As a wave travels through a medium, it will often
reach the end of the medium and encounter an obstacle or
perhaps another medium through which it could travel. One
example of this has
already been mentioned in Lesson 2. A sound wave is known
to reflect off canyon walls and other obstacles to
produce an echo. A sound wave traveling through air
within a canyon reflects off the canyon wall and returns
to its original source. What effect does reflection have
upon a wave? Does reflection of a wave effect the speed
of the wave? Does reflection of a wave effect the
wavelength and frequency of the wave? Does reflection of
a wave effect the amplitude of the wave? Or does
reflection effect other properties and characteristics of
a wave's motion? The behavior of a wave (or pulse) upon
reaching the end of a medium is referred to as
boundary behavior.
When one medium ends, another medium begins; the
interface of the two media is referred to as the
boundary and the
behavior of a wave at that boundary is described as its
boundary behavior. The questions which are listed above
are the types of questions we seek to answer when we
investigate the boundary behavior of waves.
First
consider an elastic rope stretched from end to end. One
end will be securely attached to a pole on a lab bench
while the other end will be held in hand in order to
introduce pulses into the medium. Because the right end
of the rope is attached to a pole (which is attached to a
lab bench), the last particle of the rope will be
unable to move when a disturbance reaches it. This end of
the rope is referred to as a
fixed end.
If a pulse is introduced at the left
end of the rope, it will travel through the rope towards
the right end of the medium. This pulse is called the
incident pulse since
it is incident towards (i.e., approaching) the boundary
with the pole. When the incident pulse reaches the
boundary, two things occur:
A portion of the energy carried by the pulse is
reflected and returns towards the left end of the
rope. The disturbance which returns to the left after
bouncing off the pole is known as the
reflected
pulse.
A portion of the energy carried by the pulse is
transmitted to the pole, causing the pole to
vibrate.
Because the vibrations of the pole are not visibly
obvious, the energy transmitted to it is not typically
discussed. The focus of the discussion will be the
reflected pulse. What characteristics and properties
could describe its motion?
One observes the reflected pulse off
the fixed end, there are several notable observations.
First the reflected pulse is
inverted. That is, if
a crest is incident towards a fixed end boundary, it will
reflect and return as a trough. Similarly, if a trough is
incident towards a fixed end boundary, it will reflect
and return as a crest.
The inversion of the reflected pulse
can be explained by returning to our conceptions of the
nature of a mechanical wave. When a crest reaches the end
of a medium ("medium A"), the last particle of the medium
A receives an upward displacement. This particle is
attached to the first particle of the other medium
("medium B") on the other side of the boundary. As the
last particle of medium A pulls upwards on the first
particle of medium B, the first particle of medium B
pulls downwards on the last particle of medium A. This is
merely Newton's third
law of action-reaction. For every action, there is an
equal and opposite reaction. The upward pull on the first
particle of medium B has little effect upon this particle
due to the large mass of the pole and the lab bench to
which it is attached. The effect of the downward pull on
the last particle of medium A (a pull which is in turn
transmitted to the other particles) results in causing
the upward displacement to become a downward
displacement; that is, the crest becomes a pulse. It is
important to note that it is the heaviness of the
pole and the lab bench relative to the rope which causes
the rope to become inverted upon interacting with the
wall. When two media interact by exerting pushes and
poles upon each other, the least massive medium "wins the
interaction." Just like in arm wrestling, the medium
which loses receives a change in its state of motion.
Other notable
characteristics of the reflected pulse include:
the speed of the reflected pulse is the same as
the speed of the incident pulse
the wavelength of the reflected pulse is the same
as the wavelength of the incident pulse
the amplitude of the reflected pulse is less than
the amplitude of the incident pulse
Of course, it is not surprising that
the speed of the incident and reflected pulse are
identical since the two pulses are traveling in the same
medium. Since the speed of a wave (or pulse) is dependent
upon the medium through which it travels, two
pulses in the same medium will have the same speed. A
similar line of reasoning explains why the incident and
reflected pulses have the same wavelength. Every particle
within the rope will have the same frequency. Being
connected to one another, they must vibrate at the same
frequency. Since the wavelength of a wave depends upon
the frequency and the speed, two waves having the same
frequency and the same speed must also have the same
wavelength. Finally, the amplitude of the reflected pulse
is less than the amplitude of the incident pulse since
some of the energy of the pulse was transmitted into the
pole at the boundary. The reflected pulse is carrying
less energy away from the boundary compared to the energy
which the incident pulse carried towards the boundary.
