a tension of
100 N What is the highest frequency at which the source can operate?
36. A 6.00-m segment of a long string contains four complete waves and has a mass of 180 g The string vibrates sinu- soidally with a frequency of 50.0 Hz and a peak-to-valley displacement of 15.0 cm (The “peak-to-valley” distance is the vertical distance from the farthest positive position to the farthest negative position.) (a) Write the function that
describes this wave traveling in the positive x direction.
(b) Determine the power being supplied to the string.
37. A sinusoidal wave on a string is described by the wave 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 fre- quency, and (d) the power transmitted to the wave.
38. The wave function for a wave on a taut string is
where x is in meters and t is in seconds (a) What is the
average rate at which energy is transmitted along the string if the linear mass density is 75.0 g/m? (b) What is the energy contained in each cycle of the wave?
39. A horizontal string can transmit a maximum power 0
(without breaking) if a wave with amplitude A and
angu-lar frequency v is traveling along it To increase this imum power, a student folds the string and uses this “dou- ble string” as a medium Determine the maximum power that can be transmitted along the “double string,” assum- ing that the tension in the two strands together is the same as the original tension in the single string.
max-40. In a region far from the epicenter of an earthquake, a seismic wave can be modeled as transporting energy in a single direction without absorption, just as a string wave
m
Figure P16.28
29. The elastic limit of a piece of steel wire is 2.70 8 Pa.
What is the maximum speed at which transverse wave
pulses can propagate along this wire without exceeding
this stress? (The density of steel is 7.86 3 kg/m 3 )
30. A student taking a quiz finds on a reference sheet the
two equations
and
She has forgotten what T represents in each equation.
(a) Use dimensional analysis to determine the units
required for T in each equation (b) Explain how you can
identify the physical quantity each T represents from the
units.
31. A steel wire of length 30.0 m and a copper wire of
length 20.0 m, both with 1.00-mm diameters, are
con-nected end to end and stretched to a tension of 150 N.
During what time interval will a transverse wave travel the
entire length of the two wires?
vBTm
f 1
T
Trang 8does Suppose the seismic wave moves from granite into
mudfill with similar density but with a much smaller bulk
modulus Assume the speed of the wave gradually drops
by a factor of 25.0, with negligible reflection of the wave.
Explain whether the amplitude of the ground shaking will
increase or decrease Does it change by a predictable
fac-tor? This phenomenon led to the collapse of part of the
Nimitz Freeway in Oakland, California, during the Loma
Prieta earthquake of 1989.
Section 16.6 The Linear Wave Equation
41. (a) Evaluate A in the scalar equality (7 3)4 A.
(b) Evaluate A, B, and C in the vector equality
Explain how you arrive
at the answers to convince a student who thinks that you
cannot solve a single equation for three different
unknowns (c) What If? The functional equality or identity
is true for all values of the variables x and t, measured in
meters and in seconds, respectively Evaluate the
con-stants A, B, C, D, and E Explain how you arrive at the
answers.
42. Show that the wave function y e b(x vt)is a solution of the
linear wave equation (Eq 16.27), where b is a constant.
43. Show that the wave function y ln[b(x vt)] is a
solu-tion to Equasolu-tion 16.27, where b is a constant.
44. (a) Show that the function y(x, t) x2 v2t2 is a solution
to the wave equation (b) Show that the function in part
(a) can be written as f(x vt) g(x vt) and determine
the functional forms for f and g (c) What If? Repeat parts
(a) and (b) for the function y(x, t) sin (x) cos (vt).
Additional Problems
45. The “wave” is a particular type of pulse that can
propa-gate through a large crowd gathered at a sports arena
(Fig P16.45) The elements of the medium are the
spec-tators, with zero position corresponding to their being
seated and maximum position corresponding to their
standing and raising their arms When a large fraction of
the spectators participate in the wave motion, a somewhat
stable pulse shape can develop The wave speed depends
on people’s reaction time, which is typically on the order
of 0.1 s Estimate the order of magnitude, in minutes, of
the time interval required for such a pulse to make one
circuit around a large sports stadium State the quantities
you measure or estimate and their values.
where x is in meters and t is in seconds The mass per
length of the string is 12.0 g/m (a) Find the maximum transverse acceleration of an element on this string (b) Determine the maximum transverse force on a 1.00-cm segment of the string State how this force com- pares with the tension in the string.
