Standing wave patterns are wave patterns produced in a medium when two waves of identical frequencies interfere in such a manner to produce points along the medium that always appear to be standing still. These points that have the appearance of standing still are referred to as nodes. Standing waves are often demonstrated in a Physics class using a snakey that is vibrated by the teacher at one end and held fixed at the other end by a student. The waves reflect off the fixed end and interfere with the waves introduced by the teacher to produce this regular and repeating pattern known as a standing wave pattern. A variety of actual wave patterns could be produced, with each pattern characterized by a distinctly different number of nodes. Such standing wave patterns can only be produced within the medium when it is vibrated at certain frequencies. There are several frequencies with which the snakey can be vibrated to produce the patterns. Each frequency is associated with a different standing wave pattern. These frequencies and their associated wave patterns are referred to as harmonics. the production of standing wave patterns demand that the introduction of crests and troughs into the medium be precisely timed. If the timing is not precise, then a regular and repeating wave pattern will not be discerned within the medium - a harmonic does not exist at such a frequency. With precise timing, reflected vibrations from the opposite end of the medium will interfere with vibrations introduced into the medium in such a manner that there are points that always appear to be standing still. These points of no displacement are referred to as nodes. Positioned in between every node is a point that undergoes maximum displacement from a positive position to a negative position. These points of maximum displacement are referred to as antinodes.

Examples of Standing Wave Patterns

The simplest standing wave pattern that could be produced within a snakey is one that has points of no displacement (nodes) at the two ends of the snakey and one point of maximum displacement (antinode) in the middle. The animation below depicts the vibrational pattern observed when the medium is seen vibrating in this manner.

First Harmonic Standing Wave Pattern

The above standing wave pattern is known as the first harmonic. It is the simplest wave pattern produced within the snakey and is obtained when the teacher introduced vibrations into the end of the medium at low frequencies.

Other wave patterns can be observed within the snakey when it is vibrated at greater frequencies. For instance, if the teacher vibrates the end with twice the frequency as that associated with the first harmonic, then a second standing wave pattern can be achieved. This standing wave pattern is characterized by nodes on the two ends of the snakey and an additional node in the exact center of the snakey. As in all standing wave patterns, every node is separated by an antinode. This pattern with three nodes and two antinodes is referred to as the second harmonic and is depicted in the animation shown below.

Second Harmonic Standing Wave Pattern

If the frequency at which the teacher vibrates the snakey is increased even more, then the third harmonic wave pattern can be produced within the snakey. The standing wave pattern for the third harmonic has an additional node and antinode between the ends of the snakey. The pattern is depicted in the animation shown below.

Third Harmonic Standing Wave Pattern

Numerical Patterns Associated with Standing Wave Diagrams

Observe that each consecutive harmonic is characterized by having one additional node and antinode compared to the previous one. The table below summarizes the features of the standing wave patterns for the first several harmonics.

Harmonic	# of Nodes	# of Antinodes	Pattern
1st	2	1
2nd	3	2
3rd	4	3
4th	5	4
5th	6	5
6th	7	6
nth	n + 1	N	--

As one studies harmonics and their standing wave patterns, it becomes evident that there is a predictability about them. Not surprisingly, this predictability expresses itself in a series of mathematical relationships that relate the wavelength of the wave pattern to the length of the medium. Additionally, the frequency of each harmonic is mathematically related to the frequency of the first harmonic. The next part of Lesson 4 will explore these mathematical relationships.

Flickr Physics Photo

A home-made wave machine was made using string, PVC pipe and connections, a battery, two motors and some wire. The wave machine does a great job producing the second and third harmonic standing wave patterns. The third harmonic is shown here. Observe the two nodes and the three anti nodes positioned between the ends of the string.

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 Interactive. We would like to suggest that you combine the reading of this page with the use of our Standing Wave Patterns Interactive. You can find it in the Physics Interactive section of our website. The Standing Wave Patterns Interactive provides the learner an environment for exploring the formation of standing waves, standing wave patterns, and mathematical relationships for standing wave patterns.

Visit: Standing Wave Patterns Interactive

Nodes and Anti-nodes

· Traveling Waves vs. Standing Waves

· Formation of Standing Waves

· Nodes and Anti-nodes

· Harmonics and Patterns

· Mathematics of Standing Waves

What are Nodes and Antinodes?

One characteristic of every standing wave pattern is that there are points along the medium that appear to be standing still. These points, sometimes described as points of no displacement, are referred to as nodes. There are other points along the medium that undergo vibrations between a large positive and large negative displacement. These are the points that undergo the maximum displacement during each vibrational cycle of the standing wave. In a sense, these points are the opposite of nodes, and so they are called antinodes. A standing wave pattern always consists of an alternating pattern of nodes and antinodes. The animation shown below depicts a rope vibrating with a standing wave pattern. The nodes and antinodes are labeled on the diagram. When a standing wave pattern is established in a medium, the nodes and the antinodes are always located at the same position along the medium; they are standing still. It is this characteristic that has earned the pattern the name standing wave.

Flickr Physics Photo

A standing wave is established upon a vibrating string using a harmonic oscillator and a frequency generator. A strobe is used to illuminate the string several times during each cycle. The finger is pointing at a nodal position.

Standing Wave Diagrams

The positioning of the nodes and antinodes in a standing wave pattern can be explained by focusing on the interference of the two waves. The nodes are produced at locations where destructive interference occurs. For instance, nodes form at locations where a crest of one wave meets a trough of a second wave; or a half-crest of one wave meets a half-trough of a second wave; or a quarter-crest of one wave meets a quarter-trough of a second wave; etc. Antinodes, on the other hand, are produced at locations where constructive interference occurs. For instance, if a crest of one wave meets a crest of a second wave, a point of large positive displacement results. Similarly, if a trough of one wave meets a trough of a second wave, a point of large negative displacement results. Antinodes are always vibrating back and forth between these points of large positive and large negative displacement; this is because during a complete cycle of vibration, a crest will meet a crest; and then one-half cycle later, a trough will meet a trough. Because antinodes are vibrating back and forth between a large positive and large negative displacement, a diagram of a standing wave is sometimes depicted by drawing the shape of the medium at an instant in time and at an instant one-half vibrational cycle later. This is done in the diagram below.

Nodes and antinodes should not be confused with crests and troughs. When the motion of a traveling wave is discussed, it is customary to refer to a point of large maximum displacement as a crest and a point of large negative displacement as a trough. These represent points of the disturbance that travel from one location to another through the medium. An antinode on the other hand is a point on the medium that is staying in the same location. Furthermore, an antinode vibrates back and forth between a large upward and a large downward displacement. And finally, nodes and antinodes are not actually part of a wave. Recall that a standing wave is not actually a wave but rather a pattern that results from the interference of two or more waves. Since a standing wave is not technically a wave, an antinode is not technically a point on a wave. The nodes and antinodes are merely unique points on the medium that make up the wave pattern.

Check Your Understanding

1. Suppose that there was a ride at an amusement park that was titled The Standing Wave. Which location - node or antinode - on the ride would give the greatest thrill?

See Answer

Answer: The antinode

The antinode is continually vibrating from a high to a low displacement - now that would be a ride.

2. A standing wave is formed when ____.

a. a wave refracts due to changes in the properties of the medium.

b. a wave reflects off a canyon wall and is heard shortly after it is formed.

c. red, orange, and yellow wavelengths bend around suspended atmospheric particles.

d. Two identical waves moving different directions along the same medium interfere.

See Answer

Answer: D

3. The number of nodes in the standing wave shown in the diagram at the right is ____.

a. 6	b. 7
c. 8	d. 14

See Answer

Answer: C (8 nodes)

There are eight positions along the medium which have no displacement. Be sure to avoid the common mistake of not counting the end positions.

4. The number of antinodes in the standing wave shown in the diagram above right is ____.

a. 6

b. 7

c. 8

d. 14

See Answer

Answer: B (7 antinodes)

There are seven positions along the medium which have vibrate between a large positive and a large negative displacement.

Be sure to avoid the common mistake of counting the antinodal positions twice. An antinode is simply a point along a medium which undergoes maximum displacement above and below the rest position. Do not count these positions twice.

Consider the standing wave pattern at the right in answering these next two questions.

5. The number of nodes in the entire pattern is ___.

a. 7	b. 8
c. 9	d. 16

See Answer

Answer: C (9 nodes)

There are nine positions along the medium which have no displacement. (Be sure to avoid the common mistake of not counting the end positions.)

6. Of all the labeled points, destructive interference occurs at point(s) ____.

a. B, C, and D	b. A, E, and F	c. A only
d. C only	e. all points

Formation of Standing Waves

· Traveling Waves vs. Standing Waves

· Formation of Standing Waves

· Nodes and Anti-nodes

· Harmonics and Patterns

· Mathematics of Standing Waves

A standing wave pattern is a vibrational pattern created within a medium when the vibrational frequency of the source causes reflected waves from one end of the medium to interfere with incident waves from the source. This interference occurs in such a manner that specific points along the medium appear to be standing still. Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a standing wave pattern. Such patterns are only created within the medium at specific frequencies of vibration. These frequencies are known as harmonic frequencies, or merely harmonics. At any frequency other than a harmonic frequency, the interference of reflected and incident waves leads to a resulting disturbance of the medium that is irregular and non-repeating.

