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What is a Wave?

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 newsmoves

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 interactingparticles.

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, theparticles or interacting parts of the medium are the individual coils of the slinky. In the case

of a sound wave in air, theparticles or interacting parts of the medium are

the individual molecules of air. And in the case of a stadium wave,

theparticles or interacting parts of the medium are the fans in the stadium.


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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 individualparticle 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 astadium

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 (watermolecules, 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 thismanner, 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 athought. 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.


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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.

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 thechain of particles until the


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sound wave reaches the ear of the listener. A detailed discussion ofsound is presented in another unit ofThe 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.


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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 ofThe 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 mechanicalwave. Sound waves are incapable of traveling through a vacuum. Slinky waves, water waves, stadium waves, andjump

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.


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Sound Waves and Music - Chapter Outline

Lesson 1: The Nature of a Sound Wavea. Sound is a Mechanical Waveb. Sound is a Longitudinal Wavec. Sound is a Pressure Wave

Lesson 2: Sound Properties and Their Perceptiona. Pitch and Frequencyb. Intensity and the Decibel Scalec. The Speed of Soundd. The Human Ear

Lesson 3: Behavior of Sound Wavesa. Interference and Beatsb. The Doppler Effect and Shock Wavesc. Boundary Behaviord. Reflection, Refraction, and Diffraction

Lesson 4: Resonance and Standing Wavesa. Natural Frequencyb. Forced Vibrationc.

Standing Wave Patternsd. Fundamental Frequency and Harmonics

Lesson 5: Musical Instrumentsa. Resonanceb. Guitar Stringsc. Open-End Air Columnsd. Closed-End Air Columns

LESSON 1

Sound is a Mechanical Wave

Sound and music are parts of our everyday sensory experience. Just as humans have eyes for the detection of light and

color, so we are equipped with ears for the detection of sound. We seldom take the time to ponder the characteristics

and behaviors of sound and the mechanisms by which sounds are produced, propagated, and detected. The basis for an

understanding of sound, music and hearing is the physics of waves. Sound is a wave that is created by vibrating objects

and propagated through a medium from one location to another. In this unit, we will investigate the nature, properties

and behaviors of sound waves and apply basic wave principles towards an understanding of music.

As discussed in the previous unit of The Physics Classroom Tutorial, a wave can be described as a disturbance that

travels through a medium, transporting energy from one location to another location. The medium is simply the

material through which the disturbance is moving; it can be thought of as a series of interacting particles. The example

of a slinky wave is often used to illustrate the nature of a wave. A disturbance is typically created within the slinky by theback and forth movement of the first coil of the 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 individualparticle acting to displace the adjacent particle. Subsequently the

disturbance travels through the slinky. As the disturbance moves from coil to coil, the energy that was originally

introduced into the first coil is transported along the medium from one location to another.

A sound wave is similar in nature to a slinky wave for a variety of reasons. First, there is a medium that carries the

disturbance from one location to another. Typically, this medium is air, though it could be any material such as water or


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steel. The medium is simply a series of interconnected and interacting particles. Second, there is an original source of

the wave, some vibrating object capable of disturbing the first particle of the medium. The disturbance could be created

by the vibrating vocal cords of a person, the vibrating string and soundboard of a guitar or violin, the vibrating tines of a

tuning fork, or the vibrating diaphragm of a radio speaker. Third, the sound wave is transported from one location to

another by means of particle-to-particle interaction. If the sound wave is moving through air, then as one air particle is

displaced from its equilibrium position, it exerts a push or pull on its nearest neighbors, causing them to be displaced

from their equilibrium position. This particle interaction continues throughout the entire medium, with each particle

interacting and causing a disturbance of its nearest neighbors. Since a sound wave is a disturbance that is transported

through a medium via the mechanism of particle-to-particle interaction, a sound wave is characterized as a mechanical

wave.

The creation and propagation of sound waves are often demonstrated in class through the use of a tuning fork. A tuning

fork is a metal object consisting of two tines capable of vibrating if struck by a rubber hammer or mallet. As the tines of

the tuning forks vibrate back and forth, they begin to disturb surrounding air molecules. These disturbances are passed

on to adjacent air molecules by the mechanism of particle interaction. The motion of the disturbance, originating at the

tines of the tuning fork and traveling through the medium (in this case, air) is what is referred to as a sound wave. The

generation and propagation of a sound wave is demonstrated in the animation below.

Many Physics demonstration tuning forks are mounted on a sound box. In such

instances, the vibrating tuning fork, being connectedto the sound box, sets the sound

box into vibrational motion. In turn, the sound box, being connectedto the air inside of

it, sets the air inside of the sound box into vibrational motion. As the tines of the tuning

fork, the structure of the sound box, and the air inside of the sound box begin vibrating

at the same frequency, a louder sound is produced. In fact, the more particles that can

be made to vibrate, the louder or more amplified the sound. This concept is oftendemonstrated by the placement of a vibrating tuning fork against the glass panel of an

overhead projector or on the wooden door of a cabinet. The vibrating tuning fork sets

the glass panel or wood door into vibrational motion and results in an amplified sound.

We know that a tuning fork is vibrating because we hear the sound that is produced by its vibration. Nonetheless, we do

not actually visibly detect any vibrations of the tines. This is because the tines are vibrating

at a very high frequency. If the tuning fork that is being used corresponds to middle C on the

piano keyboard, then the tines are vibrating at a frequency of 256 Hertz; that is, 256

vibrations per second. We are unable to visibly detect vibrations of such high frequency. A

common physics demonstration involves slowing down the vibrations by through the use of

a strobe light. If the strobe light puts out a flash of light at a frequency of 512 Hz (two times

the frequency of the tuning fork), then the tuning fork can be observed to be moving in aback and forth motion. With the room darkened, the strobe would allow us to view the

position of the tines two times during their vibrational cycle. Thus we would see the tines

when they are displaced far to the left and again when they are displaced far to the right. This would be convincing

proof that the tines of the tuning fork are indeed vibrating to produce sound.

In a previous unit ofThe Physics Classroom Tutorial, a distinction was made between two categories of

waves:mechanical waves and electromagnetic waves. Electromagnetic waves are

waves that have an electric and magnetic nature and are capable of traveling

through a vacuum. Electromagnetic waves do not require a medium in order to

transport their energy. Mechanical waves are waves that require a medium in order

to transport their energy from one location to another. Because mechanical wavesrely on particle interaction in order to transport their energy, they cannot travel

through regions of space that are void of particles. That is, mechanical waves cannot


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travel through a vacuum. This feature of mechanical waves is often demonstrated in a Physics class. A ringing bell is

placed in a jar and air inside the jar is evacuated. Once air is removed from the jar, the sound of the ringing bell can no

longer be heard. The clapper is seen striking the bell; but the sound that it produces cannot be heard because there are

no particles inside of the jar to transport the disturbance through the vacuum. Sound is a mechanical wave and cannot

travel through a vacuum.

