Chapter 11
Electromagnetic Waves
11.1 Mechanical Waves
11.1.1 Qualitative Definition of Mechanical Waves
The definition of a
wave: any disturbance of a system
which displaces the system from its equilibrium state and has
the characteristic that the disturbance can propagate
from one part of the system to another. This seems like a
quite general statement, and it is, but it can be reduced to
a quite rigorous mathematical treatment which allows us to
describe an incredibly large number of physical phenomena.
Sound, light, ocean waves, radio and television transmission,
earthquakes and more are all describable by the same mathematics
which we refer to as wave mechanics
. There are also
inherent properties of all wave phenomena which can be observed.
The most important of these is
interference, the behavior
of a system when two or more waves occupy the same space at the
same time.
For mechanical waves, we can be more restrictive in our definition
and state that these disturbances we referred to must propagate
in a medium, a material
substance which is deformable and
capable of transmitting a disturbance. Note that electromagnetic
waves do not need a medium - they travel through empty
space.
Types of Mechanical Waves
There are two kinds of disturbances of a medium that can
propagate. The first is one in which the displacements
of the medium are perpendicular or transverse to
the direction of travel of the disturbance. An example
of such a
transverse wave
is the kind you see when
you throw a rock into a pond. The other kind of wave is
one in which the particles of the medium move back and forth
along the direction the wave travels (e.g. sound waves
or a horizontal slinky in which one end is suddenly
pushed). This kind of wave is a longitudinal
wave. You see both kinds of waves in
figure 11.1.
Figure 11.1: Examples of a transverse and a longitudinal
wave.
Periodic Transverse Waves
For this course we will only discuss transverse waves
and, in particular, transverse waves which are
periodic. For such a wave, we can quickly
define terms for the features that distinguish it.
These terms are shown graphically in
figure 11.2.
Figure 11.2: A periodic transverse wave and terms used to
characterize it.
We have the following qualitative definitions. You can see
them visually with
this Java applet.
- amplitude:
- the magnitude of the maximum displacement
of the medium from its equilibrium position.
Generally denoted as A.
- wavelength:
- the distance between points on a wave that
have identical positions and time derivatives,
e.g. the distance between crests.
Generally denoted by the Greek letter l.
- period:
- the time necessary for a wave to completely
repeat its pattern. Generally denoted as
T. If the velocity of the wave is v,
then v = l/T.
- frequency:
- defined as the inverse of the period and
denoted by f. Therefore, f = 1/T. For
periodic waves with velocity v,
v = lf. All points on a periodic wave
oscillate with the same frequency.
- angular frequency:
- denoted by the Greek letter w.
We define w =
2pf, hence w = 2p/T.
- wave number:
- denoted by k, it is defined as k = 2p/l.
Note that for such a wave, any particular particle
will move transversely to the direction of
motion of the wave itself, i.e. the wave might
move with some velocity along a particular direction,
but the particles making up the medium will move
with simple harmonic motion back and forth along a
direction perpendicular to the velocity direction.
As an example of this, consider the correspondence
between uniform circular motion and transverse
sinusoidal motion shown in figure
11.3.
Figure 11.3: Correspondence between uniform circular
motion and a particle that traces out
motion for a tranverse sinusoidal wave.
11.1.2 Quantitative Description of Mechanical Waves
Wave Function for a Sinusoidal Wave
The correspondence between uniform circular motion and
transverse periodic wave motion leads to an immediate
mathematical description of a so-called
traveling wave,
i.e. a periodic transverse wave with velocity v. For the
wave as shown in figure 11.3,
we have a wave function
|
y(x, t) = A sinw |
æ
ç
è
|
t + |
x
v
|
+ f |
ö
÷
ø
|
= A sin2pf |
æ
ç
è
|
t + |
x
v
|
+ f |
ö
÷
ø
|
. |
| (11.1.2.1) |
for a wave moving in the -x direction. For a wave moving
in the +x direction, change the +x/v sign to -x/v
inside the parentheses. The angle f is a phase angle
and allows for a completely general description of the value of
the displacement from equilibrium at t = 0.
Note that this mathematical description
yields the position transverse to the wave direction
and relative to the equilibrium position of the medium.
We get a more convenient form for the traveling sinusoidal
wave if we make use of the wave number, k. Note that
So we can rewrite the expression for a wave moving in the
+x direction as
|
y(x, t) = A sin(wt - kx + f) |
| (11.1.2.3) |
We often refer to the
phase of a
wave. This is defined as
The Wave Equation
We have defined a function which we can easily verify
gives the position of a particle transverse to a traveling
wave. The position is a function of wave position and
time. We can find the first and second derivatives of this
transverse position in order to get the particle velocity
and particle acceleration transverse to the wave direction.
|
|
|
| |
|
|
|
¶y(x, t)
¶t
|
= wA cos(wt - kx + f) |
| |
|
|
¶2y(x,t)
¶t2
|
= -w2A sin(wt - kx + f) |
| (11.1.2.5) |
|
where we have used partial derivatives with respect to
time since the position is a function of position and
time.
Notice that for any given values of f, x,
and t, waiting a time interval of one period,
T = 2p/w, yields the same values for the
particle position, velocity, and acceleration. This is
consistent with our definition of period. Notice also
that the results for position, velocity, and acceleration
are the same as those for a particle in simple harmonic
motion with amplitude A and angular frequency w.