47. Motion picture film is projected at 24.0 frames per ond Each frame is a photograph 19.0 mm high At what constant speed does the film pass into the projector?
sec-48. A transverse wave on a string is described by the wave function
Consider the element of the string at x 0 (a) What is the time interval between the first two instants when this
element has a position of y 0.175 m? (b) What distance does the wave travel during this time interval?
49 Review problem. A 2.00-kg block hangs from a rubber cord, being supported so that the cord is not stretched The unstretched length of the cord is 0.500 m, and its mass is 5.00 g The “spring constant” for the cord is
100 N/m The block is released and stops at the lowest point (a) Determine the tension in the cord when the block is at this lowest point (b) What is the length of the cord in this “stretched” position? (c) Find the speed of a transverse wave in the cord if the block is held in this low- est 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 The unstretched length of the cord is L0, and
its mass is m, much less than M The “spring constant” for the cord is k The block is released and stops at the lowest
point (a) Determine the tension in the string when the block is at this lowest point (b) What is the length of the cord in this “stretched” position? (c) Find the speed of a transverse wave in the cord if the block is held in this low- est position.
51. An earthquake or a landslide can produce an ocean wave of short duration carrying great energy, called a tsunami When its wavelength is large compared to the
ocean depth d, the speed of a water wave is given imately by v (a) Explain why the amplitude of the wave increases as the wave approaches shore What can you consider to be constant in the motion of any one wave crest? (b) Assume an earthquake occurs all along a tectonic plate boundary running north to south and pro- duces a straight tsunami wave crest moving everywhere to the west If the wave has amplitude 1.80 m when its speed
approx-is 200 m/s, what will be its amplitude where the water approx-is 9.00 m deep? (c) Explain why the amplitude at the shore should be expected to be still greater, but cannot be meaningfully predicted by your model.
52 Review problem. A block of mass M, supported by a
string, rests on a frictionless incline making an angle u with the horizontal (Fig P16.52) The length of the string
is L, and its mass is m M Derive an expression for the
Trang 9time interval required for a transverse wave to travel from
one end of the string to the other.
472 Chapter 16 Wave Motion
What If?(b) Show that the expression in part (a) reduces
to the result of Problem 57 when M 0 (c) Show that
for m M, the expression in part (a) reduces to
59. It is stated in Problem 57 that a pulse travels from the
bot-tom to the top of a hanging rope of length L in a time
interval Use this result to answer the ing questions (It is not necessary to set up any new inte- grations.) (a) Over what time interval does a pulse travel halfway up the rope? Give your answer as a fraction of the quantity (b) A pulse starts traveling up the rope How far has it traveled after a time interval ?
follow-60. If a loop of chain is spun at high speed, it can roll along the ground like a circular hoop without collapsing Con- sider a chain of uniform linear mass density m whose cen-
ter of mass travels to the right at a high speed v0 (a) Determine the tension in the chain in terms of m and
v0 (b) If the loop rolls over a bump, the resulting mation of the chain causes two transverse pulses to propa- gate along the chain, one moving clockwise and one mov- ing counterclockwise What is the speed of the pulses traveling along the chain? (c) Through what angle does each pulse travel during the time interval over which the loop makes one revolution?