Sound as a Longitudinal Wave

In the first part of Lesson 1, it was mentioned that sound is a mechanical wave that is created by a vibrating object. Thevibrations of the object set particles in the surrounding medium in vibrational motion, thus transporting energy through

the medium. For a sound wave traveling through air, the vibrations of the particles are best described

as longitudinal. Longitudinal waves are waves in which the motion of the individual particles of the medium is in a

direction that is parallel to the direction of energy transport. A longitudinal wave can be created in a slinky if the slinky is

stretched out in a horizontal direction and the first coils of the slinky are vibrated horizontally. In such a case, each

individual coil of the medium is set into vibrational motion in directions parallel to the direction that the energy is

transported.

Sound waves in air (and any fluid medium) are longitudinal waves because particles of the medium through which the

sound is transported vibrate parallel to the direction that the sound wave moves. A vibrating string can create

longitudinal waves as depicted in the animation below. As the vibrating string moves in theforwarddirection, it begins

to push upon surrounding air molecules, moving them to the right towards their nearest neighbor. This causes the airmolecules to the right of the string to be compressed into a small region of space. As the vibrating string moves in the

reverse direction (leftward), it lowers the pressure of the air immediately to its right, thus causing air molecules to move

back leftward. The lower pressure to the right of the string causes air molecules in that region immediately to the right

of the string to expand into a large region of space. The back and forth vibration of the string causes individual air

molecules (or a layer of air molecules) in the region immediately to the right of the string to continually vibrate back and

forth horizontally. The molecules move rightward as the string moves rightward and then leftward as the string moves

leftward. These back and forth vibrations are imparted to adjacent neighbors by particle-to-particle interaction. Other

surrounding particles begin to move rightward and leftward, thus sending a wave to the right. Since air molecules (the

particles of the medium) are moving in a direction that is parallel to the direction that the wave moves, the sound wave

is referred to as a longitudinal wave. The result of such longitudinal vibrations is the creation

ofcompressions and rarefactions within the air.

Regardless of the source of the sound wave - whether it is a vibrating string or the vibrating tines of a tuning fork -

sound waves traveling through air are longitudinal waves. And the essential characteristic of a longitudinal wave that

distinguishes it from other types of waves is that the particles of the medium move in a direction parallel to thedirection of energy transport.


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Sound is a Pressure Wave

Sound is a mechanical wave that results from the back and forth vibration of the particles of the medium through which

the sound wave is moving. If a sound wave is moving from left to right through air, then particles of air will be displaced

both rightward and leftward as the energy of the sound wave passes through it. The motion of the particles is parallel

(and anti-parallel) to the direction of the energy transport. This is what characterizes sound waves in air as longitudinal

waves.

A vibrating tuning fork is capable of creating such a longitudinal wave. As the tines of the fork vibrate back and forth,they push on neighboring air particles. The forward motion of a tine pushes air molecules horizontally to the right and

the backward retraction of the tine creates a low-pressure area allowing the air particles to move back to the left.

Because of the longitudinal motion of the air particles, there are regions in the air where the air particles are

compressed together and other regions where the air particles are spread apart. These regions are knownascompressions and rarefactions respectively. The compressions are regions of high air pressure while the rarefactions

are regions of low air pressure. The diagram below depicts a sound wave created by a tuning fork and propagated

through the air in an open tube. The compressions and rarefactions are labeled.

The wavelength of a wave is merely the distance that a disturbance travels along the medium in one complete wave

cycle. Since a wave repeats its pattern once every wave cycle, the wavelength is sometimes referred to as the length of

the repeating patterns - the length of one complete wave. For a transverse wave, this length is commonly measured

from one wave crest to the next adjacent wave crest or from one wave trough to the next adjacent wave trough. Since a

longitudinal wave does not contain crests and troughs, its wavelength must be measured differently. A longitudinal

wave consists of a repeating pattern of compressions and rarefactions. Thus, the wavelength is commonly measured as

the distance from one compression to the next adjacent compression or the distance from one rarefaction to the next

adjacent rarefaction.

Since a sound wave consists of a repeating pattern of high-pressure and low-pressure regions moving through a

medium, it is sometimes referred to as a pressure wave. If a detector, whether it is the human ear or a man-madeinstrument, were used to detect a sound wave, it would detect fluctuations in pressure as the sound wave impinges

upon the detecting device. At one instant in time, the detector would detect a high pressure; this would correspond to

the arrival of a compression at the detector site. At the next instant in time, the detector might detect normal pressure.

And then finally a low pressure would be detected, corresponding to the arrival of a rarefaction at the detector site. The

fluctuations in pressure as detected by the detector occur at periodic and regular time intervals. In fact, a plot of

pressure versus time would appear as a sine curve. The peak points of the sine curve correspond to compressions; the

low points correspond to rarefactions; and the "zero points" correspond to the pressure that the air would have if there

were no disturbance moving through it. The diagram below depicts the correspondence between the longitudinal

nature of a sound wave in air and the pressure-time fluctuations that it creates at a fixed detector location.


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The above diagram can be somewhat misleading if you are not careful. The representation of sound by a sine wave is

merely an attempt to illustrate the sinusoidal nature of the pressure-time fluctuations. Do not conclude that sound is a

transverse wave that has crests and troughs. Sound waves traveling through air are indeed longitudinal waves with

compressions and rarefactions. As sound passes through air (or any fluid medium), the particles of air do not vibrate in a

transverse manner. Do not be misled - sound waves traveling through air are longitudinal waves.

Sound Waves and the Eardrum

A sound wave traveling through a fluid medium (such as a liquid or a gaseous material) has a longitudinal nature. This

means that the particles of the medium vibrate in direction which is parallel (and anti-parallel) to the direction which

the sound wave travels. If the sound wave travels from west to east, then the particles of the medium vibrate back and

forth along the east-west axis. As a sound wave impinges upon a particle of air, that particle is temporarily disturbed

from its rest position. This particle in turn pushes upon its nearest neighbor, causing it to be displaced from its rest

position. The displacement of several nearby particles produces a region of space in which several particles are

compressed together. Such a region is known as a compression or high pressure region. A restoring force typically pulls

each particle back towards its original rest position. As the particles are pulled away from each other, a region is created

in which the particles are spread apart. Such a region is known as a rarefaction or low pressure region. Because a sound

wave consists of an alternating pattern of high pressure (compressions) and low pressure (rarefactions) regions traveling

through the medium, it is known as a pressure wave.