How is a Standing Wave Formed?

But how are standing wave formations formed? And why are they only formed when the medium is vibrated at specific frequencies? And what makes these so-called harmonic frequencies so special and magical? To answer these questions, let's consider a snakey stretched across the room, approximately 4-meters from end to end. (A "snakey" is a slinky-like device that consists of a large concentration of small-diameter metal coils.) If an upward displaced pulse is introduced at the left end of the snakey, it will travel rightward across the snakey until it reaches the fixed end on the right side of the snakey. Upon reaching the fixed end, the single pulse will reflect and undergo inversion. That is, the upward displaced pulse will become a downward displaced pulse. Now suppose that a second upward displaced pulse is introduced into the snakey at the precise moment that the first crest undergoes its fixed end reflection. If this is done with perfect timing, a rightward moving, upward displaced pulse will meet up with a leftward moving, downward displaced pulse in the exact middle of the snakey. As the two pulses pass through each other, they will undergo destructive interference. Thus, a point of no displacement in the exact middle of the snakey will be produced. The animation below shows several snapshots of the meeting of the two pulses at various stages in their interference. The individual pulses are drawn in blue and red; the resulting shape of the medium (as found by the principle of superposition) is shown in green. Note that there is a point on the diagram in the exact middle of the medium that never experiences any displacement from the equilibrium position.

An upward displaced pulse introduced at one end will destructively interfere in the exact middle of the snakey with a second upward displaced pulse introduced from the same end if the introduction of the second pulse is performed with perfect timing. The same rationale could be applied to two downward displaced pulses introduced from the same end. If the second pulse is introduced at precisely the moment that the first pulse is reflecting from the fixed end, then destructive interference will occur in the exact middle of the snakey.

The above discussion only explains why two pulses might interfere destructively to produce a point of no displacement in the middle of the snakey. A wave is certainly different than a pulse. What if there are two waves traveling in the medium? Understanding why two waves introduced into a medium with perfect timing might produce a point of displacement in the middle of the medium is a mere extension of the above discussion. While a pulse is a single disturbance that moves through a medium, a wave is a repeating pattern of crests and troughs. Thus, a wave can be thought of as an upward displaced pulse (crest) followed by a downward displaced pulse (trough) followed by an upward displaced pulse (crest) followed by a downward displaced pulse (trough) followed by... . Since the introduction of a crest is followed by the introduction of a trough, every crest and trough will destructively interfere in such a way that the middle of the medium is a point of no displacement.

The Importance of Timing

Of course, this all demands that the timing is perfect. In the above discussion, perfect timing was achieved if every wave crest was introduced into the snakey at the precise time that the previous wave crest began its reflection at the fixed end. In this situation, there will be one complete wavelength within the snakey moving to the right at every instant in time; this incident wave will meet up with one complete wavelength moving to the left at every instant in time. Under these conditions, destructive interference always occurs in the middle of the snakey. Either a full crest meets a full trough or a half-crest meets a half-trough or a quarter-crest meets a quarter-trough at this point. The animation below represents several snapshots of two waves traveling in opposite directions along the same medium. The waves are interfering in such a manner that there are points of no displacement produced at the same positions along the medium. These points along the medium are known as nodes and are labeled with an N. There are also points along the medium that vibrate back and forth between points of large positive displacement and points of large negative displacement. These points are known as antinodes and are labeled with an AN. The two individual waves are drawn in blue and green and the resulting shape of the medium is drawn in black.

There are other ways to achieve this perfect timing. The main idea behind the timing is to introduce a crest at the instant that another crest is either at the halfway point across the medium or at the end of the medium. Regardless of the number of crests and troughs that are in between, if a crest is introduced at the instant another crest is undergoing its fixed end reflection, a node (point of no displacement) will be formed in the middle of the medium. The number of other nodes that will be present along the medium is dependent upon the number of crests that were present in between the two timed crests. If a crest is introduced at the instant another crest is at the halfway point across the medium, then an antinode (point of maximum displacement) will be formed in the middle of the medium by means of constructive interference. In such an instance, there might also be nodes and antinodes located elsewhere along the medium.

A standing wave pattern is an interference phenomenon. It is formed as the result of the perfectly timed interference of two waves passing through the same medium. A standing wave pattern is not actually a wave; rather it is the pattern resulting from the presence of two waves (sometimes more) of the same frequency with different directions of travel within the same medium. The physics of musical instruments has a basis in the conceptual and mathematical aspects of standing waves. For this reason, the topic will be revisited in the Sound and Music unit at The Physics Classroom Tutorial.

Traveling Waves vs. Standing Waves

· Traveling Waves vs. Standing Waves

· Formation of Standing Waves

· Nodes and Anti-nodes

· Harmonics and Patterns

· Mathematics of Standing Waves

A mechanical wave is a disturbance that is created by a vibrating object and subsequently travels through a medium from one location to another, transporting energy as it moves. The mechanism by which a mechanical wave propagates itself through a medium involves particle interaction; one particle applies a push or pull on its adjacent neighbor, causing a displacement of that neighbor from the equilibrium or rest position. As a wave is observed traveling through a medium, a crest is seen moving along from particle to particle. This crest is followed by a trough that is in turn followed by the next crest. In fact, one would observe a distinct wave pattern (in the form of a sine wave) traveling through the medium. This sine wave pattern continues to move in uninterrupted fashion until it encounters another wave along the medium or until it encounters a boundary with another medium. This type of wave pattern that is seen traveling through a medium is sometimes referred to as a traveling wave.

Traveling waves are observed when a wave is not confined to a given space along the medium. The most commonly observed traveling wave is an ocean wave. If a wave is introduced into an elastic cord with its ends held 3 meters apart, it becomes confined in a small region. Such a wave has only 3 meters along which to travel. The wave will quickly reach the end of the cord, reflect and travel back in the opposite direction. Any reflected portion of the wave will then interfere with the portion of the wave incident towards the fixed end. This interference produces a new shape in the medium that seldom resembles the shape of a sine wave. Subsequently, a traveling wave (a repeating pattern that is observed to move through a medium in uninterrupted fashion) is not observed in the cord. Indeed there are traveling waves in the cord; it is just that they are not easily detectable because of their interference with each other. In such instances, rather than observing the pure shape of a sine wave pattern, a rather irregular and non-repeating pattern is produced in the cord that tends to change appearance over time. This irregular looking shape is the result of the interference of an incident sine wave pattern with a reflected sine wave pattern in a rather non-sequenced and untimely manner. Both the incident and reflected wave patterns continue their motion through the medium, meeting up with one another at different locations in different ways. For example, the middle of the cord might experience a crest meeting a half crest; then moments later, a crest meeting a quarter trough; then moments later, a three-quarters crest meeting a one-fifth trough, etc. This interference leads to a very irregular and non-repeating motion of the medium. The appearance of an actual wave pattern is difficult to detect amidst the irregular motions of the individual particles.

What is a Standing Wave Pattern?

It is however possible to have a wave confined to a given space in a medium and still produce a regular wave pattern that is readily discernible amidst the motion of the medium. For instance, if an elastic rope is held end-to-end and vibrated at just the right frequency, a wave pattern would be produced that assumes the shape of a sine wave and is seen to change over time. The wave pattern is only produced when one end of the rope is vibrated at just the right frequency. When the proper frequency is used, the interference of the incident wave and the reflected wave occur in such a manner that there are specific points along the medium that appear to be standing still. Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a standing wave pattern. There are other points along the medium whose displacement changes over time, but in a regular manner. These points vibrate back and forth from a positive displacement to a negative displacement; the vibrations occur at regular time intervals such that the motion of the medium is regular and repeating. A pattern is readily observable.

The diagram at the right depicts a standing wave pattern in a medium. A snapshot of the medium over time is depicted using various colors. Note that point A on the medium moves from a maximum positive to a maximum negative displacement over time. The diagram only shows one-half cycle of the motion of the standing wave pattern. The motion would continue and persist, with point A returning to the same maximum positive displacement and then continuing its back-and-forth vibration between the up to the down position. Note that point B on the medium is a point that never moves. Point B is a point of no displacement. Such points are known as nodes and will be discussed in more detail later in this lesson. The standing wave pattern that is shown at the right is just one of many different patterns that could be produced within the rope. Other patterns will be discussed later in the lesson.

The Doppler Effect

· Boundary Behavior

· Reflection, Refraction, and Diffraction

· Interference of Waves

· 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.

Interference of Waves

· Boundary Behavior

· Reflection, Refraction, and Diffraction

· Interference of Waves

· The Doppler Effect

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.

Check Your Understanding

1. Several positions along the medium are labeled with a letter. Categorize each labeled position along the medium as being a position where either constructive or destructive interference occurs.

See Answer

Constructive Interference: G, J, M and N

Destructive Interference: H, I, K, L, and O

2. Twin water bugs Jimminy and Johnny are both creating a series of circular waves by jiggling their legs in the water. The waves undergo interference and create the pattern represented in the diagram at the right. The thick lines in the diagram represent wave crests and the thin lines represent wave troughs. Several of positions in the water are labeled with a letter. Categorize each labeled position as being a position where either constructive or destructive interference occurs.

