Behavior of Waves
Boundary Behavior
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 affect does
reflection have upon a wave? Does reflection of a wave affect the speed of the
wave? Does reflection of a wave affect the wavelength and frequency of the
wave? Does reflection of a wave affect the amplitude of the wave? Or does
reflection affect 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 that are listed above are the types of questions we seek to
answer when we investigate the boundary behavior of
waves.
Fixed
End Reflection
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 the 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) (which is
attached to the floor that is attached to the building that is attached to the
Earth), 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 that 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 on the reflected pulse. What characteristics and properties
could describe its motion?
When one observes the reflected pulse off the fixed end,
there are several notable observations. First the reflected pulse is inverted. That is, if an upward displaced pulse is incident towards a fixed end
boundary, it will reflect and return as a downward displaced pulse. Similarly,
if a downward displaced pulse is incident towards a fixed end boundary, it will
reflect and return as an upward displaced pulse.
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 that is in turn transmitted to the other
particles) results in causing the upward displacement to become a downward
displacement. The upward displaced incident pulse thus returns as a downward
displaced reflected pulse. It is important to note that it is the heaviness of the
pole and the lab bench relative to the rope that causes the rope to become
inverted upon interacting with the wall. When two media interact by exerting
pushes and pulls upon each other, the most massive medium wins the
interaction. Just like in arm wrestling, the medium that
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 that 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.
Flickr Physics Photo
This sequence photography photo shows an upward
displaced pulse traveling from the left end of a wave machine towards the right
end. The right end is held tightly; it is a fixed end. The wave reflects off
this fixed end and returns as a downward displaced pulse. Reflection off a
fixed end results in inversion.
Free
End Reflection
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 that 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 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 an
upward displaced pulse is incident upon a free end, it returns as an upward
displaced pulse after reflection. And when a downward displaced pulse is
incident upon a free end, it returns as a downward displaced pulse after
reflection. Inversion is not observed in free end reflection.
Watch It!
A pulse is introduced into the left end of a
wave machine. The incident pulse is displaced upward. When it reaches the right
end, it reflects back. The reflected pulse is not
inverted. It is also displaced upward.
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?
Transmission
of a Pulse Across a Boundary from Less to
More Dense
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 (the thin
rope) towards the boundary with a more dense medium
(the 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 that 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 that 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 an upward displaced pulse to become a downward displaced pulse.
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 densemedium 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.
One 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 pulses 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, 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 also have the same wavelength.
Flickr Physics Photo
A wave machine is used to demonstrate the behavior of a
wave at a boundary.
TOP: An incident pulse is introduced into the right end
of the wave machine. It travels through the less dense medium until it reaches
the boundary with a more dense medium.
MIDDLE: At the boundary, both reflection and transmission
occur.
BOTTOM: The reflected pulse is inverted and of about
the same length (though a smaller amplitude) as the incident pulse. The
transmitted pulse is shorter and slower than the incident and transmitted pulse.
Transmission
of a Pulse Across a Boundary from More to
Less Dense
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 again 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 does not have sufficient
mass to overpower the last particle of the more dense medium.
The result is that an upward displaced pulse incident towards the boundary will
reflect as an upward displaced pulse. For the same reasons, a downward
displaced pulse incident towards the boundary will reflect as a downward
displaced pulse.
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.
Flickr Physics Photo
A wave machine is used to demonstrate the behavior of a
wave at a boundary.
TOP: An incident pulse is introduced into the left end
of the wave machine. It travels through the more
dense medium until it reaches the boundary
with a less dense medium.
MIDDLE: At the boundary, both reflection and
transmission occur.
BOTTOM: The reflected pulse is NOT inverted and of
about the same length (though a smaller amplitude) as the incident pulse. The
transmitted pulse is longer and faster than the incident and transmitted pulse.
The boundary behavior of
waves in ropes can be summarized by the following principles:
· The wave
speed is always greatest in the least dense rope.
· The
wavelength is always greatest in the least dense rope.
· The
frequency of a wave is not altered by crossing a boundary.
· The
reflected pulse becomes inverted when a wave in a less dense rope is heading
towards a boundary with a more dense rope.
· 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.
1. will not... because the
reflection occurs for a wave in a more dense medium heading towards a less
dense medium.
2. faster
3. the same as
4. greater than
5. the same as
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 1 ________ (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.
6. will... because the reflection occurs for a wave in a less
dense medium heading towards a more dense medium.
7. less than
8. the same as
9. less than
10. the same as