When a pressure wave reaches the ear, a series of high and low pressure regions impinge upon the eardrum. The arrival

of a compression or high pressure region pushes the eardrum inward; the arrival of a low pressure regions serves to pull

the eardrum outward. The continuous arrival of high and low pressure regions sets the eardrum into vibrational motion.

This is depicted in the animation below.

The eardrum is attached to the bones of the middle ear - the hammer, anvil, and stirrup. As these bones begin vibrating,

the sound signal is transformed from a pressure wave traveling through air to the mechanical vibrations of the bone

structure of the middle ear. These vibrations are then transmitted to the fluid of the inner ear where they are converted

to electrical nerve impulses which are sent to the brain.

Since the eardrum is set into vibration by the incoming pressure wave, the vibrations occur at the same frequency as the

pressure wave. If the incoming compressions and rarefactions arrive more frequently, then the eardrum vibrates more


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frequently. This frequency is transmitted through the middle and inner ear and provides the perception of pitch. Higher

frequency vibrations are perceived as higher pitch sounds and lower frequency vibrations are perceived as lower pitch

sounds.

The intensity of the incoming sound wave can also be transmitted through the middle ear to the inner ear and

interpreted by the brain. A high intensity sound wave is characterized by vibrations of air particles with a high

amplitude. When these high amplitude vibrations impinge upon the eardrum, they produce a very forceful displacement

of the eardrum from its rest position. This high intensity sound wave causes a large vibration of the eardrum and

subsequently a large and forceful vibration of the bones of the middle ear. This high amplitude vibration is transmitted

to the fluid of the inner ear and encoded in the nerve signal which is sent to the brain. A high intensity sound is

perceived as a relatively loud sound by the brain.

LESSON 2

Sound Properties and Their Perception

Pitch and Frequency

A sound wave, like any other wave, is introduced into a medium by a vibrating object. The vibrating object is the source

of the disturbance that moves through the medium. The vibrating object that creates thedisturbance could be the vocal chords of a person, the vibrating string and sound board of a

guitar or violin, the vibrating tines of a tuning fork, or the vibrating diaphragm of a radio

speaker. Regardless of what vibrating object is creating the sound wave, the particles of the

medium through which the sound moves is vibrating in a back and forth motion at a given frequency. The frequency of

a wave refers to how often the particles of the medium vibrate when a wave passes through the medium. The

frequency of a wave is measured as the number of complete back-and-forth vibrations of a particle of the medium per

unit of time. If a particle of air undergoes 1000 longitudinal vibrations in 2 seconds, then the frequency of the wave

would be 500 vibrations per second. A commonly used unit for frequency is the Hertz (abbreviated Hz), where

1 Hertz = 1 vibration/second

As a sound wave moves through a medium, each particle of the medium vibrates at the same frequency. This is sensible

since each particle vibrates due to the motion of its nearest neighbor. The first particle of the medium begins vibrating,

at say 500 Hz, and begins to set the second particle into vibrational motion at the same frequency of 500 Hz. The second

particle begins vibrating at 500 Hz and thus sets the third particle of the medium into vibrational motion at 500 Hz. The

process continues throughout the medium; each particle vibrates at the same frequency. And of course the frequency at

which each particle vibrates is the same as the frequency of the original source of the sound wave. Subsequently, a

guitar string vibrating at 500 Hz will set the air particles in the room vibrating at the same frequency of 500 Hz, which

carries a sound signalto the ear of a listener, which is detected as a 500 Hz sound wave.

The back-and-forth vibrational motion of the particles of the medium would not be the only observable phenomenon

occurring at a given frequency. Since a sound wave is a pressure wave, a detector could be used to detect oscillations in

pressure from a high pressure to a low pressure and back to a high pressure. As the compressions (high pressure) and

rarefactions (low pressure) move through the medium, they would reach the detector at a given frequency. For

example, a compression would reach the detector 500 times per second if the frequency of the wave were 500 Hz.

Similarly, a rarefaction would reach the detector 500 times per second if the frequency of the wave were 500 Hz. The

frequency of a sound wave not only refers to the number of back-and-forth vibrations of the particles per unit of time,

but also refers to the number of compressions or rarefactions that pass a given point per unit of time. A detector could

be used to detect the frequency of these pressure oscillations over a given period of time. The typical output provided

by such a detector is a pressure-time plot as shown below.


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Since a pressure-time plot shows the fluctuations in pressure over time, the period of the sound wave can be found by

measuring the time between successive high pressure points (corresponding to the compressions) or the time between

successive low pressure points (corresponding to the rarefactions). As discussed in an earlier unit, the frequency is

simply the reciprocal of the period. For this reason, a sound wave with a high frequency would correspond to a pressure

time plot with a small period - that is, a plot corresponding to a small amount of time between successive high pressure

points. Conversely, a sound wave with a low frequency would correspond to a pressure time plot with a large period -

that is, a plot corresponding to a large amount of time between successive high pressure points. The diagram below

shows two pressure-time plots, one corresponding to a high frequency and the other to a low frequency.

The ears of a human (and other animals) are sensitive detectors capable of detecting the fluctuations in air pressure

that impinge upon the eardrum. The mechanics of the ear's detection ability will be discussed later in this lesson. For

now, it is sufficient to say that the human ear is capable of detecting sound waves with a wide range of frequencies,

ranging between approximately 20 Hz to 20 000 Hz. Any sound with a frequency below the audible range of hearing

(i.e., less than 20 Hz) is known as an infrasound and any sound with a frequency above the audible range of hearing (i.e.,

more than 20 000 Hz) is known as an ultrasound. Humans are not alone in their ability to detect a wide range of

frequencies. Dogs can detect frequencies as low as approximately 50 Hz and as high as 45 000 Hz. Cats can detect

frequencies as low as approximately 45 Hz and as high as 85 000 Hz. Bats, being nocturnal creature, must rely on sound

echolocation for navigation and hunting. Bats can detect frequencies as high as 120 000 Hz. Dolphins can detect

frequencies as high as 200 000 Hz. While dogs, cats, bats, and dolphins have an unusual ability to detect ultrasound, anelephant possesses the unusual ability to detect infrasound, having an audible range from approximately 5 Hz to

approximately 10 000 Hz.

