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Waves - Lesson 3 - Behavior of Waves

Boundary Behavior

Boundary Behavior

Reflection, Refraction, and Diffraction

Interference of Waves

The Doppler Effect

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.

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 to the right.

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.

Reflection, Refraction, and Diffraction

Previously in Lesson 3, the behavior of waves traveling along a rope from a more dense medium to a less dense medium (and vice versa) was discussed. The wave doesn't just stop when it reaches the end of the medium. Rather, a wave will undergo certain behaviors when it encounters the end of the medium. Specifically, there will be some reflection off the boundary and some transmission into the new medium. But what if the wave is traveling in a two-dimensional medium such as a water wave traveling through ocean water? Or what if the wave is traveling in a three-dimensional medium such as a sound wave or a light wave traveling through air? What types of behaviors can be expected of such two- and three-dimensional waves?

The study of waves in two dimensions is often done using a ripple tank. A ripple tank is a large glass-bottomed tank of water that is used to study the behavior of water waves. A light typically shines upon the water from above and illuminates a white sheet of paper placed directly below the tank. A portion of light is absorbed by the water as it passes through the tank. A crest of water will absorb more light than a trough. So the bright spots represent wave troughs and the dark spots represent wave crests. As the water waves move through the ripple tank, the dark and bright spots move as well. As the waves encounter obstacles in their path, their behavior can be observed by watching the movement of the dark and bright spots on the sheet of paper. Ripple tank demonstrations are commonly done in a Physics class in order to discuss the principles underlying the reflection, refraction, and diffraction of waves.

Reflection of Waves

If a linear object attached to an oscillator bobs back and forth within the water, it becomes a source of straight waves. These straight waves have alternating crests and troughs. As viewed on the sheet of paper below the tank, the crests are the dark lines stretching across the paper and the troughs are the bright lines. These waves will travel through the water until they encounter an obstacle - such as the wall of the tank or an object placed within the water. The diagram at the right depicts a series of straight waves approaching a long barrier extending at an angle across the tank of water. The direction that these wavefronts (straight-line crests) are traveling through the water is represented by the blue arrow. The blue arrow is called a ray and is drawn perpendicular to the wavefronts. Upon reaching the barrier placed within the water, these waves bounce off the water and head in a different direction. The diagram below shows the reflected wavefronts and the reflected ray. Regardless of the angle at which the wavefronts approach the barrier, one general law of reflection holds true: the waves will always reflect in such a way that the angle at which they approach the barrier

equals the angle at which they reflect off the barrier. This is known as the law of reflection. This law will be discussed in more detail in Unit 13 of The Physics Classroom.

The discussion above pertains to the reflection of waves off of straight surfaces. But what if the surface is curved, perhaps in the shape of a parabola? What generalizations can be made for the reflection of water waves off parabolic surfaces? Suppose that a rubber tube having the shape of a parabola is placed within the water. The diagram at the right depicts such a parabolic barrier in the ripple tank. Several wavefronts are approaching the barrier; the ray is drawn for these wavefronts. Upon reflection off the parabolic barrier, the water waves will change direction and head towards a point. This is depicted in the diagram below. It is as though all the energy being carried by the water waves is converged at a single point - the point is known as the focal point. After passing through the focal point, the waves spread out through the water. Reflection of waves off of curved surfaces will be discussed in more detail in Unit 13 of The Physics Classroom.

Refraction of Waves

Reflection involves a change in direction of waves when they bounce off a barrier. Refraction of waves involves a change in the direction of waves as they pass from one medium to another. Refraction, or the bending of the path of the waves, is accompanied by a change in speed and wavelength of the waves. In Lesson 2, it was mentioned that the speed of a wave is dependent upon the properties of the medium through which the waves travel. So if the medium (and its properties) is changed, the speed of the waves is changed. The most significant property of water that would affect the speed of waves traveling on its surface is the depth of the water. Water waves travel fastest when the medium is the deepest. Thus, if water waves are passing from deep water into shallow water, they will slow down. And as mentioned in the previous section of Lesson 3, this decrease in speed will also be accompanied by a decrease in wavelength. So as water waves are transmitted from deep water into shallow water, the speed decreases, the wavelength decreases, and the direction changes.

