Today's Assignment (Oct. 3rd)
Open a word document and make your own notes from the information provided. Once you have collected the desired data, show it to Mr. Sklar. If he agrees that the task as been adequately completed, move on to the slinky exercise.
1. Define the following terms using the notes provided below.
boundary behavior
incident pulse
fixed end reflection
reflected pulse
boundary
inverted
free end reflection
transmitted pulse
2. Cut, paste, and label the diagram of fixed end reflection and free end reflection. What is the difference between the two. Answer in complete sentences and use the terminology as defined in question 1.
3. What happens to the wave length, frequency, and velocity as the wave moves to the more dense medium? as it moves from to a less dense medium?
4. With the slinkys, produce a wave that runs into a fixed end and record your observations and include a sketch. What is the amplitude of the fixed end as the wave reverses itself? How is this possible if a wave is moving through that point?
5. Obtain a small coil slinky and a large coil slinky and attach the two. Pass a wave from one slinky to the other and determine through observation which slinky is "more dense" and which is "less dense." Record your observations and include a sketch.
6. Copy the following questions into your notes accurately completing the sentence.
*7. With a slinky, or other pre-approved
material, try to create a free-ended wave as depicted in the notes.
| A:
A pulse in a more dense medium is traveling towards the boundary with a
less dense medium.
i) The reflected pulse in medium 1 ________
(will, will not) be inverted because _______. B: A pulse in a less dense medium is traveling towards the boundary with a more dense medium.
i) The reflected pulse in medium 2 ________ (will, will not) be
inverted because _____________. |
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 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.
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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.
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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.
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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.