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[ Pobierz całość w formacie PDF ]Chapter 12 Wave Motion
"Query 17. If a stone be thrown into stagnating water, the waves excited thereby
continue some time to arise in the place where the stone fell into the water, and are
propagated from thence in concentric circles upon the surface of the water to great
distances." Isaac Newton
12.1 Introduction
Everyone has observed that when a rock is thrown into a pond of water, waves are produced that move out from
the point of the disturbance in a series of concentric circles.
The
wave
is a propagation of the disturbance through
the medium without any net displacement of the
medium.
In this case the rock hitting the water
initiates the disturbance and the water is the
medium through which the wave travels. Of the
many possible kinds of waves, the simplest to
understand, and the one that we will analyze, is
the wave that is generated by an object executing
simple harmonic motion. As an example,
consider the mass
m
executing simple harmonic
motion in figure 12.1. Attached to the right of
m
is a very long spring. The spring is so long that it
is not necessary to consider what happens to the
spring at its far end at this time. When the mass
m
is pushed out to the position
x
=
A,
the portion
of the spring immediately to the right of
A
is
compressed. This
compression
exerts a force on
the portion of the spring immediately to its right,
thereby compressing it. It in turn compresses
part of the spring to its immediate right. The
process continues with the compression moving
along the spring, as shown in figure 12.1. As the
mass
m
moves in simple harmonic motion to the
displacement
x
= −
A,
the spring immediately to
its right becomes elongated. We call the
elongation of the spring a
rarefaction;
it is the
converse of a compression. As the mass
m
returns to its equilibrium position, the
rarefaction moves down the length of the spring.
The combination of a compression and
rarefaction comprise part of a longitudinal wave.
A
longitudinal wave
is a wave in which the
particles of the medium oscillate in simple
Figure 12.1
Generation of a longitudinal wave.
harmonic motion parallel to the direction of the wave propagation.
The compressions and rarefactions propagate
down the spring, as shown in figure 12.1(f). The mass
m
in simple harmonic motion generated the wave and the
wave moves to the right with a velocity
v
. Every portion of the medium, in this case the spring, executes simple
harmonic motion around its equilibrium position. The medium oscillates back and forth with motion parallel to the
wave velocity. Sound is an example of a longitudinal wave.
Another type of wave, and one easier to visualize, is a transverse wave.
A
transverse wave
is a wave in
which the particles of the medium execute simple harmonic motion in a direction perpendicular to its direction of
propagation.
A transverse wave can be generated by a mass having simple harmonic motion in the vertical
direction, as shown in figure 12.2. A horizontal string is connected to the mass as shown. As the mass executes
simple harmonic motion in the vertical direction, the end of the string does likewise. As the end moves up and
down, it causes the particle next to it to follow suit. It, in turn, causes the next particle to move. Each particle
transmits the motion to the next particle along the entire length of the string. The resulting wave propagates in
the horizontal direction with a velocity
v
, while any one particle of the string executes simple harmonic motion in
the vertical direction. The
particle of the string is moving
perpendicular to the direction
of wave propagation, and is
not moving in the direction of
the wave.
Using figure 12.3, let us
now define the characteristics
of a transverse wave moving
in a horizontal direction.
The
displacement
of any
particle of the wave is the
displacement of that particle
from its equilibrium position
and is measured by the
vertical distance
y
.
Figure 12.2
A transverse wave.
The
amplitude
of the wave is the maximum value of the
displacement
and is denoted by
A
in figure 12.3.
The
wavelength
of a wave is the distance, in the direction of
propagation, in which the wave repeats itself
and is denoted by λ.
The
period
T of a wave is the time it takes for one complete
wave to pass a particular point.
The
frequency
f of a wave is defined as the number of waves
passing a particular point per second
. It is obvious from the
definitions that the frequency is the reciprocal of the period, that is,
f
=
1
(12.1)
T
Figure 12.3
Characteristics of a simple wave.
The speed of propagation of the wave is the distance the wave travels in unit time. Because a wave of one
wavelength passes a point in a time of one period, its speed of propagation is
v
=
distance traveled
=
λ
(12.2)
time
T
Using equation 12.1, this becomes
v
= λ
f
(12.3)
Equation 12.3 is the fundamental equation of wave propagation.
It relates the speed of the wave to its wavelength
and frequency.
Example 12.1
Wavelength of sound.
The human ear can hear sounds from a low of 20.0 Hz up to a maximum frequency of about
20,000 Hz. If the speed of sound in air at a temperature of 0
0
C is 331 m/s, find the wavelengths associated with
these frequencies.
Solution
The wavelength of a sound wave, determined from equation 12.3, is
λ =
v
f
= 331 m/s
20.0 cycles/s
= 16.6 m
λ =
v
f
=
331 m/s
20,000 cycles/s
= 0.0166 m
To go to this Interactive Example click on this sentence.
The types of waves we consider in this chapter are called mechanical waves.
The wave causes a transfer of
energy from one point in the medium to another point in the medium without the actual transfer of matter between
these points.
Another type of wave, called an electromagnetic wave, is capable of traveling through empty space
without the benefit of a medium. This type of wave is extremely unusual in this respect and we will treat it in
more detail in chapters 25 and 29.