Since the amplitude of a pulse is indicative of the
energy carried by the pulse, the reflected pulse has a
smaller amplitude than the incident pulse.
Now
consider what would happen if the end of the rope were
free to move. Instead of being securely attached to a lab
pole, suppose it is attached to a ring which is loosely
fit around the pole. Because the right end of the rope is
no longer secured to the pole, the last particle
of the rope will be able to move when a disturbance
reaches it. This end of the rope is referred to as a
free end.
Once more if a pulse is introduced at
the left end of the rope, it will travel through the rope
towards the right end of the medium. When the incident
pulse reaches the the end of the medium, the last
particle of the rope can no longer interact with the
first particle of the pole. Since the rope and pole are
no longer attached and interconnected, they will slide
past each other. So when a crest reaches the end of the
rope, the last particle of the rope receives the same
upward displacement; only now there is no adjoining
particle to pull downward upon the last particle of the
rope to cause it to be inverted. The result is that the
reflected pulse is not inverted. When a crest is incident
upon a free end, it returns as a crest after reflection;
and when a trough is incident upon a free end, it returns
as a trough after reflection. Inversion is not observed
in free end reflection.
The above discussion of free end and
fixed end reflection focuses upon the reflected pulse. As
was mentioned, the transmitted portion of the pulse is
difficult to observe when it is transmitted into a pole.
But what if the original medium were attached to another
rope with different properties? How could the reflected
pulse and transmitted pulse be described in situations in
which an incident pulse reflects off and transmits into a
second medium?
Let's consider a thin rope attached to
a thick rope, with each rope held at opposite ends by
people. And suppose that a pulse is introduced by the
person holding the end of the thin rope. If this is the
case, there will be an incident pulse traveling in the
less dense medium (thin rope) towards the boundary with a
more dense medium (thick rope).
Upon reaching the boundary, the usual
two behaviors will occur.
A portion of the energy carried by the incident
pulse is reflected and returns towards the left end of
the thin rope. The disturbance which returns to the
left after bouncing off the boundary is known as the
reflected
pulse.
A portion of the energy carried by the incident
pulse is transmitted into the thick rope. The
disturbance which continues moving to the right is
known as the transmitted
pulse.
The reflected pulse will be found to
be inverted in situations such as this. During the
interaction between the two media at the boundary, the
first particle of the more dense medium overpowers the
smaller mass of the last particle of the less dense
medium. This causes the crest to become a trough. The
more dense medium on the other hand was at rest prior to
the interaction. The first particle of this medium
receives an upward pull when the incident pulse reaches
the boundary. Since the more dense medium was originally
at rest, an upward pull can do nothing but cause an
upward displacement. For this reason, the transmitted
pulse is not inverted. In fact transmitted pulses can
never be inverted; since the particles in this medium are
originally at rest, any change in their state of motion
would be in the same direction as the displacement of the
particles of the incident pulse.
The "Before" and "After" snapshots of
the two media are shown in the diagram below.
Comparisons can also be made between
the characteristics of the transmitted pulse and those of
the reflected pulse. Once more there are several
noteworthy characteristics.