defor-61 Review problem.An aluminum wire is clamped at each end under zero tension at room temperature Reducing the temperature, which results in a decrease in the wire’s equilibrium length, increases the tension in the wire What strain (L/L) results in a transverse wave speed of
100 m/s? Take the cross-sectional area of the wire to be equal to 5.00 6 m 2 , the density to be 2.70 3 kg/m 3 , and Young’s modulus to be 7.00 10 N/m 2
62. (a) Show that the speed of longitudinal waves along a
spring of force constant k is , where L is the
unstretched length of the spring and m is the mass per unit length (b) A spring with a mass of 0.400 kg has an unstretched length of 2.00 m and a force constant of
100 N/m Using the result you obtained in part (a), mine the speed of longitudinal waves along this spring.
deter-63. A pulse traveling along a string of linear mass density m is described by the wave function
where the factor in brackets is said to be the amplitude (a) What is the power (x) carried by this wave at a point
x ? (b) What is the power carried by this wave at the
ori-gin? (c) Compute the ratio (x)/(0).
64. An earthquake on the ocean floor in the Gulf of Alaska produces a tsunami that reaches Hilo, Hawaii, 4 450 km away, in a time interval of 9 h 30 min Tsunamis have enormous wavelengths (100 to 200 km), and the propaga- tion speed for these waves is , where is the aver- age depth of the water From the information given, find the average wave speed and the average ocean depth between Alaska and Hawaii (This method was used in
1856 to estimate the average depth of the Pacific Ocean
53. A string with linear density 0.500 g/m is held under
tension 20.0 N As a transverse sinusoidal wave propagates
on the string, elements of the string move with maximum
speed v y, max (a) Determine the power transmitted by the
wave as a function of v y, max (b) State how the power
depends on v y, max (c) Find the energy contained in a
sec-tion of string 3.00 m long Express it as a funcsec-tion of v y, max
and the mass m3of this section (d) Find the energy that
the wave carries past a point in 6.00 s.
54. A sinusoidal wave in a rope is described by the wave
function
where x and y are in meters and t is in seconds The rope
has a linear mass density of 0.250 kg/m The tension in
the rope is provided by an arrangement like the one
illus-trated in Figure 16.12 What is the mass of the suspended
object?
55. A block of mass 0.450 kg is attached to one end of a cord
of mass 0.003 20 kg; the other end of the cord is attached
to a fixed point The block rotates with constant angular
speed in a circle on a horizontal, frictionless table.
Through what angle does the block rotate in the time
interval during which a transverse wave travels along the
string from the center of the circle to the block?
56. A wire of density r is tapered so that its cross-sectional
area varies with x according to
(a) The tension in the wire is T Derive a relationship for
the speed of a wave as a function of position (b) What If?
Assume the wire is aluminum and is under a tension of
24.0 N Determine the wave speed at the origin and at x
10.0 m.
57. A rope of total mass m and length L is suspended
verti-cally Show that a transverse pulse travels the length of the
rope in a time interval Suggestion: First find
an expression for the wave speed at any point a distance x
from the lower end by considering the rope’s tension as
resulting from the weight of the segment below that
point.
58. Assume an object of mass M is suspended from the
bot-tom of the rope in Problem 57 (a) Show that the time
interval for a transverse pulse to travel the length of the
rope is
¢t 21L>g
A 3x 0.0102 cm 2
y 10.20 m2 sin 10.75px 18pt2
Trang 10long before soundings were made to give a direct
deter-mination.)
65. A string on a musical instrument is held under tension T
and extends from the point x 0 to the point x L The
string is overwound with wire in such a way that its mass
per unit length m(x) increases uniformly from m0at x 0
to mL at x L (a) Find an expression for m(x) as a
func-Answers to Quick Quizzes 473
tion of x over the range 0 x L (b) Show that the
time interval required for a transverse pulse to travel the length of the string is given by
¢t2L1mL m 0 2mLm0 2
3 2T 12mL 2m 0 2
2 = intermediate; 3 = challenging; = SSM/SG; = ThomsonNOW; = symbolic reasoning; = qualitative reasoning
Answers to Quick Quizzes
16.1 (i), (b) It is longitudinal because the disturbance (the
shift of position of the people) is parallel to the
direc-tion in which the wave travels (ii), (a) It is transverse
because the people stand up and sit down (vertical
motion), whereas the wave moves either to the left or to
the right.
