Vibrations and Waves - Lesson 3 - 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 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.

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.

### We Would Like to Suggest ...

Why just read about it and when you could be interacting with it? Interact - that's exactly what you do when you use one of The Physics Classroom's Interactives. We would like to suggest that you combine the reading of this page with the use of either our Boundary Behavior of Waves Interactive or our Slinky Lab Interactive. You can find it in the Physics Interactives section of our website.

### 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 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.

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