This boundary behavior of water waves can be observed in a ripple tank if the tank is partitioned into a deep and a shallow section. If a pane of glass is placed in the bottom of the tank, one part of the tank will be deep and the other part of the tank will be shallow. Waves traveling from the deep end to the shallow end can be seen to refract (i.e., bend), decrease wavelength (the wavefronts get closer together), and slow down

(they take a longer time to travel the same distance). When traveling from deep water to shallow water, the waves are seen to bend in such a manner that they seem to be traveling more perpendicular to the surface. If traveling from shallow water to deep water, the waves bend in the opposite direction. The refraction of light waves will be discussed in more detail in a later unit of The Physics Classroom.

Diffraction of Waves

Reflection involves a change in direction of waves when they bounce off a barrier; refraction of waves involves a change in the direction of waves as they pass from one medium to another; and diffraction involves a change in direction of waves as they pass through an opening or around a barrier in their path. Water waves have the ability to travel around corners, around obstacles and through openings. This ability is most obvious for water waves with longer wavelengths. Diffraction can be demonstrated by placing small barriers and obstacles in a ripple tank and observing the path of the water waves as they encounter the obstacles. The waves are seen to pass around the barrier into the regions behind it; subsequently the water behind the barrier is disturbed. The amount of diffraction (the sharpness of the bending) increases with increasing wavelength and decreases with decreasing wavelength. In fact, when the wavelength of the waves is smaller than the obstacle, no noticeable diffraction occurs.

Diffraction of water waves is observed in a harbor as waves bend around small boats and are found to disturb the water behind them. The same waves however are unable to diffract around larger boats since their wavelength is smaller than the boat. Diffraction of sound waves is commonly observed; we notice sound diffracting around corners, allowing us to hear others who are speaking to us from adjacent rooms. Many forest-dwelling birds take advantage of the diffractive ability of long-wavelength sound waves. Owls for instance are able to communicate across long distances due to the fact that their long-wavelength hoots are able to diffract around forest trees and carry farther than the short-wavelength tweets of songbirds. Diffraction is observed of light waves but only when the waves encounter obstacles with extremely small wavelengths (such as particles suspended in our atmosphere). Diffraction of sound waves and of light waves will be discussed in a later unit of The Physics Classroom Tutorial.

Reflection, refraction and diffraction are all boundary behaviors of waves associated with the bending of the path of a wave. The bending of the path is an observable behavior when the medium is a two- or three-dimensional medium. Reflection occurs when there is a bouncing off of a barrier. Reflection of waves off straight barriers follows the law of reflection. Reflection of waves off parabolic barriers results in the convergence of the waves at a focal point. Refraction is the change in direction of waves that occurs when waves travel from one medium to another. Refraction is always accompanied by a wavelength and speed change. Diffraction is the bending of waves around obstacles and openings. The amount of diffraction increases with increasing wavelength.

Interference of Waves

What happens when two waves meet while they travel through the same medium? What effect will the meeting of the waves have upon the appearance of the medium? Will the two waves bounce off each other upon meeting (much like two billiard balls would) or will the two waves pass through each other? These questions involving the meeting of two or more waves along the same medium pertain to the topic of wave interference.

What is Interference?

Wave interference is the phenomenon that occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium. To begin our exploration of wave interference, consider two pulses of the same amplitude traveling in different directions along the same medium. Let's suppose that each displaced upward 1 unit at its crest and has the shape of a sine wave. As the sine pulses move towards each other, there will eventually be a moment in time when they are completely overlapped. At that moment, the resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2 units. The diagrams below depict the before and during interference snapshots of the medium for two such pulses. The individual sine pulses are drawn in red and blue and the resulting displacement of the medium is drawn in green.

