As a Pulse Travels Along a Rope, the Pulse Loses Energy and Its Amplitude Does What?

Boundary Behavior

As a moving ridge travels through a medium, it will oftentimes reach the finish of the medium and meet an obstacle or perhaps another medium through which it could travel. Ane case of this has already been mentioned in Lesson 2. A audio wave is known to reverberate off canyon walls and other obstacles to produce an echo. A sound wave traveling through air within a canyon reflects off the coulee wall and returns to its original source. What touch on does reflection have upon a moving ridge? Does reflection of a wave bear upon the speed of the wave? Does reflection of a moving ridge affect the wavelength and frequency of the wave? Does reflection of a moving ridge touch on the aamplitude of the moving ridge? Or does reflection impact other properties and characteristics of a wave's movement? The behavior of a wave (or pulse) upon reaching the end of a medium is referred to equally boundary beliefs . When i medium ends, another medium begins; the interface of the two media is referred to equally the purlieus and the behavior of a wave at that purlieus is described every bit its boundary beliefs. The questions that are listed in a higher place are the types of questions nosotros seek to answer when we investigate the purlieus behavior of waves.

Stock-still Stop Reflection

Starting time consider an rubberband rope stretched from end to cease. One end will be securely attached to a pole on a lab bench while the other finish will be held in the hand in social club to introduce pulses into the medium. Because the correct finish of the rope is attached to a pole (which is attached to a lab bench) (which is fastened to the floor that is attached to the building that is fastened 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 cease of the rope, it will travel through the rope towards the correct end of the medium. This pulse is called the incident pulse since information technology is incident towards (i.e., budgeted) the purlieus with the pole. When the incident pulse reaches the boundary, two things occur:

• A portion of the free energy carried by the pulse is reflected and returns towards the left stop 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 free energy carried by the pulse is transmitted to the pole, causing the pole to vibrate.

Because the vibrations of the pole are non visibly obvious, the energy transmitted to it is not typically discussed. The focus of the discussion volition exist on the reflected pulse. What characteristics and properties could describe its movement?

 

When one observes the reflected pulse off the stock-still end, there are several notable observations. First the reflected pulse is inverted . That is, if an up displaced pulse is incident towards a stock-still end boundary, it will reverberate and return as a downwardly displaced pulse. Similarly, if a downward displaced pulse is incident towards a stock-still end boundary, it volition reverberate and return equally 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 cease of a medium ("medium A"), the last particle of the medium A receives an upwards displacement. This particle is attached to the first particle of the other medium ("medium B") on the other side of the purlieus. Every bit the concluding particle of medium A pulls upwards on the offset particle of medium B, the first particle of medium B pulls downwards on the final particle of medium A. This is merely Newton's third law of activity-reaction. For every action, at that place is an equal and opposite reaction. The upwardly pull on the first particle of medium B has little result upon this particle due to the large mass of the pole and the lab bench to which it is fastened. The effect of the downwards pull on the last particle of medium A (a pull that is in turn transmitted to the other particles) results in causing the upward deportation to go a down 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 demote relative to the rope that causes the rope to get inverted upon interacting with the wall. When two media collaborate past 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 land of move.

Other notable characteristics of the reflected pulse include:

• The speed of the reflected pulse is the same equally the speed of the incident pulse.
• The wavelength of the reflected pulse is the same as the wavelength of the incident pulse.
• The aamplitude 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, ii pulses in the same medium will have the same speed. A similar line of reasoning explains why the incident and reflected pulses accept the same wavelength. Every particle within the rope volition take the aforementioned frequency. Being connected to ane another, they must vibrate at the same frequency. Since the wavelength of a wave depends upon the frequency and the speed, ii waves having the aforementioned frequency and the same speed must also take the same wavelength. Finally, the amplitude of the reflected pulse is less than the amplitude of the incident pulse since some of the free energy of the pulse was transmitted into the pole at the boundary. The reflected pulse is carrying less free energy abroad from the purlieus compared to the energy that the incident pulse carried towards the boundary. Since the aamplitude 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 photograph shows an up displaced pulse traveling from the left end of a moving ridge auto towards the correct terminate. 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

At present consider what would happen if the stop of the rope were free to motility. Instead of being deeply fastened to a lab pole, suppose it is attached to a ring that is loosely fit around the pole. Because the right cease of the rope is no longer secured to the pole, the concluding particle of the rope will be able to move when a disturbance reaches it. This end of the rope is referred to equally a gratis end .

Again if a pulse is introduced at the left end of the rope, it will travel through the rope towards the correct end of the medium. When the incident pulse reaches the end of the medium, the last particle of the rope tin can no longer interact with the kickoff particle of the pole. Since the rope and pole are no longer attached and interconnected, they volition slide past each other. So when a crest reaches the stop of the rope, the last particle of the rope receives the aforementioned upward deportation; simply now there is no adjoining particle to pull downwardly upon the last particle of the rope to cause it to exist inverted. The result is that the reflected pulse is not inverted. When an upwardly displaced pulse is incident upon a free end, it returns equally an up displaced pulse after reflection. And when a downward displaced pulse is incident upon a gratis stop, it returns every bit a downward displaced pulse subsequently reflection. Inversion is not observed in free end reflection.

Scout It!