Constructive Interference: A and B

Destructive Interference: C, D, E, and F

Reflection, Refraction, and Diffraction

· Boundary Behavior

· Reflection, Refraction, and Diffraction

· Interference of Waves

· The Doppler Effect

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. . 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 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.

The Wave Equation

· The Anatomy of a Wave

· Frequency and Period of a Wave

· Energy Transport and the Amplitude of a Wave

· The Speed of a Wave

· The Wave Equation

As was discussed in Lesson 1, a wave is produced when a vibrating source periodically disturbs the first particle of a medium. This creates a wave pattern that begins to travel along the medium from particle to particle. The frequency at which each individual particle vibrates is equal to the frequency at which the source vibrates. Similarly, the period of vibration of each individual particle in the medium is equal to the period of vibration of the source. In one period, the source is able to displace the first particle upwards from rest, back to rest, downwards from rest, and finally back to rest. This complete back-and-forth movement constitutes one complete wave cycle.

The diagrams at the right show several "snapshots" of the production of a wave within a rope. The motion of the disturbance along the medium after every one-fourth of a period is depicted. Observe that in the time it takes from the first to the last snapshot, the hand has made one complete back-and-forth motion. A period has elapsed. Observe that during this same amount of time, the leading edge of the disturbance has moved a distance equal to one complete wavelength. So in a time of one period, the wave has moved a distance of one wavelength. Combining this information with the equation for speed (speed = distance/time), it can be said that the speed of a wave is also the wavelength/period.

Since the period is the reciprocal of the frequency, the expression 1/f can be substituted into the above equation for period. Rearranging the equation yields a new equation of the form:

Speed = Wavelength • Frequency

The above equation is known as the wave equation. It states the mathematical relationship between the speed (v) of a wave and its wavelength (λ) and frequency (f). Using the symbols v, λ, and f, the equation can be rewritten as

v = f • λ

As a test of your understanding of the wave equation and its mathematical use in analyzing wave motion, consider the following three-part question:

Stan and Anna are conducting a slinky experiment. They are studying the possible effect of several variables upon the speed of a wave in a slinky. Their data table is shown below. Fill in the blanks in the table, analyze the data, and answer the following questions.

Medium	Wavelength	Frequency	Speed
Zinc, 1-in. dia. coils	1.75 m	2.0 Hz	______
Zinc, 1-in. dia. coils	0.90 m	3.9 Hz	______
Copper, 1-in. dia. coils	1.19 m	2.1 Hz	______
Copper, 1-in. dia. coils	0.60 m	4.2 Hz	______
Zinc, 3-in. dia. coils	0.95 m	2.2 Hz	______
Zinc, 3-in. dia. coils	1.82 m	1.2 Hz	______

See Speed Values

Multiply the frequency by the wavelength to determine the speed.

Row 1: speed = 3.5 m/s

Row 2: speed = 3.5 m/s

Row 3: speed = 2.5 m/s

Row 4: speed = 2.5 m/s

Row 5: speed = 2.1 m/s

Row 6: speed = 2.2 m/s

1. As the wavelength of a wave in a uniform medium increases, its speed will _____.

a. decrease

b. increase

c. remain the same

See Answer

Answer: C

In rows 1 and 2, the wavelength was altered but the speed remained the same. The same can be said about rows 3 and 4 and rows 5 and 6. The speed of a wave is not affected by the wavelength of the wave.

2. As the wavelength of a wave in a uniform medium increases, its frequency will _____.

a. decrease

b. increase

c. remain the same

See Answer

Answer: A

In rows 1 and 2, the wavelength was increased and the frequency was decreased. Wavelength and frequency are inversely proportional to each other.

3. The speed of a wave depends upon (i.e., is causally affected by) ...

a. the properties of the medium through which the wave travels

b. the wavelength of the wave.

c. the frequency of the wave.

d. both the wavelength and the frequency of the wave.

See Answer

Answer: A

Whenever the medium is the same, the speed of the wave is the same. However, when the medium changes, the speed changes. The speed of these waves were dependent upon the properties of the medium.

The above example illustrates how to use the wave equation to solve mathematical problems. It also illustrates the principle that wave speed is dependent upon medium properties and independent of wave properties. Even though the wave speed is calculated by multiplying wavelength by frequency, an alteration in wavelength does not affect wave speed. Rather, an alteration in wavelength affects the frequency in an inverse manner. A doubling of the wavelength results in a halving of the frequency; yet the wave speed is not changed.

Check Your Understanding

1. Two waves on identical strings have frequencies in a ratio of 2 to 1. If their wave speeds are the same, then how do their wavelengths compare?

a. 2:1

b. 1:2

c. 4:1

d. 1:4

See Answer

Answer: B

Frequency and wavelength are inversely proportional to each other. The wave with the greatest frequency has the shortest wavelength. Twice the frequency means one-half the wavelength. For this reason, the wavelength ratio is the inverse of the frequency ratio.

2. Mac and Tosh stand 8 meters apart and demonstrate the motion of a transverse wave on a snakey. The wave e can be described as having a vertical distance of 32 cm from a trough to a crest, a frequency of 2.4 Hz, and a horizontal distance of 48 cm from a crest to the nearest trough. Determine the amplitude, period, and wavelength and speed of such a wave.

See Answer

Amplitude = 16 cm

(Amplitude is the distance from the rest position to the crest position which is half the vertical distance from a trough to a crest.)

Wavelength = 96 cm

(Wavelength is the distance from crest to crest, which is twice the horizontal distance from crest to nearest trough.)

Period = 0.42 s

(The period is the reciprocal of the frequency. T = 1 / f)

Speed = 230 cm/s

(The speed of a wave is calculated as the product of the frequency times the wavelength.)

3. Dawn and Aram have stretched a slinky between them and begin experimenting with waves. As the frequency of the waves is doubled,

a. the wavelength is halved and the speed remains constant

b. the wavelength remains constant and the speed is doubled

c. both the wavelength and the speed are halved.

d. both the wavelength and the speed remain constant.

See Answer

Answer: A

Doubling the frequency will not alter the wave speed. Rather, it will halve the wavelength. Wavelength and frequency are inversely related.

4. A ruby-throated hummingbird beats its wings at a rate of about 70 wing beats per second.

a. What is the frequency in Hertz of the sound wave?

b. Assuming the sound wave moves with a velocity of 350 m/s, what is the wavelength of the wave?

See Answer

Answer: f = 70 Hz and λ = 5.0 m

The frequency is given and the wavelength is the v/f ratio.

5. Ocean waves are observed to travel along the water surface during a developing storm. A Coast Guard weather station observes that there is a vertical distance from high point to low point of 4.6 meters and a horizontal distance of 8.6 meters between adjacent crests. The waves splash into the station once every 6.2 seconds. Determine the frequency and the speed of these waves.

See Answer

The wavelength is 8.6 meters and the period is 6.2 seconds.

The frequency can be determined from the period. If T = 6.2 s, then

f =1 /T = 1 / (6.2 s)

f = 0.161 Hz

Now find speed using the v = f • λ equation.

v = f • λ = (0.161 Hz) • (8.6 m)

v = 1.4 m/s

6. Two boats are anchored 4 meters apart. They bob up and down, returning to the same up position every 3 seconds. When one is up the other is down. There are never any wave crests between the boats. Calculate the speed of the waves.

Frequency and Period of a Wave

· The Anatomy of a Wave

· Frequency and Period of a Wave

· Energy Transport and the Amplitude of a Wave

· The Speed of a Wave

· The Wave Equation

The nature of a wave was discussed in Lesson 1 of this unit. In that lesson, it was mentioned that a wave is created in a slinky by the periodic and repeating vibration of the first coil of the slinky. This vibration creates a disturbance that moves through the slinky and transports energy from the first coil to the last coil. A single back-and-forth vibration of the first coil of a slinky introduces a pulse into the slinky. But the act of continually vibrating the first coil with a back-and-forth motion in periodic fashion introduces a wave into the slinky.

Suppose that a hand holding the first coil of a slinky is moved back-and-forth two complete cycles in one second. The rate of the hand's motion would be 2 cycles/second. The first coil, being attached to the hand, in turn would vibrate at a rate of 2 cycles/second. The second coil, being attached to the first coil, would vibrate at a rate of 2 cycles/second. The third coil, being attached to the second coil, would vibrate at a rate of 2 cycles/second. In fact, every coil of the slinky would vibrate at this rate of 2 cycles/second. This rate of 2 cycles/second is referred to as the frequency of the wave. The frequency of a wave refers to how often the particles of the medium vibrate when a wave passes through the medium. Frequency is a part of our common, everyday language. For example, it is not uncommon to hear a question like "How frequently do you mow the lawn during the summer months?" Of course the question is an inquiry about how often the lawn is mowed and the answer is usually given in the form of "1 time per week." In mathematical terms, the frequency is the number of complete vibrational cycles of a medium per a given amount of time. Given this definition, it is reasonable that the quantity frequency would have units of cycles/second, waves/second, vibrations/second, or something/second. Another unit for frequency is the Hertz (abbreviated Hz) where 1 Hz is equivalent to 1 cycle/second. If a coil of slinky makes 2 vibrational cycles in one second, then the frequency is 2 Hz. If a coil of slinky makes 3 vibrational cycles in one second, then the frequency is 3 Hz. And if a coil makes 8 vibrational cycles in 4 seconds, then the frequency is 2 Hz (8 cycles/4 s = 2 cycles/s).