Constructive Interference

This type of interference is sometimes called constructive interference. Constructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction. In this case, both waves have an upward displacement; consequently, the medium has an upward displacement that is greater than the displacement of the two interfering pulses. Constructive interference is observed at any location where the two interfering waves are displaced upward. But it is also observed when both interfering waves are displaced downward. This is shown in the diagram below for two downward displaced pulses.

In this case, a sine pulse with a maximum displacement of -1 unit (negative means a downward displacement) interferes with a sine pulse with a maximum displacement of -1 unit. These two pulses are drawn in red and blue. The resulting shape of the medium is a sine pulse with a maximum displacement of -2 units.

Destructive Interference

Destructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum displacement of -1 unit, destructive interference occurs. This is depicted in the diagram below.

In the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped. At the instant of complete overlap, there is no resulting displacement of the particles of the medium. This "destruction" is not a permanent condition. In fact, to say that the two waves destroy each other can be partially misleading. When it is said that the two pulses destroy each other, what is

meant is that when overlapped, the effect of one of the pulses on the displacement of a given particle of the medium is destroyed or canceled by the effect of the other pulse. Recall from Lesson 1 that waves transport energy through a medium by means of each individual particle pulling upon its nearest neighbor. When two pulses with opposite displacements (i.e., one pulse displaced up and the other down) meet at a given location, the upward pull of one pulse is balanced (canceled or destroyed) by the downward pull of the other pulse. Once the two pulses pass through each other, there is still an upward displaced pulse and a downward displaced pulse heading in the same direction that they were heading before the interference. Destructive interference leads to only a momentary condition in which the medium's displacement is less than the displacement of the largest-amplitude wave.

The two interfering waves do not need to have equal amplitudes in opposite directions for destructive interference to occur. For example, a pulse with a maximum displacement of +1 unit could meet a pulse with a maximum displacement of -2 units. The resulting displacement of the medium during complete overlap is -1 unit.

This is still destructive interference since the two interfering pulses have opposite displacements. In this case, the destructive nature of the interference does not lead to complete cancellation.

Interestingly, the meeting of two waves along a medium does not alter the individual waves or even deviate them from their path. This only becomes an astounding behavior when it is compared to what happens when two billiard balls meet or two football players meet. Billiard balls might crash and bounce off each other and football players might crash and come to a stop. Yet two waves will meet, produce a net resulting shape of the medium, and then continue on doing what they were doing before the interference.

The Principle of Superposition

The task of determining the shape of the resultant demands that the principle of superposition is applied. The principle of superposition is sometimes stated as follows:

When two waves interfere, the resulting displacement of the medium at any location is the algebraic

sum of the displacements of the individual waves at that same location.

In the cases above, the summing the individual displacements for locations of complete overlap was made out to be an easy task - as easy as simple arithmetic:

Displacement of Pulse 1 Displacement of Pulse 2 = Resulting Displacement

+1 +1 = +2

-1 -1 = -2

+1 -1 = 0

+1 -2 = -1

In actuality, the task of determining the complete shape of the entire medium during interference demands that the principle of superposition be applied for every point (or nearly every point) along the medium. As an example of the complexity of this task, consider the two interfering waves at the right. A snapshot of the shape of each individual wave at a particular instant in time is shown. To determine the precise shape of the medium at this given instant in time, the principle of superposition must be applied to several locations along the medium. A short cut involves measuring the displacement from equilibrium at a few strategic locations. Thus, approximately 20 locations have been picked and labeled as A, B, C, D, etc. The actual displacement of each individual wave can be counted by measuring from the equilibrium position up to the particular wave. At position A, there is no displacement for either individual wave; thus, the resulting displacement of the medium at position will be 0 units. At position B, the smaller wave has a displacement of approximately 1.4 units (indicated by the red dot); the larger wave has a displacement of approximately 2 units (indicated by the blue dot). Thus, the resulting displacement of the medium will be approximately 3.4 units. At position C, the smaller wave has a displacement of approximately 2 units; the larger wave has a displacement of approximately 4 units; thus, the resulting displacement of the medium will be approximately 6 units. At position D, the smaller wave has a displacement of approximately 1.4 units; the larger wave has a displacement of approximately 2 units; thus, the resulting displacement of the medium will be approximately 3.4 units. This process can be repeated for every position. When finished, a dot (done in green below) can be marked on the graph to note the displacement of the medium at each given location. The actual shape of the medium can then be sketched by estimating the position between the various marked points and sketching the wave. This is shown as the green line in the diagram below.