16.2 (i), (c) The wave speed is determined by the medium,
so it is unaffected by changing the frequency (ii), (b).
Because the wave speed remains the same, the result of
doubling the frequency is that the wavelength is half as
large (iii), (d) The amplitude of a wave is unrelated to
the wave speed, so we cannot determine the new
ampli-tude without further information.
16.3 (c) With a larger amplitude, an element of the string
has more energy associated with its simple harmonic
motion, so the element passes through the equilibrium
position with a higher maximum transverse speed.
16.4 Only answers (f) and (h) are correct Choices (a) and (b) affect the transverse speed of a particle of the string, but not the wave speed along the string Choices (c) and (d) change the amplitude Choices (e) and (g) increase the time interval by decreasing the wave speed.
16.5 (d) Doubling the amplitude of the wave causes the power to be larger by a factor of 4 In choice (a), halving the linear mass density of the string causes the power to change by a factor of 0.71, and the rate decreases In choice (b), doubling the wavelength of the wave halves the frequency and causes the power to change by a fac- tor of 0.25, and the rate decreases In choice (c), dou- bling the tension in the string changes the wave speed and causes the power to change by a factor of 1.4, which
is not as large as in choice (d).
Trang 11Sound waves travel through any material medium with a speed that depends on
the properties of the medium As sound waves travel through air, the elements ofair vibrate to produce changes in density and pressure along the direction ofmotion of the wave If the source of the sound waves vibrates sinusoidally, the pres-sure variations are also sinusoidal The mathematical description of sinusoidalsound waves is very similar to that of sinusoidal waves on strings, which were dis-cussed in Chapter 16
Sound waves are divided into three categories that cover different frequency
ranges (1) Audible waves lie within the range of sensitivity of the human ear They
can be generated in a variety of ways, such as by musical instruments, human
voices, or loudspeakers (2) Infrasonic waves have frequencies below the audible
range Elephants can use infrasonic waves to communicate with one another, even
when separated by many kilometers (3) Ultrasonic waves have frequencies above
the audible range You may have used a “silent” whistle to retrieve your dog Dogseasily hear the ultrasonic sound this whistle emits, although humans cannot detect
it at all Ultrasonic waves are also used in medical imaging
This chapter begins with a discussion of the speed of sound waves and thenwave intensity, which is a function of wave amplitude We then provide an alterna-
Human ears have evolved to detect sound waves and interpret them as
music or speech Some animals, such as this young bat-eared fox, have
ears adapted for the detection of very weak sounds (Getty Images)
17.1 Speed of Sound Waves 17.2 Periodic Sound Waves 17.3 Intensity of Periodic Sound Waves 17.4 The Doppler Effect
17.5 Digital Sound Recording 17.6 Motion Picture Sound
Sound Waves 17
474
Trang 12tive description of the intensity of sound waves that compresses the wide range of
intensities to which the ear is sensitive into a smaller range for convenience The
effects of the motion of sources and listeners on the frequency of a sound are also
investigated Finally, we explore digital reproduction of sound, focusing in
particu-lar on sound systems used in modern motion pictures
Let us describe pictorially the motion of a one-dimensional longitudinal pulse
moving through a long tube containing a compressible gas as shown in Figure
17.1 A piston at the left end can be moved to the right to compress the gas and
create the pulse Before the piston is moved, the gas is undisturbed and of
uni-form density as represented by the uniuni-formly shaded region in Figure 17.1a When
the piston is suddenly pushed to the right (Fig 17.1b), the gas just in front of it is
compressed (as represented by the more heavily shaded region); the pressure and
density in this region are now higher than they were before the piston moved
When the piston comes to rest (Fig 17.1c), the compressed region of the gas
con-tinues to move to the right, corresponding to a longitudinal pulse traveling
through the tube with speed v.