The quantity frequency is often confused with the quantity period. Period refers to the time that it takes to do something. When an event occurs repeatedly, then we say that the event is periodic and refer to the time for the event to repeat itself as the period. The period of a wave is the time for a particle on a medium to make one complete vibrational cycle. Period, being a time, is measured in units of time such as seconds, hours, days or years. The period of orbit for the Earth around the Sun is approximately 365 days; it takes 365 days for the Earth to complete a cycle. The period of a typical class at a high school might be 55 minutes; every 55 minutes a class cycle begins (50 minutes for class and 5 minutes for passing time means that a class begins every 55 minutes). The period for the minute hand on a clock is 3600 seconds (60 minutes); it takes the minute hand 3600 seconds to complete one cycle around the clock.

Frequency and period are distinctly different, yet related, quantities. Frequency refers to how often something happens. Period refers to the time it takes something to happen. Frequency is a rate quantity. Period is a time quantity. Frequency is the cycles/second. Period is the seconds/cycle. As an example of the distinction and the relatedness of frequency and period, consider a woodpecker that drums upon a tree at a periodic rate. If the woodpecker drums upon a tree 2 times in one second, then the frequency is 2 Hz. Each drum must endure for one-half a second, so the period is 0.5 s. If the woodpecker drums upon a tree 4 times in one second, then the frequency is 4 Hz; each drum must endure for one-fourth a second, so the period is 0.25 s. If the woodpecker drums upon a tree 5 times in one second, then the frequency is 5 Hz; each drum must endure for one-fifth a second, so the period is 0.2 s. Do you observe the relationship? Mathematically, the period is the reciprocal of the frequency and vice versa. In equation form, this is expressed as follows.

Since the symbol f is used for frequency and the symbol T is used for period, these equations are also expressed as:

The quantity frequency is also confused with the quantity speed. The speed of an object refers to how fast an object is moving and is usually expressed as the distance traveled per time of travel. For a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given period of time. So while wave frequency refers to the number of cycles occurring per second, wave speed refers to the meters traveled per second. A wave can vibrate back and forth very frequently, yet have a small speed; and a wave can vibrate back and forth with a low frequency, yet have a high speed. Frequency and speed are distinctly different quantities.

Investigate!

How do changes in the frequency of a wave affect the wavelength of a wave? Use the Wave plotter widget below to find out. Alter the frequency and observe how the pattern changes.

Wave Plotter

Check Your Understanding

Throughout this unit, internalize the meaning of terms such as period, frequency, and wavelength. Utilize the meaning of these terms to answer conceptual questions; avoid a formula fixation.

1. A wave is introduced into a thin wire held tight at each end. It has an amplitude of 3.8 cm, a frequency of 51.2 Hz and a distance from a crest to the neighboring trough of 12.8 cm. Determine the period of such a wave.

The Anatomy of a Wave

· The Anatomy of a Wave

· Frequency and Period of a Wave

· Energy Transport and the Amplitude of a Wave

· The Speed of a Wave

· The Wave Equation

A transverse wave is a wave in which the particles of the medium are displaced in a direction perpendicular to the direction of energy transport. A transverse wave can be created in a rope if the rope is stretched out horizontally and the end is vibrated back-and-forth in a vertical direction. If a snapshot of such a transverse wave could be taken so as to freeze the shape of the rope in time, then it would look like the following diagram.

The dashed line drawn through the center of the diagram represents the equilibrium or rest position of the string. This is the position that the string would assume if there were no disturbance moving through it. Once a disturbance is introduced into the string, the particles of the string begin to vibrate upwards and downwards. At any given moment in time, a particle on the medium could be above or below the rest position. Points A, E and H on the diagram represent the crests of this wave. The crest of a wave is the point on the medium that exhibits the maximum amount of positive or upward displacement from the rest position. Points C and J on the diagram represent the troughs of this wave. The trough of a wave is the point on the medium that exhibits the maximum amount of negative or downward displacement from the rest position.

The wave shown above can be described by a variety of properties. One such property is amplitude. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. In a sense, the amplitude is the distance from rest to crest. Similarly, the amplitude can be measured from the rest position to the trough position. In the diagram above, the amplitude could be measured as the distance of a line segment that is perpendicular to the rest position and extends vertically upward from the rest position to point A.

The wavelength is another property of a wave that is portrayed in the diagram above. The wavelength of a wave is simply the length of one complete wave cycle. If you were to trace your finger across the wave in the diagram above, you would notice that your finger repeats its path. A wave is a repeating pattern. It repeats itself in a periodic and regular fashion over both time and space. And the length of one such spatial repetition (known as a wave cycle) is the wavelength. The wavelength can be measured as the distance from crest to crest or from trough to trough. In fact, the wavelength of a wave can be measured as the distance from a point on a wave to the corresponding point on the next cycle of the wave. In the diagram above, the wavelength is the horizontal distance from A to E, or the horizontal distance from B to F, or the horizontal distance from D to G, or the horizontal distance from E to H. Any one of these distance measurements would suffice in determining the wavelength of this wave.

A longitudinal wave is a wave in which the particles of the medium are displaced in a direction parallel to the direction of energy transport. A longitudinal wave can be created in a slinky if the slinky is stretched out horizontally and the end coil is vibrated back-and-forth in a horizontal direction. If a snapshot of such a longitudinal wave could be taken so as to freeze the shape of the slinky in time, then it would look like the following diagram.

Because the coils of the slinky are vibrating longitudinally, there are regions where they become pressed together and other regions where they are spread apart. A region where the coils are pressed together in a small amount of space is known as a compression. A compression is a point on a medium through which a longitudinal wave is traveling that has the maximum density. A region where the coils are spread apart, thus maximizing the distance between coils, is known as a rarefaction. A rarefaction is a point on a medium through which a longitudinal wave is traveling that has the minimum density. Points A, C and E on the diagram above represent compression and points B, D, and F represent rarefaction. While a transverse wave has an alternating pattern of crests and troughs, a longitudinal wave has an alternating pattern of compression and rarefaction.

As discussed above, the wavelength of a wave is the length of one complete cycle of a wave. For a transverse wave, the wavelength is determined by measuring from crest to crest. A longitudinal wave does not have crest; so how can its wavelength be determined? The wavelength can always be determined by measuring the distance between any two corresponding points on adjacent waves. In the case of a longitudinal wave, a wavelength measurement is made by measuring the distance from a compression to the next compression or from a rarefaction to the next rarefaction. On the diagram above, the distance from point A to point C or from point B to point D would be representative of the wavelength.

Check Your Understanding

Consider the diagram below in order to answer questions #1-2.

1. The wavelength of the wave in the diagram above is given by letter ______.

Categories of Waves

· Waves and Wavelike Motion

· What is a Wave?

· Categories of Waves

Waves come in many shapes and forms. While all waves share some basic characteristic properties and behaviors, some waves can be distinguished from others based on some observable (and some non-observable) characteristics. It is common to categorize waves based on these distinguishing characteristics.

Longitudinal versus Transverse Waves versus Surface Waves

One way to categorize waves is on the basis of the direction of movement of the individual particles of the medium relative to the direction that the waves travel. Categorizing waves on this basis leads to three notable categories: transverse waves, longitudinal waves, and surface waves.

A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction that the wave moves. Suppose that a slinky is stretched out in a horizontal direction across the classroom and that a pulse is introduced into the slinky on the left end by vibrating the first coil up and down. Energy will begin to be transported through the slinky from left to right. As the energy is transported from left to right, the individual coils of the medium will be displaced upwards and downwards. In this case, the particles of the medium move perpendicular to the direction that the pulse moves. This type of wave is a transverse wave. Transverse waves are always characterized by particle motion being perpendicular to wave motion.

A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction that the wave moves. Suppose that a slinky is stretched out in a horizontal direction across the classroom and that a pulse is introduced into the slinky on the left end by vibrating the first coil left and right. Energy will begin to be transported through the slinky from left to right. As the energy is transported from left to right, the individual coils of the medium will be displaced leftwards and rightwards. In this case, the particles of the medium move parallel to the direction that the pulse moves. This type of wave is a longitudinal wave. Longitudinal waves are always characterized by particle motion being parallel to wave motion.

A sound wave traveling through air is a classic example of a longitudinal wave. As a sound wave moves from the lips of a speaker to the ear of a listener, particles of air vibrate back and forth in the same direction and the opposite direction of energy transport. Each individual particle pushes on its neighboring particle so as to push it forward. The collision of particle #1 with its neighbor serves to restore particle #1 to its original position and displace particle #2 in a forward direction. This back and forth motion of particles in the direction of energy transport creates regions within the medium where the particles are pressed together and other regions where the particles are spread apart. Longitudinal waves can always be quickly identified by the presence of such regions. This process continues along the chain of particles until the sound wave reaches the ear of the listener. A detailed discussion of sound is presented in another unit of The Physics Classroom Tutorial.