The sensation of a frequency is commonly referred to as the pitch of a sound. A high pitch sound corresponds to a high

frequency sound wave and a low pitch sound corresponds to a low frequency sound wave. Amazingly, many people,

especially those who have been musically trained, are capable of detecting a difference in frequency between two

separate sounds that is as little as 2 Hz. When two sounds with a frequency difference of greater than 7 Hz are played

simultaneously, most people are capable of detecting the presence of a complex wave pattern resulting from

the interference and superposition of the two sound waves. Certain sound waves when played (and heard)

simultaneously will produce a particularly pleasant sensation when heard, are said to be consonant. Such sound waves

form the basis ofintervals in music. For example, any two sounds whose frequencies make a 2:1 ratio are said to beseparated by an octave and result in a particularly pleasing sensation when heard. That is, two sound waves sound good

when played together if one sound has twice the frequency of the other. Similarly two sounds with a frequency ratio of


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5:4 are said to be separated by an interval of a third; such sound waves also sound good when played

together. Examples of other sound wave intervals and their respective frequency ratios are listed in the table below.

Interval Frequency Ratio Examples

Octave 2:1 512 Hz and 256 Hz

Third 5:4 320 Hz and 256 Hz

Fourth 4:3 342 Hz and 256 Hz

Fifth 3:2 384 Hz and 256 Hz

The ability of humans to perceive pitch is associated with the frequency of the sound wave that impinges upon the ear.

Because sound waves traveling through air are longitudinal waves that produce high- and low-pressure disturbances of

the particles of the air at a given frequency, the ear has an ability to detect such frequencies and associate them with

the pitch of the sound. But pitch is not the only property of a sound wave detectable by the human ear. In the next part

of Lesson 2, we will investigate the ability of the ear to perceive the intensity of a sound wave.

Intensity and the Decibel Scale

Sound waves are introduced into a medium by the vibration of an object. For example, a vibrating guitar string forces

surrounding air molecules to be compressed and expanded, creating a pressure

disturbance consisting of an alternating pattern ofcompressions and rarefactions.The disturbance then travels from particle to particle through the medium,

transporting energy as it moves. The energy that is carried by the disturbance

was originally imparted to the medium by the vibrating string. The amount of

energy that is transferred to the medium is dependent upon the amplitude of

vibrations of the guitar string. If more energy is put into the plucking of the string

(that is, more work is done to displace the string a greater amount from its rest

position), then the string vibrates with a greater amplitude. The greater

amplitude of vibration of the guitar string thus imparts more energy to the medium, causing air particles to be displaced

a greater distance from their rest position. Subsequently, the amplitude of vibration of the particles of the medium is

increased, corresponding to an increased amount of energy being carried by the particles. This relationship between

energy and amplitude was discussed in more detail in a previous unit.

The amount of energy that is transported past a given area of the medium per unit of time is known as the intensityof

the sound wave. The greater the amplitude of vibrations of the particles of the medium, the greater the rate at which

energy is transported through it, and the more intense that the sound wave is. Intensity is the energy/time/area; and

since the energy/time ratio is equivalent to the quantity power, intensity is simply the power/area.

Typical units for expressing the intensity of a sound wave are Watts/meter2.

As a sound wave carries its energy through a two-dimensional or three-dimensional medium,the intensity of the sound wave decreases with increasing distance from the source. The

decrease in intensity with increasing distance is explained by the fact that the wave is

spreading out over a circular (2 dimensions) or spherical (3 dimensions) surface and thus the

energy of the sound wave is being distributed over a greater surface area. The diagram at the

right shows that the sound wave in a 2-dimensional medium is spreading out in space over a

circular pattern. Since energy is conserved and the area through which this energy is

transported is increasing, the power (being a quantity that is measured on a per areabasis)

must decrease. The mathematical relationship between intensity and distance is sometimes

referred to as aninverse square relationship. The intensity varies inversely with the square of

the distance from the source. So if the distance from the source is doubled (increased by a

factor of 2), then the intensity is quartered (decreased by a factor of 4). Similarly, if thedistance from the source is quadrupled, then the intensity is decreased by a factor of 16. Applied to the diagram at the

right, the intensity at point B is one-fourth the intensity as point A and the intensity at point C is one-sixteenth the


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intensity at point A. Since the intensity-distance relationship is an inverse relationship, an increase in one quantity

corresponds to a decrease in the other quantity. And since the intensity-distance relationship is an inverse square

relationship, whatever factor by which the distance is increased, the intensity is decreased by a factor equal to the

square of the distance change factor. The sample data in the table below illustrate the inverse square relationship

between power and distance.

Distance Intensity

1 m 160 units

2 m 40 units3 m 17.8 units

4 m 10 units

Humans are equipped with very sensitive ears capable of detecting sound waves of extremely low intensity. The faintest

sound that the typical human ear can detect has an intensity of 1*10-12

W/m2. This intensity corresponds to a pressure

wave in which a compression of the particles of the medium increases the air pressure in that compressional region by a

mere 0.3 billionth of an atmosphere. A sound with an intensity of 1*10-12

W/m2

corresponds to a sound that will

displace particles of air by a mere one-billionth of a centimeter. The human ear can detect such a sound. WOW! This

faintest sound that a human ear can detect is known as the threshold of hearing. The most intense sound that the ear

can safely detect without suffering any physical damage is more than one billion times more intense than the threshold

of hearing.

Since the range of intensities that the human ear can detect is so large, the scale that is frequently used by physicists to

measure intensity is a scale based on multiples of 10. This type of scale is sometimes referred to as a logarithmic scale.

The scale for measuring intensity is the decibel scale. The threshold of hearing is assigned a sound level of 0 decibels

(abbreviated 0 dB); this sound corresponds to an intensity of 1*10-12

W/m2. A sound that is 10 times more intense (

1*10-11

W/m2) is assigned a sound level of 10 dB. A sound that is 10*10 or 100 times more intense (1*10

-10W/m

2) is

assigned a sound level of 20 db. A sound that is 10*10*10 or 1000 times more intense (1*10-9

W/m2) is assigned a sound

level of 30 db. A sound that is 10*10*10*10 or 10000 times more intense (1*10-8

W/m2) is assigned a sound level of 40

db. Observe that this scale is based on powers or multiples of 10. If one sound is 10xtimes more intense than another

sound, then it has a sound level that is 10*x more decibels than the less intense sound. The table below lists some

common sounds with an estimate of their intensity and decibel level.

