traveling waves vs standing wave

7/27/2019 Traveling Waves Vs standing wave

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Traveling Waves vs. 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 movingalong 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 encountersanother

wave along the mediumor until it encountersa 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 theninterferewith 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 crestmeeting 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.

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 readilyobservable.

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
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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 detaillater 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 discussedlater in the lesson.

Formation of Standing Waves

A standing wave pattern is a vibrational pattern created within a medium when the vibrationalfrequency of the source causes reflected waves from one end of the medium tointerferewith 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.

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 fromend 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 itsfixed 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 undergodestructive 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 ofsuperposition) 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
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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.

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-crestmeets a half-trough or a quarter-crestmeets 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 endof 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 timedcrests. 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 ofmusical instruments has a basis in the conceptual and mathematical aspects of standing waves. For

this reason, the topic will be revisited in theSound and Music unitat The Physics Classroom Tutorial.

Nodes and Anti-nodes

As mentionedearlier in Lesson 4, 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
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presence of two waves of the same frequency with different directions of travel within the same

medium.

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 undergovibrations 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 namestanding 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.


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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-crestof one wave meets a half-trough of a second wave; or a quarter-

crestof 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 atraveling

waveis discussed, it is customary to refer to a point of large maximum displacement as acrestand a

point of large negative displacement as atrough. 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
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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.

Mathematics of Standing Waves

As discussed in Lesson 4,standing wave patternsare wave patterns produced in a medium when two

waves of identical frequencies interfere in such a manner to produce points along the medium thatalways appear to be standing still. Such standing wave patterns are produced within the medium

when it is vibrated at certain frequencies. Each frequency is associated with a different standing wave

pattern. These frequencies and their associated wave patterns are referred to asharmonics. A careful

study of the standing wave patterns reveal a clear mathematical relationship between the wavelength

of the wave that produces the pattern and the length of the medium in which the pattern is displayed.

Furthermore, there is a predictability about this mathematical relationship that allows one to

generalize and deduce a statement concerning this relationship. To illustrate, consider the first

harmonic standing wave pattern for a vibrating rope as shown below.

The pattern for the first harmonic reveals a single antinode in the middle of the rope. This antinodeposition along the rope vibrates up and down from a maximum upward displacement from rest to a

maximum downward displacement as shown. The vibration of the rope in this manner creates the

appearance of a loop within the string. A complete wave in a pattern could be described as starting at

the rest position, rising upward to a peak displacement, returning back down to a rest position, then

descending to a peak downward displacement and finally returning back to the rest position. The

animation below depicts this familiar pattern. As shown in the animation, one complete wave in a

standing wave pattern consists of two loops. Thus, one loop is equivalent to one-half of a wavelength.

In comparing the standing wave pattern for the first harmonic with its single loop to the diagram of a

complete wave, it is evident that there is only one-half of a wave stretching across the length of the

string. That is, the length of the string is equal to one-half the length of a wave. Put in the form of an

equation:

Now consider the string being vibrated with a frequency that establishes the standing wave pattern for

the second harmonic.
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The second harmonic pattern consists of two anti-nodes. Thus, there are two loops within the length

of the string. Since each loop is equivalent to one-half a wavelength, the length of the string is equal

to two-halves of a wavelength. Put in the form of an equation:

The same reasoning pattern can be applied to the case of the string being vibrated with a frequency

that establishes the standing wave pattern for the third harmonic.

The third harmonic pattern consists of three anti-nodes. Thus, there are three loops within the length

of the string. Since each loop is equivalent to one-half a wavelength, the length of the string is equal

to three-halves of a wavelength. Put in the form of an equation:

When inspecting the standing wave patterns and the length-wavelength relationships for the first

three harmonics, a clear pattern emerges. The number of antinodes in the pattern is equal to

the harmonic number of that pattern. The first harmonic has one antinode; the second harmonic has

two antinodes; and the third harmonic has three antinodes. Thus, it can be generalized that the nth

harmonic has n antinodes where n is an integer representing the harmonic number. Furthermore, one

notices that there are n halves wavelengths present within the length of the string. Put in the form of

an equation:

This information is summarized in the table below.

Harmonic Pattern # of Loops

Length-Wavelength

Relationship

1st 1 L= 1 / 2

2nd 2 L= 2 / 2

3rd 3 L= 3 / 2

4th 4 L= 4 / 2


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5th 5 L= 5 / 2

6th 6 L= 6 / 2

nth -- n L= n / 2

For standing wave patterns, there is a clear mathematical relationship between the length of a string

and the wavelength of the wave that creates the pattern. The mathematical relationship simply

emerges from the inspection of the pattern and the understanding that each loop in the pattern is

equivalent to one-half of a wavelength. The general equation that describes this length-wavelength

relationship for any harmonic is:

traveling waves vs standing wave

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