12.2 Mathematical Representation of a Wave
The simple wave shown in figure 12.3 is a picture of a transverse wave in a string at a particular time, let us say
at
t
= 0. The wave can be described as a sine wave and can be expressed mathematically as
y
=
A
sin
x
(12.4)
The value of
y
represents the displacement of the string at every position
x
along the string, and
A
is the
maximum displacement, and is called the amplitude of the wave. Equation 12.4 is plotted in figure 12.4. We see
that the wave repeats itself for
x
= 360
0
= 2π rad. Also plotted
in figure 12.4 is
y
=
A
sin 2
x
and
y
=
A
sin 3
x
. Notice from
the figure that
y
=
A
sin 2
x
repeats itself twice in the same
interval of 2π that
y
=
A
sin
x
repeats itself only once. Also
note that
y
=
A
sin 3
x
repeats
itself three times in that same
interval of 2π. The wave
y
=
A
sin
kx
would repeat itself
k
times in the interval of 2π. We
call the space interval in which
y
=
A
sin
x
repeats itself its
wavelength, denoted by λ
1
.
Thus, when
x
= λ
1
= 2π, the
wave starts to repeat itself.
The wave represented by
y
=
A
sin 2
x
repeats itself for 2
x
= 2π,
Figure 12.4
Plot of
A
sin
x, A
sin 2
x
, and
A
sin 3
x
.
and hence its wavelength is
λ
2
=
x
= 2π = π
2
The wave
y
=
A
sin 3
x
repeats itself when 3
x
= 2π, hence its wavelength is
λ
3
=
x
=
2π
3
Using this notation any wave can be represented as
y
=
A
sin
kx
(12.5)
where
k
is a number, called the
wave number.
The wave repeats itself whenever
kx
= 2π (12.6)
Because the value of
x
for a wave to repeat itself is its wavelength λ, equation 12.6 can be written as
k
λ = 2π (12.7)
We can obtain the wavelength λ from equation 12.7 as
λ =
2π
(12.8)
k
Note that equation 12.8 gives the wavelengths in figure 12.4 by letting
k
have the values 1, 2, 3, and so forth, that
λ
1
=
2π
1
λ
2
=
2π
2
λ
3
=
2π
3
λ
4
=
2π
4
We observe that the wave number k is the number of waves contained in the interval of 2
π
.
We can express the wave
number
k
in terms of the wavelength λ by rearranging equation 12.8 into the form
k
=
2π
(12.9)
λ
Note that in order for the units to be consistent, the wave number must have units of m
−
1
. The quantity
x
in
equation 12.5 represents the location of any point on the string and is measured in meters. The quantity
kx
in
equation 12.5 has the units (m
−
1
m = 1) and is thus a dimensionless quantity and represents an angle measured in
radians. Also note that the wave number
k
is a different quantity than the spring constant
k,
discussed in chapter
Equation 12.5 represents a snapshot of the wave at
t
= 0. That is, it gives the displacement of every
particle of the string at time
t
= 0. As time passes, this wave, and every point on it, moves. Since each particle of
the string executes simple harmonic motion in the vertical, we can look at the particle located at the point
x
= 0
and see how that particle moves up and down with time. Because the particle executes simple harmonic motion in
the vertical, it is reasonable to represent the displacement of the particle of the string at any time
t
as
y
=
A
sin ω
t
(12.10)
just as a simple harmonic motion on the
x
-axis was represented as
x
=
A
cos ω
t
in chapter 11. The quantity
t
is the
time and is measured in seconds, whereas the quantity ω is an angular velocity or an angular frequency and is
measured in radians per second. Hence the quantity ω
t
represents an angle measured in radians. The
displacement
y
repeats itself when
t
=
T,
the period of the wave. Since the sine function repeats itself when the
argument is equal to 2π, we have
ω
T
= 2π (12.11)
The period of the wave is thus
T
=
2π
ω
but the period of the wave is the reciprocal of the frequency. Therefore,
T
=
1 =
2π
f
ω
Solving for the angular frequency ω, in terms of the frequency
f,
we get
ω = 2π
f
(12.12)
Notice that the wave is periodic in both space and time. The space period is represented by the wavelength
λ
, and the
time by the period T.
Equation 12.5 represents every point on the string at
t
= 0, while equation 12.10 represents the point
x
= 0
for every time
t
. Obviously the general equation for a wave must represent every point
x
of the wave at every time
t
. We can arrange this by combining equations 12.5 and 12.10 into the one equation for a wave given by
y
=
A
sin(
kx
− ω
t
) (12.13)
The reason for the minus sign for ω
t
is explained below. We can find the relation between the wave number
k
and
the angular frequency ω by combining equations 12.7 and 12.11 as
k
λ = 2π (12.7)
ω
T
= 2π (12.11)
Thus,
ω
T
=
k
λ
ω =
k
λ
T
However, the wavelength λ, divided by the period
T
is equal to the velocity of propagation of the wave
v
, equation
12.2. Therefore, the angular frequency becomes
ω =
kv
(12.14)
Now we can write equation 12.13 as
y
=
A
sin(
kx
−
kv
t) (12.15)
y
=
A
sin
k
(
x
−
vt
) (12.16)
The minus sign before the velocity
v
determines the direction of propagation of the wave. As an example,
consider the wave
y
1
=
A
sin
k
(
x
−
vt
) (12.17)
We will now see that this is the equation of a wave traveling to the right with a speed
v
at any time
t
. A little later
in time, ∆
t,
the wave has moved a distance ∆
x
to the right such that the same point of the wave now has the
coordinates
x
+ ∆
x
and
t
+ ∆
t,
figure 12.5(a). Then we represent the wave as
y
2
=
A
sin
k
[(
x
+ ∆
x
) −
v
(
t
+ ∆
t
)]
y
2
=
A
sin
k
[(
x
−
vt
) + ∆
x
−
v
∆
t
] (12.18)
If this equation for
y
2
is to represent the same wave as
y
1
, then
y
2
must be equal to
y
1
. It is clear from equations
12.18 and 12.17 that if
v
=
∆
x
(12.19)
∆
t
the velocity of the wave to the right, then
∆
x
−
v
∆
t
= ∆
x
−
∆
x
∆
t
= 0
∆
t
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