Waves - Theory - General Physics

CHAPTER 16
Waves—I
16-1TRANSVERSE WAVES
After reading this module, you should be able to . . .
16.01Identify the three main types of waves.
16.02Distinguish between transverse waves and longitudi-
nal waves.
16.03Given a displacement function for a traverse wave,
determine amplitude , angular wave number k, angular
frequencyv, phase constant f, and direction of travel,
and calculate the phase kx●vtfand the displace-
ment at any given time and position.
16.04Given a displacement function for a traverse
wave, calculate the time between two given displace-
ments.
16.05Sketch a graph of a transverse wave as a function
of position, identifying amplitude , wavelength l, where
the slope is greatest, where it is zero, and where the
string elements have positive velocity, negative velocity,
and zero velocity.
16.06Given a graph of displacement versus time for
a transverse wave, determine amplitude and
periodT.
y
m
y
m
y
m
16.07Describe the effect on a transverse wave of changing
phase constant f.
16.08Apply the relation between the wave speed v, the
distance traveled by the wave, and the time required for
that travel.
16.09Apply the relationships between wave speed v,
angular frequency v, angular wave number k, wavelength
l, period T, and frequency f.
16.10Describe the motion of a string element as a trans-
verse wave moves through its location, and identify
when its transverse speed is zero and when it is maxi-
mum.
16.11Calculate the transverse velocity u(t) of a string
element as a transverse wave moves through its location.
16.12Calculate the transverse acceleration a(t) of a
string element as a transverse wave moves through its
location.
16.13Given a graph of displacement, transverse velocity,
or transverse acceleration, determine the phase con-
stant f.
Key Ideas
Learning Objectives
444
●Mechanical waves can exist only in material media and are
governed by Newton’s laws. Transverse mechanical waves,
like those on a stretched string, are waves in which the
particles of the medium oscillate perpendicular to the wave’s
direction of travel. Waves in which the particles of the
medium oscillate parallel to the wave’s direction of travel are
longitudinal waves.
●A sinusoidal wave moving in the positive direction of an
xaxis has the mathematical form
y(x,t)y
msin(kxvt),
wherey
mis the amplitude (magnitude of the maximum dis-
placement) of the wave, kis the angular wave number, vis
the angular frequency, and kxvtis the phase. The wave-
lengthlis related to kby
k
2p
l
.
●The period Tand frequency fof the wave are related to vby
●The wave speed v(the speed of the wave along the string) is
related to these other parameters by
●Any function of the form
y(x,t)h(kx●vt)
can represent a traveling wave with a wave speed as given
above and a wave shape given by the mathematical form of h.
The plus sign denotes a wave traveling in the negative
direction of the xaxis, and the minus sign a wave traveling in
the positive direction.
v
v
k

l
T
lf.
v
2p
f
1
T
.

44516-1TRANSVERSE WAVES
What Is Physics?
One of the primary subjects of physics is waves. To see how important waves are
in the modern world, just consider the music industry. Every piece of music you
hear, from some retro-punk band playing in a campus dive to the most eloquent
concerto playing on the web, depends on performers producing waves and your
detecting those waves. In between production and detection, the information
carried by the waves might need to be transmitted (as in a live performance on
the web) or recorded and then reproduced (as with CDs, DVDs, or the other
devices currently being developed in engineering labs worldwide). The
financial importance of controlling music waves is staggering, and the rewards to
engineers who develop new control techniques can be rich.
This chapter focuses on waves traveling along a stretched string, such as on
a guitar. The next chapter focuses on sound waves, such as those produced by
a guitar string being played. Before we do all this, though, our first job is to
classify the countless waves of the everyday world into basic types.
Types of Waves
Waves are of three main types:
1.Mechanical waves.These waves are most familiar because we encounter them
almost constantly; common examples include water waves, sound waves, and
seismic waves. All these waves have two central features: They are governed
by Newton’s laws, and they can exist only within a material medium, such as
water, air, and rock.
2.Electromagnetic waves.These waves are less familiar, but you use them
constantly; common examples include visible and ultraviolet light, radio and
television waves, microwaves, x rays, and radar waves. These waves require no
material medium to exist. Light waves from stars, for example, travel through
the vacuum of space to reach us. All electromagnetic waves travel through a
vacuum at the same speed c299 792 458 m/s.
3.Matter waves.Although these waves are commonly used in modern technol-
ogy, they are probably very unfamiliar to you. These waves are associated
with electrons, protons, and other fundamental particles, and even atoms and
molecules. Because we commonly think of these particles as constituting
matter, such waves are called matter waves.
Much of what we discuss in this chapter applies to waves of all kinds.
However, for specific examples we shall refer to mechanical waves.
Transverse and Longitudinal Waves
A wave sent along a stretched, taut string is the simplest mechanical wave. If you
give one end of a stretched string a single up-and-down jerk, a wave in the form
of a single pulsetravels along the string. This pulse and its motion can occur
because the string is under tension. When you pull your end of the string upward,
it begins to pull upward on the adjacent section of the string via tension between
the two sections. As the adjacent section moves upward, it begins to pull the next
section upward, and so on. Meanwhile, you have pulled down on your end of the
string. As each section moves upward in turn, it begins to be pulled back
downward by neighboring sections that are already on the way down. The net
result is that a distortion in the string’s shape (a pulse, as in Fig. 16-1a) moves
along the string at some velocity .v
:
Figure 16-1(a) A single pulse is sent along
a stretched string.A typical string element
(marked with a dot) moves up once and
then down as the pulse passes.The ele-
ment’s motion is perpendicular to the
wave’s direction of travel, so the pulse is a
transverse wave.(b) A sinusoidal wave is
sent along the string.A typical string
element moves up and down continuously
as the wave passes.This too is a transverse
wave.
y
x
y
x
(a)
(b)
Sinusoidal
wave
Pulse
v
v