Source IntensityIntensity

Level

# of Times

Greater Than TOH

Threshold of Hearing (TOH) 1*10-12

W/m2

0 dB 100

Rustling Leaves 1*10-11

W/m2

10 dB 101

Whisper 1*10-10

W/m2

20 dB 102

Normal Conversation 1*10-6

W/m2

60 dB 106

Busy Street Traffic 1*10-5

W/m2

70 dB 107

Vacuum Cleaner 1*10-4

W/m2

80 dB 108

Large Orchestra 6.3*10-3

W/m2

98 dB 109.8

Walkman at Maximum Level 1*10

-2

W/m

2

100 dB 10

10

Front Rows of Rock Concert 1*10-1

W/m2

110 dB 1011

Threshold of Pain 1*101

W/m2

130 dB 1013

Military Jet Takeoff 1*102

W/m2

140 dB 1014

Instant Perforation of Eardrum 1*104

W/m2

160 dB 1016

The Speed of Sound

A sound wave is a pressure disturbance that travels through a medium by means of particle-to-particle interaction. As

one particle becomes disturbed, it exerts a force on the next adjacent particle, thus disturbing that particle from rest

and transporting the energy through the medium. Like any wave, the speed of asound wave refers to how fast the disturbance is passed from particle to particle.

While frequency refers to the number of vibrations that an individual particle makes per


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unit of time, speed refers to the distance that the disturbance travels per unit of time. Always be cautious to distinguish

between the two often-confused quantities of speed (how fast...) and frequency (how often...).

Since the speed of a wave is defined as the distance that a point on a wave (such as a compression or a rarefaction)

travels per unit of time, it is often expressed in units of meters/second (abbreviated m/s). In equation form, this is

speed = distance/time

The faster a sound wave travels, the more distance it will cover in the same period of time. If a sound wave were

observed to travel a distance of 700 meters in 2 seconds, then the speed of the wave would be 350 m/s. A slower wave

would cover less distance - perhaps 660 meters - in the same time period of 2 seconds and thus have a speed of 330

m/s. Faster waves cover more distance in the same period of time.

Factors Affecting Wave Speed

The speed of any wave depends upon the properties of the medium through which the wave is traveling. Typically there

are two essential types of properties that effect wave speed - inertial properties and elastic properties. Elastic

properties are those properties related to the tendency of a material to maintain its shape and not deform whenever a

force or stress is applied to it. A material such as steel will experience a very small deformation of shape (and

dimension) when a stress is applied to it. Steel is a rigid material with a high elasticity. On the other hand, a material

such as a rubber band is highly flexible; when a force is applied to stretch the rubber band, it deforms or changes its

shape readily. A small stress on the rubber band causes a large deformation. Steel is considered to be a stiff or rigid

material, whereas a rubber band is considered a flexible material. At the particle level, a stiff or rigid material is

characterized by atoms and/or molecules with strong attractions for each other. When a force is applied in an attempt

to stretch or deform the material, its strong particle interactions prevent this deformation and help the material

maintain its shape. Rigid materials such as steel are considered to have a high elasticity. (Elastic modulus is the technical

term). The phase of matter has a tremendous impact upon the elastic properties of the medium. In general, solids have

the strongest interactions between particles, followed by liquids and then gases. For this reason, longitudinal sound

waves travel faster in solids than they do in liquids than they do in gases. Even though the inertial factor may favor

gases, the elastic factor has a greater influence on the speed (v) of a wave, thus yielding this general pattern:

vsolids > vliquids > vgases

Inertial properties are those properties related to the material's tendency to be sluggish to changes in its state of

motion. The density of a medium is an example of an inertial property. The greater the inertia (i.e., mass density) of

individual particles of the medium, the less responsive they will be to the interactions between neighboring particles

and the slower that the wave will be. As stated above, sound waves travel faster in solids than they do in liquids than

they do in gases. However, within a single phase of matter, the inertial property of density tends to be the property that

has a greatest impact upon the speed of sound. A sound wave will travel faster in a less dense material than a more

dense material. Thus, a sound wave will travel nearly three times faster in Helium than it will in air. This is mostly due to

the lower mass of Helium particles as compared to air particles.

The speed of a sound wave in air depends upon the properties of the air, mostly the temperature, and to a lesser

degree, the humidity. Humidity is the result of water vapor being present in air. Like any liquid, water has a tendency to

evaporate. As it does, particles of gaseous water become mixed in the air. This additional matter will affect the mass

density of the air (an inertial property). The temperature will affect the strength of the particle interactions (an elastic

property). At normal atmospheric pressure, the temperature dependence of the speed of a sound wave through dry

airis approximated by the following equation:

v = 331 m/s + (0.6 m/s/C)T

where T is the temperature of the air in degrees Celsius. Using this equation to determine the speed of a sound wave in

air at a temperature of 20 degrees Celsius yields the following solution.


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v = 331 m/s + (0.6 m/s/C)T

v = 331 m/s + (0.6 m/s/C)(20 C)

v = 331 m/s + 12 m/s

v = 343 m/s

(The above equation relating the speed of a sound wave in air to the temperature provides reasonably accurate speed

values for temperatures between 0 and 100 Celsius. The equation itself does not have any theoretical basis; it is simply

the result of inspecting temperature-speed data for this temperature range. Other equations do exist that are based

upon theoretical reasoning and provide accurate data for all temperatures. Nonetheless, the equation above will be

sufficient for our use as introductory Physics students.)

The Human Ear

Understanding how humans hear is a complex subject involving the fields of physiology, psychology and acoustics. In

this part of Lesson 2, we will focus on the acoustics (the branch of physics pertaining to sound) of hearing. We will

attempt to understand how the human ear serves as an astounding transducer, converting sound energy to mechanical

energy to a nerve impulse that is transmitted to the brain. The ear's ability to do this allows us to perceive the pitch of

sounds by detection of the wave's frequencies, the loudness of sound by detection of the wave's amplitude and the

timbre of the sound by the detection of the various frequencies that make up a complex sound wave.

The ear consists of three basic parts - the outer ear, the middle ear, and the inner ear. Each part of the ear serves a

specific purpose in the task of detecting and interpreting sound. The outer ear serves to collect and channel sound to

the middle ear. The middle ear serves to transform the energy of a sound wave into the internal vibrations of the bone

structure of the middle ear and ultimately transform these vibrations into a compressional wave in the inner ear. The

inner ear serves to transform the energy of a compressional wave within the inner ear fluid into nerve impulses that can

be transmitted to the brain. The three parts of the ear are shown below.

The outer ear consists of an earflap and an approximately 2-cm long ear canal. The earflap provides protection for the

middle ear in order to prevent damage to the eardrum. The outer ear also channels sound waves that reach the ear

through the ear canal to the eardrum of the middle ear. Because of the length of the ear canal, it is capable of

amplifying sounds with frequencies of approximately 3000 Hz. As sound travels through the outer ear, the sound is still

in the form of a pressure wave, with an alternating pattern of high and low pressure regions. It is not until the sound

reaches the eardrum at the interface of the outer and the middle ear that the energy of the mechanical wavebecomes

converted into vibrations of the inner bone structure of the ear.