The Doppler Effect

Suppose that there is a happy bug in the center of a circular water puddle. The bug is periodically shaking its legs in order to produce disturbances that travel through the water. If these disturbances originate at a point, then they would travel outward from that point in all directions. Since each disturbance is traveling in the same medium, they would all travel in every direction at the same speed. The pattern produced by the bug's shaking would be a series of concentric circles as shown in the diagram at the right. These circles would reach the edges of the water puddle at the same frequency. An observer at point A (the left edge of the puddle) would observe the disturbances to strike the puddle's edge at the same frequency that would be observed by an observer at point B (at the right edge of the puddle). In fact, the frequency at which disturbances reach the edge of the puddle would be the same as the

frequency at which the bug produces the disturbances. If the bug produces disturbances at a frequency of 2 per second, then each observer would observe them approaching at a frequency of 2 per second.

Now suppose that our bug is moving to the right across the puddle of water and producing disturbances at the same frequency of 2 disturbances per second. Since the bug is moving towards the right, each consecutive disturbance originates from a position that is closer to observer B and farther from observer A. Subsequently, each consecutive disturbance has a shorter distance to travel before reaching observer B and thus takes less time to reach observer B. Thus, observer B observes that the frequency of arrival of the disturbances is higher than the frequency at which disturbances are produced. On the other hand, each consecutive disturbance has a further distance to travel before reaching observer A. For this reason, observer A observes a frequency of arrival that is less than the frequency at which the disturbances are produced. The net effect of the motion of the bug (the source of waves) is that the observer towards whom the bug is moving observes a frequency that is higher than 2 disturbances/second; and the observer away from whom the bug is moving observes a frequency that is less than 2 disturbances/second. This effect is known as the Doppler effect.

What is the Doppler Effect?

The Doppler effect is observed whenever the source of waves is moving with respect to an observer. The Doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for observers towards whom the source is approaching and an apparent downward shift in frequency for observers from whom the source is receding. It is important to note that the effect does not result because of an actual change in the frequency of the source. Using the example above, the bug is still producing disturbances at a rate of 2 disturbances per second; it just appears to the observer whom the bug is approaching that the disturbances are being produced at a frequency greater than 2 disturbances/second. The effect is only observed because the distance between observer B and the bug is decreasing and the distance between observer A and the bug is increasing.

The Doppler effect can be observed for any type of wave - water wave, sound wave, light wave, etc. We are most familiar with the Doppler effect because of our experiences with sound waves. Perhaps you recall an instance in which a police car or emergency vehicle was traveling towards you on the highway. As the car approached with its siren blasting, the pitch of the siren sound (a measure of the siren's frequency) was high; and then suddenly after the car passed by, the pitch of the siren sound was low. That was the Doppler effect - an apparent shift in frequency for a sound wave produced by a moving source.

The Doppler Effect in Astronomy

The Doppler effect is of intense interest to astronomers who use the information about the shift in frequency of electromagnetic waves produced by moving stars in our galaxy and beyond in order to derive information about those stars and galaxies. The belief that the universe is expanding is based in part upon observations of electromagnetic waves emitted by stars in distant galaxies. Furthermore, specific information about stars within galaxies can be determined by application of the Doppler effect. Galaxies are clusters of stars that typically rotate about some center of mass point. Electromagnetic radiation emitted by such stars in a distant galaxy would appear to be shifted downward in frequency (a red shift) if the star is rotating in its cluster in a direction that is away from the Earth. On the other hand, there is an upward shift in frequency (a blue shift) of such observed radiation if the star is rotating in a direction that is towards the Earth.