The speed of sound waves in a medium depends on the compressibility and
density of the medium If the medium is a liquid or a gas and has a bulk modulus
B (see Section 12.4) and density r, the speed of sound waves in that medium is
(17.1)
It is interesting to compare this expression with Equation 16.18 for the speed of
transverse waves on a string, In both cases, the wave speed depends
on an elastic property of the medium (bulk modulus B or string tension T ) and
on an inertial property of the medium (r or m) In fact, the speed of all
mechani-cal waves follows an expression of the general form
For longitudinal sound waves in a solid rod of material, for example, the speed of
sound depends on Young’s modulus Y and the density r Table 17.1 (page 476)
provides the speed of sound in several different materials
The speed of sound also depends on the temperature of the medium For
sound traveling through air, the relationship between wave speed and air
tempera-ture is
where 331 m/s is the speed of sound in air at 0°C and TCis the air temperature in
degrees Celsius Using this equation, one finds that at 20°C, the speed of sound in
air is approximately 343 m/s
This information provides a convenient way to estimate the distance to a
thun-derstorm First count the number of seconds between seeing the flash of lightning
and hearing the thunder Dividing this time by 3 gives the approximate distance to
the lightning in kilometers because 343 m/s is approximately km/s Dividing the
time in seconds by 5 gives the approximate distance to the lightning in miles
because the speed of sound is approximately mi/s.15
1 3
v 1331 m>s2B1 TC
273°C
vB elastic propertyinertial property
Undisturbed gas
Figure 17.1 Motion of a nal pulse through a compressible gas The compression (darker region) is produced by the moving piston.
Trang 13longitudi-17.2 Periodic Sound Waves
One can produce a one-dimensional periodic sound wave in a long, narrow tubecontaining a gas by means of an oscillating piston at one end as shown in ActiveFigure 17.2 The darker parts of the colored areas in this figure represent regions
in which the gas is compressed and the density and pressure are above their librium values A compressed region is formed whenever the piston is pushed into
equi-the tube This compressed region, called a compression, moves through equi-the tube,
continuously compressing the region just in front of itself When the piston ispulled back, the gas in front of it expands and the pressure and density in thisregion fall below their equilibrium values (represented by the lighter parts of the
colored areas in Active Fig 17.2) These low-pressure regions, called rarefactions,
also propagate along the tube, following the compressions Both regions move atthe speed of sound in the medium
As the piston oscillates sinusoidally, regions of compression and rarefaction arecontinuously set up The distance between two successive compressions (or twosuccessive rarefactions) equals the wavelength l of the sound wave As theseregions travel through the tube, any small element of the medium moves with sim-
ple harmonic motion parallel to the direction of the wave If s(x, t) is the position
of a small element relative to its equilibrium position,1 we can express this monic position function as
har-(17.2)
where smax is the maximum position of the element relative to equilibrium This
parameter is often called the displacement amplitude of the wave The parameter
k is the wave number, and v is the angular frequency of the wave Notice that the displacement of the element is along x, in the direction of propagation of the
sound wave, which means we are describing a longitudinal wave
The variation in the gas pressure P measured from the equilibrium value is
also periodic For the position function in Equation 17.2, P is given by
A longitudinal wave propagating
through a gas-filled tube The source
of the wave is an oscillating piston at
the left.
Sign in at www.thomsonedu.comand
go to ThomsonNOW to adjust the
frequency of the piston.
1We use s(x, t) here instead of y(x, t) because the displacement of elements of the medium is not pendicular to the x direction.