446 CHAPTER 16 WAVES—I
If you move your hand up and down in continuous simple harmonic motion, a
continuous wave travels along the string at velocity . Because the motion of your
hand is a sinusoidal function of time, the wave has a sinusoidal shape at any given in-
stant, as in Fig. 16-1b; that is, the wave has the shape of a sine curve or a cosine curve.
We consider here only an “ideal” string, in which no friction-like forces
within the string cause the wave to die out as it travels along the string. In
addition, we assume that the string is so long that we need not consider a wave
rebounding from the far end.
One way to study the waves of Fig. 16-1 is to monitor the wave forms(shapes of
the waves) as they move to the right. Alternatively, we could monitor the motion of
an element of the string as the element oscillates up and down while a wave passes
through it. We would find that the displacement of every such oscillating string ele-
ment is perpendicularto the direction of travel of the wave, as indicated in Fig. 16-1b.
This motion is said to be transverse,and the wave is said to be a transverse wave.
Longitudinal Waves.Figure 16-2 shows how a sound wave can be produced
by a piston in a long, air-filled pipe. If you suddenly move the piston rightward
and then leftward, you can send a pulse of sound along the pipe. The rightward
motion of the piston moves the elements of air next to it rightward, changing the
air pressure there. The increased air pressure then pushes rightward on the
elements of air somewhat farther along the pipe. Moving the piston leftward
reduces the air pressure next to it. As a result, first the elements nearest the
piston and then farther elements move leftward. Thus, the motion of the air and
the change in air pressure travel rightward along the pipe as a pulse.
If you push and pull on the piston in simple harmonic motion, as is being
done in Fig. 16-2, a sinusoidal wave travels along the pipe. Because the motion of
the elements of air is parallel to the direction of the wave’s travel, the motion
is said to be longitudinal,and the wave is said to be a longitudinal wave.In this
chapter we focus on transverse waves, and string waves in particular; in
Chapter 17 we focus on longitudinal waves, and sound waves in particular.
Both a transverse wave and a longitudinal wave are said to be traveling
wavesbecause they both travel from one point to another, as from one end of the
string to the other end in Fig. 16-1 and from one end of the pipe to the other end
in Fig. 16-2. Note that it is the wave that moves from end to end, not the material
(string or air) through which the wave moves.
Wavelength and Frequency
To completely describe a wave on a string (and the motion of any element along
its length), we need a function that gives the shape of the wave. This means that
we need a relation in the form
yh(x,t), (16-1)
in which yis the transverse displacement of any string element as a function hof
the time tand the position xof the element along the string. In general, a sinu-
soidal shape like the wave in Fig. 16-1bcan be described with hbeing either a sine
or cosine function; both give the same general shape for the wave. In this chapter
we use the sine function.
Sinusoidal Function.Imagine a sinusoidal wave like that of Fig. 16-1btraveling
in the positive direction of an xaxis. As the wave sweeps through succeeding ele-
ments (that is, very short sections) of the string, the elements oscillate parallel to the y
axis.At time t, the displacement yof the element located at position xis given by
y(x,t)y
msin(kxvt). (16-2)
Because this equation is written in terms of position x, it can be used to find the
displacements of all the elements of the string as a function of time. Thus, it can
tell us the shape of the wave at any given time.
v
:
Figure 16-2A sound wave is set up in an air-
filled pipe by moving a piston back and
forth. Because the oscillations of an ele-
ment of the air (represented by the dot) are
parallel to the direction in which the wave
travels, the wave is a longitudinal wave.
Air
v

44716-1TRANSVERSE WAVES
The names of the quantities in Eq. 16-2 are displayed in Fig. 16-3 and defined
next. Before we discuss them, however, let us examine Fig. 16-4, which shows five
“snapshots” of a sinusoidal wave traveling in the positive direction of an xaxis.
The movement of the wave is indicated by the rightward progress of the short
arrow pointing to a high point of the wave. From snapshot to snapshot, the short
arrow moves to the right with the wave shape, but the string moves onlyparallel
to the yaxis.To see that, let us follow the motion of the red-dyed string element at
xφ0. In the first snapshot (Fig. 16-4a), this element is at displacement yφ0.
In the next snapshot, it is at its extreme downward displacement because a valley
(or extreme low point) of the wave is passing through it. It then moves back up
throughyφ0. In the fourth snapshot, it is at its extreme upward displacement
because a peak(or extreme high point) of the wave is passing through it. In the
fifth snapshot, it is again at yφ0, having completed one full oscillation.
Amplitude and Phase
The amplitudey
mof a wave, such as that in Fig. 16-4 , is the magnitude of the
maximum displacement of the el