Waves traveling through a solid medium can be either transverse waves or longitudinal waves. Yet waves traveling through the bulk of a fluid (such as a liquid or a gas) are always longitudinal waves. Transverse waves require a relatively rigid medium in order to transmit their energy. As one particle begins to move it must be able to exert a pull on its nearest neighbor. If the medium is not rigid as is the case with fluids, the particles will slide past each other. This sliding action that is characteristic of liquids and gases prevents one particle from displacing its neighbor in a direction perpendicular to the energy transport. It is for this reason that only longitudinal waves are observed moving through the bulk of liquids such as our oceans. Earthquakes are capable of producing both transverse and longitudinal waves that travel through the solid structures of the Earth. When seismologists began to study earthquake waves they noticed that only longitudinal waves were capable of traveling through the core of the Earth. For this reason, geologists believe that the Earth's core consists of a liquid - most likely molten iron.

While waves that travel within the depths of the ocean are longitudinal waves, the waves that travel along the surface of the oceans are referred to as surface waves. A surface wave is a wave in which particles of the medium undergo a circular motion. Surface waves are neither longitudinal nor transverse. In longitudinal and transverse waves, all the particles in the entire bulk of the medium move in a parallel and a perpendicular direction (respectively) relative to the direction of energy transport. In a surface wave, it is only the particles at the surface of the medium that undergo the circular motion. The motion of particles tends to decrease as one proceeds further from the surface.

Any wave moving through a medium has a source. Somewhere along the medium, there was an initial displacement of one of the particles. For a slinky wave, it is usually the first coil that becomes displaced by the hand of a person. For a sound wave, it is usually the vibration of the vocal chords or a guitar string that sets the first particle of air in vibrational motion. At the location where the wave is introduced into the medium, the particles that are displaced from their equilibrium position always moves in the same direction as the source of the vibration. So if you wish to create a transverse wave in a slinky, then the first coil of the slinky must be displaced in a direction perpendicular to the entire slinky. Similarly, if you wish to create a longitudinal wave in a slinky, then the first coil of the slinky must be displaced in a direction parallel to the entire slinky.

Electromagnetic versus Mechanical Waves

Another way to categorize waves is on the basis of their ability or inability to transmit energy through a vacuum (i.e., empty space). Categorizing waves on this basis leads to two notable categories: electromagnetic waves and mechanical waves.

An electromagnetic wave is a wave that is capable of transmitting its energy through a vacuum (i.e., empty space). Electromagnetic waves are produced by the vibration of charged particles. Electromagnetic waves that are produced on the sun subsequently travel to Earth through the vacuum of outer space. Were it not for the ability of electromagnetic waves to travel to through a vacuum, there would undoubtedly be no life on Earth. All light waves are examples of electromagnetic waves. Light waves are the topic of another unit at The Physics Classroom Tutorial. While the basic properties and behaviors of light will be discussed, the detailed nature of an electromagnetic wave is quite complicated and beyond the scope of The Physics Classroom Tutorial.

A mechanical wave is a wave that is not capable of transmitting its energy through a vacuum. Mechanical waves require a medium in order to transport their energy from one location to another. A sound wave is an example of a mechanical wave. Sound waves are incapable of traveling through a vacuum. Slinky waves, water waves, stadium waves, and jump rope waves are other examples of mechanical waves; each requires some medium in order to exist. A slinky wave requires the coils of the slinky; a water wave requires water; a stadium wave requires fans in a stadium; and a jump rope wave requires a jump rope.

The above categories represent just a few of the ways in which physicists categorize waves in order to compare and contrast their behaviors and characteristic properties. This listing of categories is not exhaustive; there are other categories as well. The five categories of waves listed here will be used periodically throughout this unit on waves as well as the units on sound and light.

Earthquakes and other geologic disturbances sometimes result in the formation of seismic waves. Seismic waves are waves of energy that are transported through the earth and over its surface by means of both transverse and longitudinal waves. Just how common are seismic waves? Use the Recent Earthquakes widget below to explore the frequency of earthquakes. Search the past week or the past 24 hours or by Richter scale magnitude.

Recent Earthquakes

What is a Wave?

· Waves and Wavelike Motion

· What is a Wave?

· Categories of Waves

So waves are everywhere. But what makes a wave a wave? What characteristics, properties, or behaviors are shared by the phenomena that we typically characterize as being a wave? How can waves be described in a manner that allows us to understand their basic nature and qualities?

A wave can be described as a disturbance that travels through a medium from one location to another location. Consider a slinky wave as an example of a wave. When the slinky is stretched from end to end and is held at rest, it assumes a natural position known as the equilibrium or rest position. The coils of the slinky naturally assume this position, spaced equally far apart. To introduce a wave into the slinky, the first particle is displaced or moved from its equilibrium or rest position. The particle might be moved upwards or downwards, forwards or backwards; but once moved, it is returned to its original equilibrium or rest position. The act of moving the first coil of the slinky in a given direction and then returning it to its equilibrium position creates a disturbance in the slinky. We can then observe this disturbance moving through the slinky from one end to the other. If the first coil of the slinky is given a single back-and-forth vibration, then we call the observed motion of the disturbance through the slinky a slinky pulse. A pulse is a single disturbance moving through a medium from one location to another location. However, if the first coil of the slinky is continuously and periodically vibrated in a back-and-forth manner, we would observe a repeating disturbance moving within the slinky that endures over some prolonged period of time. The repeating and periodic disturbance that moves through a medium from one location to another is referred to as a wave.

What is a Medium?

But what is meant by the word medium? A medium is a substance or material that carries the wave. You have perhaps heard of the phrase news media. The news media refers to the various institutions (newspaper offices, television stations, radio stations, etc.) within our society that carry the news from one location to another. The news moves through the media. The media doesn't make the news and the media isn't the same as the news. The news media is merely the thing that carries the news from its source to various locations. In a similar manner, a wave medium is the substance that carries a wave (or disturbance) from one location to another. The wave medium is not the wave and it doesn't make the wave; it merely carries or transports the wave from its source to other locations. In the case of our slinky wave, the medium through that the wave travels is the slinky coils. In the case of a water wave in the ocean, the medium through which the wave travels is the ocean water. In the case of a sound wave moving from the church choir to the pews, the medium through which the sound wave travels is the air in the room. And in the case of the stadium wave, the medium through which the stadium wave travels is the fans that are in the stadium.

Particle-to-Particle Interaction

To fully understand the nature of a wave, it is important to consider the medium as a collection of interacting particles. In other words, the medium is composed of parts that are capable of interacting with each other. The interactions of one particle of the medium with the next adjacent particle allow the disturbance to travel through the medium. In the case of the slinky wave, the particles or interacting parts of the medium are the individual coils of the slinky. In the case of a sound wave in air, the particles or interacting parts of the medium are the individual molecules of air. And in the case of a stadium wave, the particles or interacting parts of the medium are the fans in the stadium.

Consider the presence of a wave in a slinky. The first coil becomes disturbed and begins to push or pull on the second coil; this push or pull on the second coil will displace the second coil from its equilibrium position. As the second coil becomes displaced, it begins to push or pull on the third coil; the push or pull on the third coil displaces it from its equilibrium position. As the third coil becomes displaced, it begins to push or pull on the fourth coil. This process continues in consecutive fashion, with each individual particle acting to displace the adjacent particle. Subsequently, the disturbance travels through the medium. The medium can be pictured as a series of particles connected by springs. As one particle moves, the spring connecting it to the next particle begins to stretch and apply a force to its adjacent neighbor. As this neighbor begins to move, the spring attaching this neighbor to its neighbor begins to stretch and apply a force on its adjacent neighbor.

A Wave Transports Energy and Not Matter

When a wave is present in a medium (that is, when there is a disturbance moving through a medium), the individual particles of the medium are only temporarily displaced from their rest position. There is always a force acting upon the particles that restores them to their original position. In a slinky wave, each coil of the slinky ultimately returns to its original position. In a water wave, each molecule of the water ultimately returns to its original position. And in a stadium wave, each fan in the bleacher ultimately returns to its original position. It is for this reason, that a wave is said to involve the movement of a disturbance without the movement of matter. The particles of the medium (water molecules, slinky coils, stadium fans) simply vibrate about a fixed position as the pattern of the disturbance moves from one location to another location.

Waves are said to be an energy transport phenomenon. As a disturbance moves through a medium from one particle to its adjacent particle, energy is being transported from one end of the medium to the other. In a slinky wave, a person imparts energy to the first coil by doing work upon it. The first coil receives a large amount of energy that it subsequently transfers to the second coil. When the first coil returns to its original position, it possesses the same amount of energy as it had before it was displaced. The first coil transferred its energy to the second coil. The second coil then has a large amount of energy that it subsequently transfers to the third coil. When the second coil returns to its original position, it possesses the same amount of energy as it had before it was displaced. The third coil has received the energy of the second coil. This process of energy transfer continues as each coil interacts with its neighbor. In this manner, energy is transported from one end of the slinky to the other, from its source to another location.