Now note something else. If we look at the second partial
derivative of the particle position with respect to x,
we have
|
|
¶2y(x, t)
¶x2
|
= -k2A sin(wt -kx) = -k2y(x, t) |
| (11.1.2.6) |
Using our derivation for the particle acceleration and the
relation w = v k gives
|
|
¶2y(x, t)/¶t2
¶2y(x, t)/¶x2
|
= |
-w2y(x, t)
-k2y(x, t)
|
= |
w2
k2
|
= v2 |
| (11.1.2.7) |
Hence we derive the
wave equation.
|
|
¶2y(x, t)
¶x2
|
= |
1
v2
|
|
¶2y(x, t)
¶t2
|
|
| (11.1.2.8) |
11.1.3 Energy and Power of a Traveling Wave
on a String
Kinetic and Potential Energy
Imagine a string with mass per unit length, m,
which is stretched along the x axis as in
figure 11.4. Any element
of the string with mass dm and length dx oscillates
transversely (along the y axis) in simple harmonic
motion as the wave passes its horizontal position. This
oscillation gives the string element a kinetic energy due
to imparting a velocity u. When the
element is at the y = 0 position, the velocity, and
therefore the kinetic energy, is at a maximum. When
the element is at its maximum displacement (either amplitude
location), the potential energy is maximum and the
kinetic energy is zero. The potential energy comes
from the fact that the wave must stretch the string
in order to displace a taut string from its equilibrium
position.
Figure 11.4: A taut string is stretched from its equilibrium
position at one point and then released. The
displacement travels from the point of stretch
and release as a traveling sinusoidal wave. Note
that stretching the string delivers potential energy
to the system. This energy is propagated, in the
form of kinetic and potential energy of the particles
making up the string, along with the wave.
The Rate of Energy Transmission
As the wave moves along the x direction, forces due
to tension in the string continuously do work to transfer
kinetic and potential energy from one part of the string
to another. The kinetic energy dK associated with
a string element dm is given by
where u is the transverse speed of the oscillating
string element. We use the velocity from
equation 5 to find u.
|
u = |
¶y(x, t)
¶t
|
= wA cos(wt - kx + f) |
| (11.1.3.10) |
In terms of the mass density, dm = m dx, we have
|
dK = |
1
2
|
(m dx)(wA)2cos2(wt - kx + f) |
| (11.1.3.11) |
Divide this quantity by dt to get the rate at which
kinetic energy is transported as
|
|
|
|
|
1
2
|
m |
dx
dt
|
w2A2cos2(wt - kx + f) |
| |
|
|
1
2
|
mvw2A2cos2(wt - kx +f). |
| (11.1.3.12) |
|
This is the instantaneous power transferred by the
transverse wave. The average rate of kinetic
energy transfer is
|
|
æ
ç
è
|
dK
dt
|
ö
÷
ø
|
avg
|
= |
1
2
|
mvw2A2[cos2(wt - kx + f)]avg. = |
1
4
|
mvw2A2 |
| (11.1.3.13) |
We will not present the proof here, but for simple
harmonic systems, the average total energy is just
twice the average kinetic energy (i.e. the average
potential energy equals the average kinetic energy).
Therefore, the average power transmitted by the
wave is
|
Pavg. = 2 |
æ
ç
è
|
dK
dt
|
ö
÷
ø
|
avg.
|
= |
1
2
|
mvw2A2. |
| (11.1.3.14) |
The average power depends on the square of the amplitude
and the square of the angular frequency. This result
is general and true for waves of all types.
By similar arguments, we can show that a traveling
wave also carries linear momentum. We shall explicitly
derive this for the case of electromagnetic waves.
11.1.4 Superposition of Mechanical Waves
Two or more waves can pass simultaneously through the
same region of space. We can ask what the result of
this is. Waves follow the
principle of superposition
which says that the net effect of the waves is the sum
of their individual effects. Overlapping waves do not
alter the travel of each other.
The mathematics for describing overlapping waves of the
same frequency and wavelength, we call this
interference, is not
difficult. For example, imagine
two transverse waves with the same amplitude, frequency,
direction, and wavelength traveling along a string.
We can specify different spatial positions for these waves
through different phase angles. For simplicity, we assume
the phase angle of one of these waves is zero as this does
not change the mathematics or physics of the description.
According to the principle of superposition, the overlap
of these two waves can be described as follows.
|
|
|
| |
|
|
Asin(wt - kx) + Asin(wt - kx + f) |
| |
|
| [2Acos |
1
2
|
f]sin(wt -kx + |
1
2
|
f). |
| (11.1.4.15) |
|
If we plot this equation, we see that it is also a
sinusoidal wave traveling in the +x direction.
This net wave is the only wave you would see
traveling along the string. The two interfering
waves would not be visible. We note that by changing
the value of f we can get a resultant wave which
shows constructive
interference
or destructive interference.
For completely constructive interference,
we want crests of one wave to coincide with crests of
the other and troughs to coincide with troughs. For
completely destructive interference, we want crests of
one wave to coincide with troughs of the other, thereby
leading to complete cancellation of the net wave.
We see examples of these cases in
figure 11.5.
Figure 11.5: Two waves with the same amplitude and frequency
but independent phase angles are depicted in blue.
The top curve has phase angle equal to zero and the
bottom curve has the phase angle set as shown by the
label of the graph. The red curve is the result of
superposing the two waves.
Send comments to larryg@upenn5.hep.upenn.edu.
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