the transmitted pulse (in the more dense medium)
is traveling slower than the reflected pulse (in the
less dense medium)
the transmitted pulse (in the more dense medium)
has a smaller wavelength than the reflected pulse (in
the less dense medium)
the speed and the wavelength of the reflected
pulse are the same as the speed and the wavelength of
the incident pulse
The goal of
physics is to use physical models and ideas to explain
the observations made of the physical world. So how can
these three characteristics be explained? First recall
from Lesson 2 that the
speed of a wave is dependent upon the properties of the
medium. In this case, the transmitted and reflected pulse
are traveling in two distinctly different media. Waves
always travel fastest in the least dense medium. Thus,
the reflected pulse will be traveling faster than the
transmitted pulse. Second, particles in the more dense
medium will be vibrating with the same frequency as
particles in the less dense medium. Since the transmitted
pulse was introduced into the more dense medium by the
vibrations of particles in the less dense medium, they
must be vibrating at the same frequency. So the reflected
and transmitted pulses have the different speeds but the
same frequency. Since the wavelength of a wave depends
upon the frequency and the speed, the wave with the
greatest speed must also have the greatest wavelength.
Finally, for the incident and the reflected pulse share
the same medium. Since the two pulses are in the same
medium, they will have the same speed. Since the
reflected pulse was created by the vibrations of the
incident pulse, they will have the same frequency. And
two waves with the same speed and the same frequency,
must have the same wavelength.
Finally, let's consider a thick rope
attached to a thin rope, with the incident pulse
originating in the thick rope. If this is the case, there
will be an incident pulse traveling in the more dense
medium (thick rope) towards the boundary with a less
dense medium (thin rope). Once more, there will be
partial reflection and partial transmission at the
boundary. The reflected pulse in this situation will not
be inverted. Similarly, the transmitted pulse is not
inverted (as is always the case). Since the incident
pulse is in a heavier medium, when it reaches the
boundary, the first particle of the less dense medium is
not have sufficient mass to overpower the last particle
of the more dense medium. The result is that a crest
incident towards the boundary will reflect as a crest;
for the same reasons, a trough incident towards the
boundary will reflect as a trough.
The "Before" and "After" snapshots of
the two media are shown in the diagram below.
Comparisons between the
characteristics of the transmitted pulse and the
reflected pulse lead to the following observations.
the transmitted pulse (in the less dense medium)
is traveling faster than the reflected pulse (in the
more dense medium)
the transmitted pulse (in the less dense medium)
has a larger wavelength than the reflected pulse (in
the more dense medium)
the speed and the wavelength of the reflected
pulse are the same as the speed and the wavelength of
the incident pulse
These three observations are explained using the same
logic as used above.
The boundary behavior of waves can be
summarized by the following principles:
the wave speed is always greatest in the least
dense medium,
the wavelength is always greatest in the least
dense medium,
the frequency of a wave is not altered by crossing
a boundary,
the reflected pulse becomes inverted when a wave
in a less dense medium is heading towards a boundary
with a more dense medium,
the amplitude of the incident pulse is always
greater than the amplitude of the reflected
pulse.
All the observations discussed here can be explained
by the simple application of these principles. Take a few
moments to use these principles to answer the following
questions.
Check
Your Understanding
Case
1: A pulse in a more dense
medium is traveling towards the boundary with a less
dense medium.
1. The reflected pulse in medium
1 ________ (will, will not) be inverted because
_______.
2. The speed of the transmitted
pulse will be ___________ (greater than, less than, the
same as) the speed of the incident pulse.
3. The speed of the reflected
pulse will be ______________ (greater than, less than,
the same as) the speed of the incident pulse.
4. The wavelength of the
transmitted pulse will be ___________ (greater than, less
than, the same as) the wavelength of the incident
pulse.
5. The frequency of the
transmitted pulse will be ___________ (greater than, less
than, the same as) the frequency of the incident
pulse.
Case
2: A pulse in a less dense
medium is traveling towards the boundary with a more
dense medium.
6. The reflected pulse in medium 2 ________ (will,
will not) be inverted because _____________.
7. The speed of the transmitted pulse will be
___________ (greater than, less than, the same as) the
speed of the incident pulse.
8. The speed of the reflected pulse will be
______________ (greater than, less than, the same as) the
speed of the incident pulse.
9. The wavelength of the transmitted pulse will be
___________ (greater than, less than, the same as) the
wavelength of the incident pulse.
10. The frequency of the transmitted pulse will be
___________ (greater than, less than, the same as) the
frequency of the incident pulse.