This characteristic of a wave as an energy transport phenomenon distinguishes waves from other types of phenomenon. Consider a common phenomenon observed at a softball game - the collision of a bat with a ball. A batter is able to transport energy from her to the softball by means of a bat. The batter applies a force to the bat, thus imparting energy to the bat in the form of kinetic energy. The bat then carries this energy to the softball and transports the energy to the softball upon collision. In this example, a bat is used to transport energy from the player to the softball. However, unlike wave phenomena, this phenomenon involves the transport of matter. The bat must move from its starting location to the contact location in order to transport energy. In a wave phenomenon, energy can move from one location to another, yet the particles of matter in the medium return to their fixed position. A wave transports its energy without transporting matter.

Waves are seen to move through an ocean or lake; yet the water always returns to its rest position. Energy is transported through the medium, yet the water molecules are not transported. Proof of this is the fact that there is still water in the middle of the ocean. The water has not moved from the middle of the ocean to the shore. If we were to observe a gull or duck at rest on the water, it would merely bob up-and-down in a somewhat circular fashion as the disturbance moves through the water. The gull or duck always returns to its original position. The gull or duck is not transported to the shore because the water on which it rests is not transported to the shore. In a water wave, energy is transported without the transport of water.

The same thing can be said about a stadium wave. In a stadium wave, the fans do not get out of their seats and walk around the stadium. We all recognize that it would be silly (and embarrassing) for any fan to even contemplate such a thought. In a stadium wave, each fan rises up and returns to the original seat. The disturbance moves through the stadium, yet the fans are not transported. Waves involve the transport of energy without the transport of matter.

In conclusion, a wave can be described as a disturbance that travels through a medium, transporting energy from one location (its source) to another location without transporting matter. Each individual particle of the medium is temporarily displaced and then returns to its original equilibrium positioned.

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 Interactive. We would like to suggest that you combine the reading of this page with the use of our Slinky Lab Interactive. You can find it in the Physics Interactives section of our website. The Slinky Lab provides the learner with a simple environment for exploring the movement of a wave along a medium and the factors that affect its speed.

Waves and Wavelike Motion

· Waves and Wavelike Motion

· What is a Wave?

· Categories of Waves

Waves are everywhere. Whether we recognize it or not, we encounter waves on a daily basis. Sound waves, visible light waves, radio waves, microwaves, water waves, sine waves, cosine waves, stadium waves, earthquake waves, waves on a string, and slinky waves and are just a few of the examples of our daily encounters with waves. In addition to waves, there are a variety of phenomena in our physical world that resemble waves so closely that we can describe such phenomenon as being wavelike. The motion of a pendulum, the motion of a mass suspended by a spring, the motion of a child on a swing, and the "Hello, Good Morning!" wave of the hand can be thought of as wavelike phenomena. Waves (and wavelike phenomena) are everywhere!

We study the physics of waves because it provides a rich glimpse into the physical world that we seek to understand and describe as students of physics. Before beginning a formal discussion of the nature of waves, it is often useful to ponder the various encounters and exposures that we have of waves. Where do we see waves or examples of wavelike motion? What experiences do we already have that will help us in understanding the physics of waves?

For many people, the first thought concerning waves conjures up a picture of a wave moving across the surface of an ocean, lake, pond or other body of water. The waves are created by some form of a disturbance, such as a rock thrown into the water, a duck shaking its tail in the water or a boat moving through the water. The water wave has a crest and a trough and travels from one location to another. One crest is often followed by a second crest that is often followed by a third crest. Every crest is separated by a trough to create an alternating pattern of crests and troughs. A duck or gull at rest on the surface of the water is observed to bob up-and-down at rather regular time intervals as the wave passes by. The waves may appear to be plane waves that travel together as a front in a straight-line direction, perhaps towards a sandy shore. Or the waves may be circular waves that originate from the point where the disturbances occur; such circular waves travel across the surface of the water in all directions. These mental pictures of water waves are useful for understanding the nature of a wave and will be revisited later when we begin our formal discussion of the topic.

The thought of waves often brings to mind a recent encounter at the baseball or football stadium when the crowd enthusiastically engaged in doing the wave. When performed with reasonably good timing, a noticeable ripple is produced that travels around the circular stadium or back and forth across a section of bleachers. The observable ripple results when a group of enthusiastic fans rise up from their seats, swing their arms up high, and then sit back down. Beginning in Section 1, the first row of fans abruptly rise up to begin the wave; as they sit back down, row 2 begins its motion; as row 2 sits back down, row 3 begins its motion. The process continues, as each consecutive row becomes involved by a momentary standing up and sitting back down. The wave is passed from row to row as each individual member of the row becomes temporarily displaced out of his or her seat, only to return to it as the wave passes by. This mental picture of a stadium wave will also provide a useful context for the discussion of the physics of wave motion.

Another picture of waves involves the movement of a slinky or similar set of coils. If a slinky is stretched out from end to end, a wave can be introduced into the slinky by either vibrating the first coil up and down vertically or back and forth horizontally. A wave will subsequently be seen traveling from one end of the slinky to the other. As the wave moves along the slinky, each individual coil is seen to move out of place and then return to its original position. The coils always move in the same direction that the first coil was vibrated. A continued vibration of the first coil results in a continued back and forth motion of the other coils. If looked at closely, one notices that the wave does not stop when it reaches the end of the slinky; rather it seems to bounce off the end and head back from where it started. A slinky wave provides an excellent mental picture of a wave and will be used in discussions and demonstrations throughout this unit.

We likely have memories from childhood of holding a long jump rope with a friend and vibrating an end up and down. The up and down vibration of the end of the rope created a disturbance of the rope that subsequently moved towards the other end. Upon reaching the opposite end, the disturbance often bounced back to return to the end we were holding. A single disturbance could be created by the single vibration of one end of the rope. On the other hand, a repeated disturbance would result in a repeated and regular vibration of the rope. The shape of the pattern formed in the rope was influenced by the frequency at which we vibrated it. If we vibrated the rope rapidly, then a short wave was created. And if we vibrated the rope less frequently (not as often), a long wave was created. While we were likely unaware of it as children, we were entering the world of the physics of waves as we contentedly played with the rope.

Then there is the "Hello, Good Morning!" wave. Whether encountered in the driveway as you begin your trip to school, on the street on the way to school, in the parking lot upon arrival to school, or in the hallway on the way to your first class, the "Hello, Good Morning!" wave provides a simple (yet excellent) example of physics in action. The simple back and forth motion of the hand is called a wave. When Mom commands us to "wave to Mr. Smith," she is telling us to raise our hand and to temporarily or even repeatedly vibrate it back and forth. The hand is raised, moved to the left, and then back to the far right and finally returns to its original position. Energy is put into the hand and the hand begins its back-and-forth vibrational motion. And we call the process of doing it "waving." Soon we will see how this simple act is representative of the nature of a physical wave.

We also encountered waves in Math class in the form of the sine and cosine function. We often plotted y = B•sine(A•x) on our calculator or by hand and observed that its graphical shape resembled the characteristic shape of a wave. There was a crest and a trough and a repeating pattern. If we changed the constant A in the equation, we noticed that we could change the length of the repeating pattern. And if we changed B in the equation, we noticed that we changed the height of the pattern. In math class, we encountered the underlying mathematical functions that describe the physical nature of waves.

Finally, we are familiar with microwaves and visible light waves. While we have never seen them, we believe that they exist because we have witnessed how they carry energy from one location to another. And similarly, we are familiar with radio waves and sound waves. Like microwaves, we have never seen them. Yet we believe they exist because we have witnessed the signals that they carry from one location to another and we have even learned how to tune into those signals through use of our ears or a tuner on a television or radio. Waves, as we will learn, carry energy from one location to another. And if the frequency of those waves can be changed, then we can also carry a complex signal that is capable of transmitting an idea or thought from one location to another. Perhaps this is one of the most important aspects of waves and will become a focus of our study in later units.

Waves are everywhere in nature. Our understanding of the physical world is not complete until we understand the nature, properties and behaviors of waves. The goal of this unit is to develop mental models of waves and ultimately apply those models to an understanding of the two most common types of waves - sound waves and light waves.

Vibrational Motion

· Vibrational Motion

· Properties of Periodic Motion

· Pendulum Motion

· Motion of a Mass on a Spring

Things wiggle. They do the back and forth. They vibrate; they shake; they oscillate. These phrases describe the motion of a variety of objects. They even describe the motion of matter at the atomic level. Even atoms wiggle - they do the back and forth. Wiggles, vibrations, and oscillations are an inseparable part of nature. In this chapter of The Physics Classroom Tutorial, we will make an effort to understand vibrational motion and its relationship to waves. An understanding of vibrations and waves is essential to understanding our physical world. Much of what we see and hear is only possible because of vibrations and waves. We see the world around us because of light waves. And we hear the world around us because of sound waves. If we can understand waves, then we will be able to understand the world of sight and sound.