The middle ear is an air-filled cavity that consists of an eardrum and three tiny, interconnected bones - the hammer,

anvil, and stirrup. The eardrum is a very durable and tightly stretched membrane that vibrates as the incoming pressure

waves reach it. As shown below, a compression forces the eardrum inward and a rarefaction forces the eardrum

outward, thus vibrating the eardrum at the same frequency of the sound wave.


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Being connected to the hammer, the movements of the eardrum will set the hammer, anvil, and stirrup into motion at

the same frequency of the sound wave. The stirrup is connected to the inner ear; and thus the vibrations of the stirrup

are transmitted to the fluid of the inner ear and create a compression wave within the fluid. The three tiny bones of the

middle ear act as levers to amplify the vibrations of the sound wave. Due to a mechanical advantage, the displacements

of the stirrup are greater than that of the hammer. Furthermore, since the pressure wave striking the large area of the

eardrum is concentrated into the smaller area of the stirrup, the force of the vibrating stirrup is nearly 15 times larger

than that of the eardrum. This feature enhances our ability of hear the faintest of sounds. The middle ear is an air-filled

cavity that is connected by the Eustachian tube to the mouth. This connection allows for the equalization of pressure

within the air-filled cavities of the ear. When this tube becomes clogged during a cold, the ear cavity is unable to

equalize its pressure; this will often lead to earaches and other pains.

The inner ear consists of a cochlea, the semicircular canals, and the auditory nerve. The cochlea and the semicircular

canals are filled with a water-like fluid. The fluid and nerve cells of the semicircular canals provide no role in the task of

hearing; they merely serve as accelerometers for detecting accelerated movements and assisting in the task of

maintaining balance. The cochlea is a snail-shaped organ that would stretch to approximately 3 cm. In addition to being

filled with fluid, the inner surface of the cochlea is lined with over 20 000 hair-like nerve cells that perform one of the

most critical roles in our ability to hear. These nerve cells differ in length by minuscule amounts; they also have different

degrees of resiliency to the fluid that passes over them. As a compressional wave moves from the interface between the

hammer of the middle ear and the oval windowof the inner ear through the cochlea, the small hair-like nerve cells will

be set in motion. Each hair cell has a natural sensitivity to a particular frequency of vibration. When the frequency of the

compressional wave matches the natural frequency of the nerve cell, that nerve cell will resonate with a larger

amplitude of vibration. This increased vibrational amplitude induces the cell to release an electrical impulse that passes

along the auditory nerve towards the brain. In a process that is not clearly understood, the brain is capable of

interpreting the qualities of the sound upon reception of these electric nerve impulses.

LESSON 3

Behavior of Sound Waves

Interference and Beats

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. As mentioned in a previous unit of The Physics Classroom Tutorial, if two upward

displaced pulses having the same shape meet up with one another while traveling in opposite directions along a

medium, the medium will take on the shape of an upward displaced pulse with twice the amplitude of the two

interfering pulses. This type of interference is known as constructive interference. If an upward displaced pulse and a

downward displaced pulse having the same shape meet up with one another while traveling in opposite directions along

a medium, the two pulses will cancel each other's effect upon the displacement of the medium and the medium will

assume the equilibrium position. This type of interference is known as destructive interference. The diagrams below

show two waves - one is blue and the other is red - interfering in such a way to produce a resultant shape in a medium;

the resultant is shown in green. In two cases (on the left and in the middle), constructive interference occurs and in the

third case (on the far right, destructive interference occurs.


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But how can sound waves that do not possess upward and downward displacements interfere constructively and

destructively? Sound is a pressure wave that consists ofcompressions and rarefactions. As a compression passes

through a section of a medium, it tends to pull particles together into a small region of space, thus creating a high-

pressure region. And as a rarefaction passes through a section of a medium, it tends to push particles apart, thus

creating a low-pressure region. The interference of sound waves causes the particles of the medium to behave in a

manner that reflects the net effect of the two individual waves upon the particles. For example, if a compression (high

pressure) of one wave meets up with a compression (high pressure) of a second wave at the same location in the

medium, then the net effect is that that particular location will experience an even greater pressure. This is a form of

constructive interference. If two rarefactions (two low-pressure disturbances) from two different sound waves meet up

at the same location, then the net effect is that that particular location will experience an even lower pressure. This is

also an example of constructive interference. Now if a particular location along the medium repeatedly experiences the

interference of two compressions followed up by the interference of two rarefactions, then the two sound waves will

continually reinforce each other and produce a very loud sound. The loudness of the sound is the result of the particles

at that location of the medium undergoing oscillations from very high to very low pressures. As mentioned ina previous

unit, locations along the medium where constructive interference continually occurs are known as anti-nodes. The

animation below shows two sound waves interfering constructively in order to produce very large oscillations in

pressure at a variety of anti-nodal locations. Note that compressions are labeled with a C and rarefactions are labeled

with an R.

Now if two sound waves interfere at a given location in such a way that the compression of one wave meets up with the

rarefaction of a second wave, destructive interference results. The net effect of a compression (which pushes particles

together) and a rarefaction (which pulls particles apart) upon the particles in a given region of the medium is to not

even cause a displacement of the particles. The tendency of the compression to push particles together is canceled by

the tendency of the rarefactions to pull particles apart; the particles would remain at their rest position as though there

wasn't even a disturbance passing through them. This is a form of destructive interference. Now if a particular location

along the medium repeatedly experiences the interference of a compression and rarefaction followed up by the

interference of a rarefaction and a compression, then the two sound waves will continually canceleach other and no

sound is heard. The absence of sound is the result of the particles remaining at rest and behaving as though there were

no disturbance passing through it. Amazingly, in a situation such as this, two sound waves would combine to produce no

sound. As mentioned in a previous unit, locations along the medium where destructive interference continually occurs

are known as nodes.

Two Source Sound Interference

A popular Physics demonstration involves the interference of two sound waves from two speakers. The speakers are set

approximately 1-meter apart and produced identical tones. The two sound waves traveled through the air in front of the

speakers, spreading our through the room in spherical fashion. A snapshot in time of the appearance of these waves is

shown in the diagram below. In the diagram, the compressions of a wavefront are represented by a thick line and the

rarefactions are represented by thin lines. These two waves interfere in such a manner as to produce locations of some

loud sounds and other locations of no sound. Of course the loud sounds are heard at locations where compressions

meet compressions or rarefactions meet rarefactions and the "no sound" locations appear wherever the compressions

of one of the waves meet the rarefactions of the other wave. If you were to plug one ear and turn the other ear towards


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the place of the speakers and then slowly walk across the room parallel to the plane of the speakers, then you would

encounter an amazing phenomenon. You would alternatively hear loud sounds as you approached anti-nodal locations

and virtually no sound as you approached nodal locations. (As would commonly be observed, the nodal locations are

not true nodal locations due to reflections of sound waves off the walls. These reflections tend to fill the entire room

with reflected sound. Even though the sound waves that reach the nodal locations directly from the speakers

destructively interfere, other waves reflecting off the walls tend to reach that same location to produce a pressure

disturbance.)