A pulse is introduced into the left terminate of a wave machine. The incident pulse is displaced upwardly. When it reaches the right end, it reflects back. The reflected pulse is not inverted. It is besides displaced upward.

The above discussion of gratuitous end and fixed terminate reflection focuses upon the reflected pulse. As was mentioned, the transmitted portion of the pulse is difficult to find when information technology is transmitted into a pole. But what if the original medium were attached to another rope with dissimilar 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 Beyond a Boundary from Less to More Dense

Allow'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 past the person belongings the cease of the thin rope. If this is the example, there volition exist an incident pulse traveling in the less dense medium (the thin rope) towards the boundary with a more than 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 stop of the thin rope. The disturbance that returns to the left after bouncing off the boundary is known equally 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 concluding particle of the less dense medium. This causes an upward displaced pulse to become a downwards displaced pulse. The more dumbo medium on the other paw was at rest prior to the interaction. The offset particle of this medium receives an upwardly pull when the incident pulse reaches the boundary. Since the more dense medium was originally at residual, an upward pull can do aught merely cause an up 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, whatever change in their state of motion would be in the aforementioned 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 tin can also exist made between the characteristics of the transmitted pulse and those of the reflected pulse. Again there are several noteworthy characteristics.

• The transmitted pulse (in the more dumbo medium) is traveling slower than the reflected pulse (in the less dumbo 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 equally the speed and the wavelength of the incident pulse.

One goal of physics is to employ physical models and ideas to explicate the observations made of the physical world. So how can these three characteristics be explained? Showtime recall from Lesson 2 that the speed of a wave is dependent upon the properties of the medium. In this example, the transmitted and reflected pulses are traveling in two distinctly different media. Waves ever travel fastest in the to the lowest degree dense medium. Thus, the reflected pulse will exist traveling faster than the transmitted pulse. 2nd, particles in the more than dense medium will be vibrating with the same frequency as particles in the less dumbo medium. Since the transmitted pulse was introduced into the more than dense medium past the vibrations of particles in the less dense medium, they must be vibrating at the same frequency. Then the reflected and transmitted pulses have the unlike speeds but the aforementioned frequency. Since the wavelength of a moving ridge depends upon the frequency and the speed, the wave with the greatest speed must also accept the greatest wavelength. Finally, the incident and the reflected pulse share the same medium. Since the 2 pulses are in the same medium, they will take the same speed. Since the reflected pulse was created past the vibrations of the incident pulse, they will have the same frequency. And ii waves with the aforementioned speed and the same frequency must besides have the same wavelength.

Flickr Physics Photo

A wave machine is used to demonstrate the beliefs of a wave at a purlieus.
Height: 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.
Eye: At the boundary, both reflection and transmission occur.
BOTTOM: The reflected pulse is inverted and of nigh the same length (though a smaller amplitude) equally the incident pulse. The transmitted pulse is shorter and slower than the incident and transmitted pulse.

Transmission of a Pulse Across a Purlieus from More to Less Dense

Finally, let'due south consider a thick rope fastened to a thin rope, with the incident pulse originating in the thick rope. If this is the case, there will exist an incident pulse traveling in the more dense medium (thick rope) towards the boundary with a less dumbo medium (thin rope). Once again there will be partial reflection and partial transmission at the boundary. The reflected pulse in this state of affairs volition non exist 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 upshot is that an upward displaced pulse incident towards the boundary volition reflect as an upwardly displaced pulse. For the same reasons, a down displaced pulse incident towards the purlieus will reflect every bit 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 dumbo 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 aforementioned logic as used higher up.

Flickr Physics Photograph

A wave machine is used to demonstrate the behavior of a wave at a boundary.
Superlative: An incident pulse is introduced into the left end of the wave machine. Information technology travels through the more than dense medium until it reaches the boundary with a less dense medium.
Eye: At the boundary, both reflection and transmission occur.
BOTTOM: The reflected pulse is NOT inverted and of nigh 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 dumbo 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 moving ridge 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.

We Would Like to Suggest ...

Why just read about it and when you could be interacting with it? Interact - that'south exactly what you do when you utilize ane of The Physics Classroom'south Interactives. Nosotros would like to suggest that y'all combine the reading of this page with the use of our Slinky Lab Interactive. You tin observe it in the Physics Interactives section of our website. The Slinky Lab provides the learner with a elementary surroundings for exploring free- and fixed-end reflection of incident pulses.

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 i ________ (will, will not) be inverted because _______.

2. The speed of the transmitted pulse will be ___________ (greater than, less than, the aforementioned as) the speed of the incident pulse.

3. The speed of the reflected pulse volition be ______________ (greater than, less than, the same as) the speed of the incident pulse.

4. The wavelength of the transmitted pulse will exist ___________ (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.

Instance 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 ________ (volition, volition not) be inverted because _____________.

7. The speed of the transmitted pulse will exist ___________ (greater than, less than, the same as) the speed of the incident pulse.

8. The speed of the reflected pulse volition exist ______________ (greater than, less than, the same every bit) the speed of the incident pulse.

9. The wavelength of the transmitted pulse will be ___________ (greater than, less than, the aforementioned as) the wavelength of the incident pulse.

ten. The frequency of the transmitted pulse will be ___________ (greater than, less than, the same as) the frequency of the incident pulse.

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