Bobblehead Dolls - An Example of a Vibrating Object

To begin our ponderings of vibrations and waves, consider one of those crazy bobblehead dolls that you've likely seen at baseball stadiums or novelty shops. A bobblehead doll consists of an oversized replica of a person's head attached by a spring to a body and a stand. A light tap to the oversized head causes it to bobble. The head wiggles; it vibrates; it oscillates. When pushed or somehow disturbed, the head does the back and forth. The back and forth doesn't happen forever. Over time, the vibrations tend to die off and the bobblehead stops bobbing and finally assumes its usual resting position.

The bobblehead doll is a good illustration of many of the principles of vibrational motion. Think about how you would describe the back and forth motion of the oversized head of a bobblehead doll. What words would you use to describe such a motion? How does the motion of the bobblehead change over time? How does the motion of one bobblehead differ from the motion of another bobblehead? What quantities could you measure to describe the motion and so distinguish one motion from another motion? How would you explain the cause of such a motion? Why does the back and forth motion of the bobblehead finally stop? These are all questions worth pondering and answering if we are to understand vibrational motion. These are the questions we will attempt to answer in Section 1 of this chapter.

What Causes Objects to Vibrate?

Like any object that undergoes vibrational motion, the bobblehead has a resting position. The resting position is the position assumed by the bobblehead when it is not vibrating. The resting position is sometimes referred to as the equilibrium position. When an object is positioned at its equilibrium position, it is in a state of equilibrium. As discussed in the Newton's Law Chapter of the Tutorial, an object which is in a state of equilibrium is experiencing a balance of forces. All the individual forces - gravity, spring, etc. - are balanced or add up to an overall net force of 0 Newtons. When a bobblehead is at the equilibrium position, the forces on the bobblehead are balanced. The bobblehead will remain in this position until somehow disturbed from its equilibrium.

If a force is applied to the bobblehead, the equilibrium will be disturbed and the bobblehead will begin vibrating. We could use the phrase forced vibration to describe the force which sets the otherwise resting bobblehead into motion. In this case, the force is a short-lived, momentary force that begins the motion. The bobblehead does its back and forth, repeating the motion over and over. Each repetition of its back and forth motion is a little less vigorous than its previous repetition. If the head sways 3 cm to the right of its equilibrium position during the first repetition, it may only sway 2.5 cm to the right of its equilibrium position during the second repetition. And it may only sway 2.0 cm to the right of its equilibrium position during the third repetition. And so on. The extent of its displacement from the equilibrium position becomes less and less over time. Because the forced vibration that initiated the motion is a single instance of a short-lived, momentary force, the vibrations ultimately cease. The bobblehead is said to experience damping. Damping is the tendency of a vibrating object to lose or to dissipate its energy over time. The mechanical energy of the bobbing head is lost to other objects. Without a sustained forced vibration, the back and forth motion of the bobblehead eventually ceases as energy is dissipated to other objects. A sustained input of energy would be required to keep the back and forth motion going. After all, if the vibrating object naturally loses energy, then it must continuously be put back into the system through a forced vibration in order to sustain the vibration.

The Restoring Force

A vibrating bobblehead often does the back and forth a number of times. The vibrations repeat themselves over and over. As such, the bobblehead will move back to (and past) the equilibrium position every time it returns from its maximum displacement to the right or the left (or above or below). This begs a question - and perhaps one that you have been thinking of yourself as you've pondered the topic of vibration. If the forces acting upon the bobblehead are balanced when at the equilibrium position, then why does the bobblehead sway past this position? Why doesn't the bobblehead stop the first time it returns to the equilibrium position? The answer to this question can be found in Newton's first law of motion. Like any moving object, the motion of a vibrating object can be understood in light of Newton's laws. According to Newton's law of inertia, an object which is moving will continue its motion if the forces are balanced. Put another way, forces, when balanced, do not stop moving objects. So every instant in time that the bobblehead is at the equilibrium position, the momentary balance of forces will not stop the motion. The bobblehead keeps moving. It moves past the equilibrium position towards the opposite side of its swing. As the bobblehead is displaced past its equilibrium position, then a force capable of slowing it down and stopping it exists. This force that slows the bobblehead down as it moves away from its equilibrium position is known as a restoring force. The restoring force acts upon the vibrating object to move it back to its original equilibrium position.

Vibrational motion is often contrasted with translational motion. In translational motion, an object is permanently displaced. The initial force that is imparted to the object displaces it from its resting position and sets it into motion. Yet because there is no restoring force, the object continues the motion in its original direction. When an object vibrates, it doesn't move permanently out of position. The restoring force acts to slow it down, change its direction and force it back to its original equilibrium position. An object in translational motion is permanently displaced from its original position. But an object in vibrational motion wiggles about a fixed position - its original equilibrium position. Because of the restoring force, vibrating objects do the back and forth. We will explore the restoring force in more detail later in this lesson.

Other Vibrating Systems

As you know, bobblehead dolls are not the only objects that vibrate. It might be safe to say that all objects in one way or another can be forced to vibrate to some extent. The vibrations might not be large enough to be visible. Or the amount of damping might be so strong that the object scarcely completes a full cycle of vibration. But as long as a force persists to restore the object to its original position, a displacement from its resting position will result in a vibration. Even a large massive skyscraper is known to vibrate as winds push upon its structure. While held fixed in place at its foundation (we hope), the winds force the length of the structure out of position and the skyscraper is forced into vibration.

A pendulum is a classic example of an object that is considered to vibrate. A simple pendulum consists of a relatively massive object hung by a string from a fixed support. It typically hangs vertically in its equilibrium position. When the mass is displaced from equilibrium, it begins its back and forth vibration about its fixed equilibrium position. The motion is regular and repeating. In the next part of this lesson, we will describe such a regular and repeating motion as a periodic motion. Because of the regular nature of a pendulum's motion, many clocks, such as grandfather clocks, use a pendulum as part of its timing mechanism.

An inverted pendulum is another classic example of an object that undergoes vibrational motion. An inverted pendulum is simply a pendulum which has its fixed end located below the vibrating mass. An inverted pendulum can be made by attaching a mass (such as a tennis ball) to the top end of a dowel rod and then securing the bottom end of the dowel rod to a horizontal support. This is shown in the diagram below. A gentle force exerted upon the tennis ball will cause it to vibrate about a fixed, equilibrium position. The vibrating skyscraper can be thought of as a type of inverted pendulum. Tall trees are often displaced from their usual vertical orientation by strong winds. As the winds cease, the trees will vibrate back and forth about their fixed positions. Such trees can be thought of as acting as inverted pendula. Even the tines of a tuning fork can be considered a type of inverted pendulum

Another classic example of an object that undergoes vibrational motion is a mass on a spring. The animation at the right depicts a mass suspended from a spring. The mass hangs at a resting position. If the mass is pulled down, the spring is stretched. Once the mass is released, it begins to vibrate. It does the back and forth, vibrating about a fixed position. If the spring is rotated horizontally and the mass is placed upon a supporting surface, the same back and forth motion can be observed. Pulling the mass to the right of its resting position stretches the spring. When released, the mass is pulled back to the left, heading towards its resting position. After passing by its resting position, the spring begins to compress. The compressions of the coiled spring result in a restoring force that again pushes rightward on the leftward moving mass. The cycle continues as the mass vibrates back and forth about a fixed position. The springs inside of a bed mattress, the suspension systems of some cars, and bathroom scales all operated as a mass on a spring system.

In all the vibrating systems just mentioned, damping is clearly evident. The simple pendulum doesn't vibrate forever; its energy is gradually dissipated through air resistance and loss of energy to the support. The inverted pendulum consisting of a tennis ball mounted to the top of a dowel rod does not vibrate forever. Like the simple pendulum, the energy of the tennis ball is dissipated through air resistance and vibrations of the support. Frictional forces also cause the mass on a spring to lose its energy to the surroundings. In some instances, damping is a favored feature. Car suspension systems are intended to dissipate vibrational energy, preventing drivers and passengers from having to do the back and forth as they also do the down the road.

Hopefully a lot of our original questions have been answered. But one question that has not yet been answered is the question pertaining to quantities that can be measured. How can we quantitatively describe a vibrating object? What measurements can be made of vibrating objects that would distinguish one vibrating object from another? We will ponder this question in the next part of this lesson on vibrational motion.

(The animation of the mass on a spring is a public domain file from WikiMedia Commons. Special thanks to Oleg Alexandrov for his creation.)

The Speed of a Wave

· The Anatomy of a Wave

· Frequency and Period of a Wave

· Energy Transport and the Amplitude of a Wave

· The Speed of a Wave

· The Wave Equation

A wave is a disturbance that moves along a medium from one end to the other. If one watches an ocean wave moving along the medium (the ocean water), one can observe that the crest of the wave is moving from one location to another over a given interval of time. The crest is observed to cover distance. The speed of an object refers to how fast an object is moving and is usually expressed as the distance traveled per time of travel. In the case of a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given interval of time. In equation form,

If the crest of an ocean wave moves a distance of 20 meters in 10 seconds, then the speed of the ocean wave is 2.0 m/s. On the other hand, if the crest of an ocean wave moves a distance of 25 meters in 10 seconds (the same amount of time), then the speed of this ocean wave is 2.5 m/s. The faster wave travels a greater distance in the same amount of time.