Destructive interference of sound waves becomes an important issue in the design of concert halls and auditoriums. Therooms must be designed in such as way as to reduce the amount of destructive interference. Interference can occur as

the result of sound from two speakers meeting at the same location as well as the result of sound from a speaker

meeting with sound reflected off the walls and ceilings. If the sound arrives at a given location such that compressions

meet rarefactions, then destructive interference will occur resulting in a reduction in the loudness of the sound at that

location. One means of reducing the severity of destructive interference is by the design of walls, ceilings, and baffles

that serve to absorb sound rather than reflect it. This will be discussed in more detail later in Lesson 3.

The destructive interference of sound waves can also be used advantageously in noise reduction systems. Earphones

have been produced that can be used by factory and construction workers to reduce the noise levels on their jobs. Such

earphones capture sound from the environment and use computer technology to produce a second sound wave that

one-half cycle out of phase. The combination of these two sound waves within the headset will result in destructiveinterference and thus reduce a worker's exposure to loud noise.

Musical Beats and Intervals

Interference of sound waves has widespread applications in the world of music. Music seldom consists of sound waves

of a single frequency played continuously. Few music enthusiasts would be impressed by an orchestra that played music

consisting of the note with a pure tone played by all instruments in the orchestra. Hearing a sound wave of 256 Hz

(middle C) would become rather monotonous (both literally and figuratively). Rather, instruments are known to produce

overtones when played resulting in a sound that consists of a multiple of frequencies. Such instruments are described as

being rich in tone color. And even the best choirs will earn their moneywhen two singers sing two notes (i.e., produce

two sound waves) that are an octave apart. Music is a mixture of sound waves that typically have whole number ratiosbetween the frequencies associated with their notes. In fact, the major distinction between music and noise is that

noise consists of a mixture of frequencies whose mathematical relationship to one another is not readily discernible. On

the other hand, music consists of a mixture of frequencies that have a clear mathematical relationshipbetween them.

While it may be true that "one person's music is another person's noise" (e.g., your music might be thought of by your

parents as being noise), a physical analysis of musical sounds reveals a mixture of sound waves that are mathematically

related.

To demonstrate this nature of music, let's consider one of the simplest mixtures of two different sound waves - two

sound waves with a 2:1 frequency ratio. This combination of waves is known as an octave. A simple sinusoidal plot of

the wave pattern for two such waves is shown below. Note that the red wave has two times the frequency of the blue

wave. Also observe that the interference of these two waves produces a resultant (in green) that has a periodic andrepeating pattern. One might say that two sound waves that have a clear whole number ratio between their frequencies

interfere to produce a wave with a regular and repeating pattern. The result is music.


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Another simple example of two sound waves with a clear mathematical relationship between frequencies is shownbelow. Note that the red wave has three-halves the frequency of the blue wave. In the music world, such waves are said

to be a fifth apart and represent a popular musical interval. Observe once more that the interference of these two

waves produces a resultant (in green) that has a periodic and repeating pattern. It should be said again: two sound

waves that have a clear whole number ratio between their frequencies interfere to produce a wave with a regular and

repeating pattern; the result is music.

Finally, the diagram below illustrates the wave pattern produced by two dissonant or displeasing sounds. The diagram

shows two waves interfering, but this time there is no simple mathematical relationship between their frequencies (in

computer terms, one has a wavelength of 37 and the other has a wavelength 20 pixels). Observe (look carefully) that the

pattern of the resultant is neither periodic nor repeating (at least not in the short sample of time that is shown). The

message is clear: if two sound waves that have no simple mathematical relationship between their frequencies interfere

to produce a wave, the result will be an irregular and non-repeating pattern. This tends to be displeasing to the ear.

A final application of physics to the world of music pertains to the topic of beats. Beats are the periodic and repeating

fluctuations heard in the intensity of a sound when two sound waves of very similar frequencies interfere with one

another. The diagram below illustrates the wave interference pattern resulting from two waves (drawn in red and blue)

with very similar frequencies. A beat pattern is characterized by a wave whose amplitude is changing at a regular rate.Observe that the beat pattern (drawn in green) repeatedly oscillates from zero amplitude to a large amplitude, back to

zero amplitude throughout the pattern. Points of constructive interference (C.I.) and destructive interference (D.I.) are

labeled on the diagram. When constructive interference occurs between two crests or two troughs, a loud sound is

heard. This corresponds to a peak on the beat pattern (drawn in green). When destructive interference between a crest

and a trough occurs, no sound is heard; this corresponds to a point of no displacement on the beat pattern. Since there

is a clear relationship between the amplitude and the loudness, this beat pattern would be consistent with a wave that

varies in volume at a regular rate.


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The beat frequency refers to the rate at which the volume is heard to be oscillating from high to low volume. For

example, if two complete cycles of high and low volumes are heard every second, the beat frequency is 2 Hz. The beat

frequency is always equal to the difference in frequency of the two notes that interfere to produce the beats. So if two

sound waves with frequencies of 256 Hz and 254 Hz are played simultaneously, a beat frequency of 2 Hz will bedetected. A common physics demonstration involves producing beats using two tuning forks with very similar

frequencies. If a tine on one of two identical tuning forks is wrapped with a rubber band, then that tuning forks

frequency will be lowered. If both tuning forks are vibrated together, then they produce sounds with slightly different

frequencies. These sounds will interfere to produce detectable beats. The human ear is capable of detecting beats with

frequencies of 7 Hz and below.

A piano tuner frequently utilizes the phenomenon of beats to tune a piano string. She will pluck the string and tap a

tuning fork at the same time. If the two sound sources - the piano string and the tuning fork - produce detectable beats

then their frequencies are not identical. She will then adjust the tension of the piano string and repeat the process until

the beats can no longer be heard. As the piano string becomes more in tune with the tuning fork, the beat frequency

will be reduced and approach 0 Hz. When beats are no longer heard, the piano string is tuned to the tuning fork; that is,they play the same frequency. The process allows a piano tuner to match the strings' frequency to the frequency of a

standardized set of tuning forks.

Important Note: Many of the diagrams on this page represent a sound wave by a sine wave. Such a wave more closely

resembles a transverse wave and may mislead people into thinking that sound is a transverse wave. Sound is not a

transverse wave, but rather a longitudinal wave. Nonetheless, the variations in pressure with time take on the pattern

of a sine wave and thus a sine wave is often used to represent the pressure-time features of a sound wave.