per-TABLE 17.1
Speed of Sound in Various Media
Gases
Hydrogen (0°C) 1 286 Helium (0°C) 972
Solidsa Pyrex glass 5 640
Trang 14where the pressure amplitude Pmax—which is the maximum change in pressure
from the equilibrium value—is given by
(17.4)
Equation 17.3 is derived in Example 17.1
A sound wave may be considered to be either a displacement wave or a pressure
wave A comparison of Equations 17.2 and 17.3 shows that the pressure wave is 90°
out of phase with the displacement wave Graphs of these functions are shown in
Figure 17.3 The pressure variation is a maximum when the displacement from
equilibrium is zero, and the displacement from equilibrium is a maximum when
the pressure variation is zero
Quick Quiz 17.1 If you blow across the top of an empty soft-drink bottle, a
pulse of sound travels down through the air in the bottle At the moment the
pulse reaches the bottom of the bottle, what is the correct description of the
dis-placement of elements of air from their equilibrium positions and the pressure of
the air at this point? (a) The displacement and pressure are both at a maximum
(b) The displacement and pressure are both at a minimum (c) The displacement
is zero, and the pressure is a maximum (d) The displacement is zero, and the
From the definition of bulk modulus (see Eq 12.8),
express the pressure variation in the element of gas as a
function of its change in volume:
Fig-Categorize This derivation combines elastic properties of a gas (Chapter 12) with the wave phenomena discussed
in this chapter
Analyze The element of gas has a thickness x in the horizontal direction and a cross-sectional area A, so its ume is V i A x When a sound wave displaces the element, the disk’s two flat faces move through different dis- tances s The change in volume V of the element when a sound wave displaces the element is equal to A s, where
vol-s is the difference between the values of s between the two flat faces of the disk.
Derivation of Equation 17.3
Substitute for the initial volume and the change in
vol-ume of the element:
¢P B A ¢s
A ¢x B ¢s
¢x
Let the thickness x of the disk approach zero so that
the ratio s/x becomes a partial derivative:
¢P B 0s
0x
Substitute the position function given by Equation 17.2: ¢P B 0
0x 3smax cos 1kx vt2 4 Bsmaxk sin 1kx vt2
Trang 1517.3 Intensity of Periodic Sound Waves
In Chapter 16, we showed that a wave traveling on a taut string transports energy.The same concept applies to sound waves Consider an element of air of mass m
and length x in front of a piston of area A oscillating with a frequency v as
shown in Figure 17.4 The piston transmits energy to this element of air in thetube, and the energy is propagated away from the piston by the sound wave Toevaluate the rate of energy transfer for the sound wave, let’s evaluate the kineticenergy of this element of air, which is undergoing simple harmonic motion A pro-cedure similar to that in Section 16.5 in which we evaluated the rate of energytransfer for a wave on a string shows that the kinetic energy in one wavelength ofthe sound wave is
As in the case of the string wave in Section 16.5, the total potential energy for onewavelength has the same value as the total kinetic energy; therefore, the totalmechanical energy for one wavelength is
As the sound wave moves through the air, this amount of energy passes by a givenpoint during one period of oscillation Hence, the rate of energy transfer is
where v is the speed of sound in air Compare this expression with Equation 16.21
for a wave on a string
We define the intensity I of a wave, or the power per unit area, as the rate at
which the energy transported by the wave transfers through a unit area A
perpen-dicular to the direction of travel of the wave:
(17.5)
In this case, the intensity is therefore
Hence, the intensity of a periodic sound wave is proportional to the square ofthe displacement amplitude and to the square of the angular frequency Thisexpression can also be written in terms of the pressure amplitude Pmax; in thiscase, we use Equation 17.4 to obtain
478 Chapter 17 Sound Waves
Because the sine function has a maximum value of 1,
identify the maximum value of the pressure variation as
Pmax rvvsmax (see Eq 17.4) and substitute for this
combination in the previous expression:
¢P ¢Pmax sin 1kx vt2
Finalize This final expression for the pressure variation of the air in a sound wave matches Equation 17.3
Use Equation 17.1 to express the bulk modulus as
B rv2and substitute:
¢P rv2
smaxk sin 1kx vt2 Use Equation 16.11 in the form k v/v and substitute: ¢P rvvsmax sin 1kx vt2
Figure 17.4 An oscillating piston
transfers energy to the air in the tube,
causing the element of air of length
x and mass m to oscillate with an
amplitude smax.
Intensity of a sound wave