Sometimes a wave encounters the end of a medium and the presence of a different medium. For example, a wave introduced by a person into one end of a slinky will travel through the slinky and eventually reach the end of the slinky and the presence of the hand of a second person. One behavior that waves undergo at the end of a medium is reflection. The wave will reflect or bounce off the person's hand. When a wave undergoes reflection, it remains within the medium and merely reverses its direction of travel. In the case of a slinky wave, the disturbance can be seen traveling back to the original end. A slinky wave that travels to the end of a slinky and back has doubled its distance. That is, by reflecting back to the original location, the wave has traveled a distance that is equal to twice the length of the slinky.

Reflection phenomena are commonly observed with sound waves. When you let out a holler within a canyon, you often hear the echo of the holler. The sound wave travels through the medium (air in this case), reflects off the canyon wall and returns to its origin (you). The result is that you hear the echo (the reflected sound wave) of your holler. A classic physics problem goes like this:

Noah stands 170 meters away from a steep canyon wall. He shouts and hears the echo of his voice one second later. What is the speed of the wave?

In this instance, the sound wave travels 340 meters in 1 second, so the speed of the wave is 340 m/s. Remember, when there is a reflection, the wave doubles its distance. In other words, the distance traveled by the sound wave in 1 second is equivalent to the 170 meters down to the canyon wall plus the 170 meters back from the canyon wall.

Variables Affecting Wave Speed

What variables affect the speed at which a wave travels through a medium? Does the frequency or wavelength of the wave affect its speed? Does the amplitude of the wave affect its speed? Or are other variables such as the mass density of the medium or the elasticity of the medium responsible for affecting the speed of the wave? These questions are often investigated in the form of a lab in a physics classroom.

Suppose a wave generator is used to produce several waves within a rope of a measurable tension. The wavelength, frequency and speed are determined. Then the frequency of vibration of the generator is changed to investigate the effect of frequency upon wave speed. Finally, the tension of the rope is altered to investigate the effect of tension upon wave speed. Sample data for the experiment are shown below.

Speed of a Wave Lab - Sample Data

Trial	Tension (N)	Frequency (Hz)	Wavelength (m)	Speed (m/s)
1	2.0	4.05	4.00	16.2
2	2.0	8.03	2.00	16.1
3	2.0	12.30	1.33	16.4
4	2.0	16.2	1.00	16.2
5	2.0	20.2	0.800	16.2
6	5.0	12.8	2.00	25.6
7	5.0	19.3	1.33	25.7
8	5.0	25.5	1.00	25.5

In the first five trials, the tension of the rope was held constant and the frequency was systematically changed. The data in rows 1-5 of the table above demonstrate that a change in the frequency of a wave does not affect the speed of the wave. The speed remained a near constant value of approximately 16.2 m/s. The small variations in the values for the speed were the result of experimental error, rather than a demonstration of some physical law. The data convincingly show that wave frequency does not affect wave speed. An increase in wave frequency caused a decrease in wavelength while the wave speed remained constant.

The last three trials involved the same procedure with a different rope tension. Observe that the speed of the waves in rows 6-8 is distinctly different than the speed of the wave in rows 1-5. The obvious cause of this difference is the alteration of the tension of the rope. The speed of the waves was significantly higher at higher tensions. Waves travel through tighter ropes at higher speeds. So while the frequency did not affect the speed of the wave, the tension in the medium (the rope) did. In fact, the speed of a wave is not dependent upon (causally affected by) properties of the wave itself. Rather, the speed of the wave is dependent upon the properties of the medium such as the tension of the rope.

One theme of this unit has been that "a wave is a disturbance moving through a medium." There are two distinct objects in this phrase - the "wave" and the "medium." The medium could be water, air, or a slinky. These media are distinguished by their properties - the material they are made of and the physical properties of that material such as the density, the temperature, the elasticity, etc. Such physical properties describe the material itself, not the wave. On the other hand, waves are distinguished from each other by their properties - amplitude, wavelength, frequency, etc. These properties describe the wave, not the material through which the wave is moving.

Check Your Understanding

1. A teacher attaches a slinky to the wall and begins introducing pulses with different amplitudes. Which of the two pulses (A or B) below will travel from the hand to the wall in the least amount of time? Justify your answer.

See Answer

They reach the wall at the same time. Don't be fooled! The amplitude of a wave does not affect the speed at which the wave travels. Both Wave A and Wave B travel at the same speed. The speed of a wave is only altered by alterations in the properties of the medium through which it travels.

2. The teacher then begins introducing pulses with a different wavelength. Which of the two pulses (C or D) will travel from the hand to the wall in the least amount of time ? Justify your answer.

See Answer

They reach the wall at the same time. Don't be fooled! The wavelength of a wave does not affect the speed at which the wave travels. Both Wave C and Wave D travel at the same speed. The speed of a wave is only altered by alterations in the properties of the medium through which it travels.

3. The time required for the sound waves (v = 340 m/s) to travel from the tuning fork to point A is ____ .

a. 0.020 second	b. 0.059 second
c. 0.59 second	d. 2.9 second

See Answer

Answer: B

GIVEN: v = 340 m/s, d = 20 m and f = 1000 Hz

Find time

Use v = d / t and rearrange to t = d / v

Substitute and solve.

4. Two waves are traveling through the same container of nitrogen gas. Wave A has a wavelength of 1.5 m. Wave B has a wavelength of 4.5 m. The speed of wave B must be ________ the speed of wave A.

a. one-ninth	b. one-third
c. the same as	d. three times larger than

See Answer

Answer: C

The medium is the same for both of these waves ("the same container of nitrogen gas"). Thus, the speed of the wave will be the same. Alterations in a property of a wave (such as wavelength) will not affect the speed of the wave. Two different waves travel with the same speed when present in the same medium.

5. An automatic focus camera is able to focus on objects by use of an ultrasonic sound wave. The camera sends out sound waves that reflect off distant objects and return to the camera. A sensor detects the time it takes for the waves to return and then determines the distance an object is from the camera. The camera lens then focuses at that distance. Now that's a smart camera! In a subsequent life, you might have to be a camera; so try this problem for practice:

If a sound wave (speed = 340 m/s) returns to the camera 0.150 seconds after leaving the camera, then how far away is the object?

See Answer

GIVEN: v = 340 m/s, t = 0.150 s (down and back time)

Find d (1-way)

If it takes 0.150 s to travel to the object and back, then it takes 0.075 s to travel the one-way distance to the object. Now solve for time using the equation d = v • t.

d = v • t = (340 m/s) • (0.075 s) = 25.5 m

6. TRUE or FALSE:

Doubling the frequency of a wave source doubles the speed of the waves.

See Answer

FALSE!

The speed of a wave is unaffected by changes in the frequency.

7. While hiking through a canyon, Noah Formula lets out a scream. An echo (reflection of the scream off a nearby canyon wall) is heard 0.82 seconds after the scream. The speed of the sound wave in air is 342 m/s. Calculate the distance from Noah to the nearby canyon wall.

See Answer

GIVEN: v = 342 m/s, t = 0.82 s (2-way)

Find d (1-way)

If it takes 0.82 s to travel to the canyon wall and back (a down-and-back time), then it takes 0.41 s to travel the one-way distance to the wall. Now use d = v • t

d = v • t = (342 m/s) • (0.41 s) = 140 m

8. Mac and Tosh are resting on top of the water near the end of the pool when Mac creates a surface wave. The wave travels the length of the pool and back in 25 seconds. The pool is 25 meters long. Determine the speed of the wave.

See Answer

GIVEN: d (1-way) =25 m, t (2-way)=25 s

Find v.

If the pool is 25 meters long, then the back-and-forth distance is 50 meters. The wave covers this distance in 25 seconds. Now use v = d / t.

v = d / t = (50 m) / (25 s) = 2 m/s

9. The water waves below are traveling along the surface of the ocean at a speed of 2.5 m/s and splashing periodically against Wilbert's perch. Each adjacent crest is 5 meters apart. The crests splash Wilbert's feet upon reaching his perch. How much time passes between each successive drenching? Answer and explain using complete sentences.

Universal Learning House

Sound Waves

Nodes and Anti-nodes

What are Nodes and Antinodes?

Standing Wave Diagrams

Check Your Understanding

Answer: C (9 nodes)

Interference of Waves

What is Interference?

Constructive Interference

The Principle of Superposition

Frequency and Period of a Wave

Investigate!

The Anatomy of a Wave

Categories of Waves

The Speed of a Wave

Variables Affecting Wave Speed

They reach the wall at the same time. Don't be fooled! The amplitude of a wave does not affect the speed at which the wave travels. Both Wave A and Wave B travel at the same speed. The speed of a wave is only altered by alterations in the properties of the medium through which it travels.

GIVEN: v = 342 m/s, t = 0.82 s (2-way)

Comments

Post a Comment

Popular posts from this blog

Irin, a mother of silent ocean

Home, my sweet home

Hooks Law and its application