The Doppler Effect and Shock Waves

The Doppler effect is a phenomenon 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 the observer and the source are approaching and an apparent downward shift in

frequency when the observer and the source is receding. The Doppler effect can be observed to occur with all types of

waves - most notably water waves, sound waves, and light waves. The application of this phenomenon to water waves

was discussed in detail in Unit 10 of The Physics Classroom Tutorial. In this unit, we will focus on the application of the

Doppler effect to sound.

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 - a shift in the apparent

frequency for a sound wave produced by a moving source.


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Another common experience is the shift in apparent frequency of the sound of a train horn. As the train approaches, the

sound of its horn is heard at a high pitch and as the train moved away, the sound of its horn is heard at a low pitch. This

is the Doppler effect.

A common Physics demonstration the use of a large Nerf ball equipped with a buzzer that produces a sound with a

constant frequency. The Nerf ball is then through around the room. As the ball approaches you, you observe a higher

pitch than when the ball is at rest. And when the ball is thrown away from you, you observe a lower pitch than when the

ball is at rest. This is the Doppler effect.

Explaining the Doppler Effect

The Doppler effect is observed because the distance between the source of sound and the observer is changing. If the

source and the observer are approaching, then the distance is decreasing and if the source and the observer are

receding, then the distance is increasing. The source of sound always emits the same frequency. Therefore, for the same

period of time, the same number of waves must fit between the source and the observer. if the distance is large, then

the waves can be spread apart; but if the distance is small, the waves must be compressed into the smaller distance. For

these reasons, if the source is moving towards the observer, the observer perceives sound waves reaching him or her at

a more frequent rate (high pitch). And if the source is moving away from the observer, the observer perceives sound

waves reaching him or her at a less frequent rate (low pitch). It is important to note that the effect does not result

because of an actual change in the frequency of the source. The source puts out the same frequency; the observer only

perceives a different frequency because of the relative motion between them. The Doppler effect is a shift in the

apparent or observed frequency and not a shift in the actual frequency at which the source vibrates.

Shock Waves and Sonic Booms

The Doppler effect is observed whenever the speed of the source is moving slower than the

speed of the waves. But if the source actually moves at the same speed as or faster than the

wave itself can move, a different phenomenon is observed. If a moving source of sound

moves at the same speed as sound, then the source will always be at the leading edge of the

waves that it produces. The diagram at the right depicts snapshots in time of a variety of

wavefronts produced by an aircraft that is moving at the same speed as sound. The circularlines represent compressional wavefronts of the sound waves. Notice that these circles

arebunched up at the front of the aircraft. This phenomenon is known as a shock wave.


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Shock waves are also produced if the aircraft moves faster than the speed of sound. If a moving source of sound moves

faster than sound, the source will always be ahead ofthe waves that it produces. The diagram at the right depicts

snapshots in time of a variety of wavefronts produced by an aircraft that is moving faster than sound. Note that the

circular compressional wavefronts fall behind the faster moving aircraft (in actuality, these circles would be spheres).

If you are standing on the ground when a supersonic (faster than sound) aircraft passes overhead, you might hear a

sonic boom. A sonic boom occurs as the result of the piling up of compressional wavefronts along the conical edge of

the wave pattern. These compressional wavefronts pile up and interfere to produce a very high-pressure zone. This is

shown below. Instead of these compressional regions (high-pressure regions) reaching you one at a time in consecutive

fashion, they all reach you at once. Since every compression is followed by a rarefaction, the high-pressure zone will be

immediately followed by a low-pressure zone. This creates a very loud noise.

If you are standing on the ground as the supersonic aircraft passes by, there will be a short time delay and then you will

hear the boom - the sonic boom. This boom is merely a loud noise resulting from the high pressure sound followed by a

low pressure sound. Do not be mistaken into thinking that this boom only happens the instant that the aircraft

surpasses the speed of sound and that it is the signature that the aircraft just attained supersonic speed. Sonic booms

are observed when any aircraft that is traveling faster than the speed of sound passes overhead. It is not a sign that the

aircraft just overcame the sound barrier, but rather a sign that the aircraft is traveling faster than sound.

Boundary Behavior

As a sound wave travels through a medium, it will often reach the end of the medium and encounter an obstacle or

perhaps another medium through which it could travel. When one medium ends, another medium begins; the interface

of the two media is referred to as the boundary and the behavior of a wave at that boundary is described as its

boundary behavior. The behavior of a wave (or pulse) upon reaching the end of a medium is referred to as boundary

behavior. There are essentially four possible behaviors that a wave could exhibit at a boundary: reflection (the bouncing

off of the boundary), diffraction (the bending around the obstacle without crossing over the boundary), transmission

(the crossing of the boundary into the new material or obstacle), and refraction (occurs along with transmission and is

characterized by the subsequent change in speed and direction). In this part of Lesson 3, the focus will be upon the

reflection behavior of sound waves. Later in Lesson 3, diffraction, transmission, and refraction will be discussed in more

detail.


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In Unit 10 of The Physics Classroom, the boundary behavior of a pulse on a rope

was discussed. In that unit, it was mentioned that there are two types of

reflection for waves on ropes: fixed end reflection and free end reflection. A

pulse moving through a rope will eventually reach its end. Upon reaching the

end of the medium, two things occur:

A portion of the energy carried by the pulse is reflected andreturns towards the left end of the rope. The disturbance that returns to the left is known as the reflected

pulse.

A portion of the energy carried by the pulse is transmitted into the new medium. If the rope isattached to a pole (as shown at the right), the pole will receive some of the energy and begin to vibrate. If the

rope is not attached to a pole but rather resting on the ground, then a portion of the energy is transmitted

into the air (the new medium), causing slight disturbances of the air particles.

The amount of energy that becomes reflected is dependent upon the dissimilarity of the two media. The more similar

that the two media on each side of the boundary are, the less reflection that occurs and the more transmission that

occurs. Conversely, the less similar that the two media on each side of the boundary are, the more reflection that occurs

and the less transmission that occurs. So if a heavy rope is attached to a light rope (two very dissimilar media), little

transmission and mostly reflection occurs. And if a heavy rope is attached to another heavy rope (two very similar

media), little reflection and mostly transmission occurs.

The more similar the medium, the more transmission that occurs.

These principles of reflection can be applied to sound waves. Though a sound wave does not consist of crests and

troughs, they do consist ofcompressions and rarefactions. If a sound wave is traveling through a cylindrical tube, it will

eventually come to the end of the tube. The end of the tube represents a boundary between the enclosed air in the

tube and the expanse of ai

Wave Sound Wave

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