1. Chapter Twenty Six Notes: Sound
Conceptual Physics
Chapter Twenty Six Notes:
Sound

2. Sound is a Mechanical Wave
The Origin of Sound
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 which 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.

3. A sound wave is similar in nature to a slinky wave for a variety of reasons. First, there is a medium which carries the disturbance from one location to another. Typically, this medium is air; though it could be any material such as water or 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 vibrating object which creates the disturbance 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. 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 which is transported through a medium via the mechanism of particle-to-particle interaction, a sound wave is characterized as a mechanical wave.

4. 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 connected to the sound box, sets the sound box into vibrational motion. In turn,

5. the sound box, being connected to 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 which can be made to vibrate, the louder or more amplified the sound. This concept is often demonstrated 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 which 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 which 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.

6. 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 a back 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.
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.

7. The ears of a human (and other animals) are sensitive detectors capable of detecting the fluctuations in air pressure which 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 Hz. Any sound with a frequency below the audible range of hearing (i.e., less than 20 Hz) is known as an infrasonic and any sound with a frequency above the audible range of hearing (i.e., more than 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 Hz. Cats can detect frequencies as low as approximately 45 Hz and as high as Hz. Bats, being nocturnal creature, must rely on sound echolocation for navigation and hunting. Bats can detect frequencies as high as Hz. Dolphins can detect frequencies as high as Hz. While dogs, cats, bats, and dolphins have an unusual ability to detect ultrasound, an elephant possesses the unusual ability to detect infrasound, having an audible range from approximately 5 Hz to approximately Hz.

8. The sensation of a frequencies 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 which 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 are said to be consonant. Such sound waves form the basis of intervals in music. For example, any two sounds whose frequencies make a 2:1 ratio are said to be separated 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 5:4 are said to be separated by an interval of a third
such sound waves also sound good when played
together.

9. 26.2 Sound in Air
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 which moves through the medium. The vibrating object which creates the disturbance 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 Hertz = 1 vibration/second

10. 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 signal to 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

11. (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 which 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.

12. Sound as a Longitudinal Wave
In the first part of Lesson 1, it was mentioned that sound is a mechanical wave which is created by a vibrating object. The vibrations 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 which 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 which 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 which the sound wave moves.

13. A vibrating string can create longitudinal waves as depicted in the animation below. As the vibrating string moves in the forward direction, it begins to push upon surrounding air molecules, moving them to the right towards their nearest neighbor. This causes the air molecules 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.

14. Since air molecules (the particles of the medium) are moving in a direction which is parallel to the direction which the wave moves, the sound wave is referred to as a longitudinal wave. The result of such longitudinal vibrations is the creation of compressions and rarefactions within the air.
Regardless of the source of the sound wave - whether it be 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 which distinguishes it from other types of waves is that the particles of the medium move in a direction parallel to the direction of energy transport.

15. 26.3 Media That Transmit Sound
Any elastic material can transmit sound.
Steel is a very good conductor of sound.
Water is not as good a conductor as steel, but is better than air.
Air is a poor conductor of sound

16. 26.4 Speed of Sound The Speed of Sound
A sound wave is a pressure disturbance which 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 a sound wave refers to how fast the disturbance is passed from particle to particle. While frequency refers to the number of vibrations which an individual particle makes per unit of time, speed refers to the distance which 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 which 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

17. Factors Affecting Wave Speed
The faster a sound wave travels, the more distance it will cover in the same period of time. If a sound wave is 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 which affect 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.

18. vsolids > vliquids > vgases
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 it's state of motion. The density of a medium is an example of an inertial property.

19. The greater the inertia (i. e
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 which 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 as 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, namely the temperature and the pressure. The pressure of air (like any gas) will affect the mass density of the air (an inertial property) and 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 air is 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.

20. 26.5 Loudness • 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
Loudness
While the intensity of a sound is a very objective quantity which can be measured with sensitive instrumentation, the loudness of a sound is more of a subjective response which will vary with a number of factors. The same sound will not be perceived to have the same loudness to all individuals. Age is one factor which effects the human ear's response to a sound. Quite obviously, your grandparents do not hear like they used to. The same intensity sound would not be perceived to have the same loudness to them as it would to you. Furthermore, two sounds with the same intensity but different frequencies will not be perceived to have the same loudness. Because of the human ear's tendency to amplify sounds having frequencies in the range from 1000 Hz to 5000 Hz, sounds with these intensities seem louder to the human ear. Despite the distinction between intensity and loudness, it is safe to state that the more intense sounds will be perceived to be the loudest sounds.

21. Natural Frequency
Nearly all objects, when hit or struck or plucked or strummed or somehow disturbed, will vibrate. If you drop a meter stick or pencil on the floor, it will begin to vibrate. If you pluck a guitar string, it will begin to vibrate. If you blow over the top of a pop bottle, the air inside will vibrate. When each of these objects vibrate, they tend to vibrate at a particular frequency or a set of frequencies. The frequency or frequencies at which an object tends to vibrate with when hit, struck, plucked, strummed or somehow disturbed is known as the natural frequency of the object. If the amplitude of the vibrations are large enough and if natural frequency is within the human frequency range, then the vibrating object will produce sound waves which are audible.
All objects have a natural frequency or set of frequencies at which they vibrate. The quality or timbre of the sound produced by a vibrating object is dependent upon the natural frequencies of the sound waves produced by the objects.

22. Some objects tend to vibrate at a single frequency and they are often said to produce a pure tone. A flute tends to vibrate at a single frequency, producing a very pure tone. Other objects vibrate and produce more complex waves with a set of frequencies which have a whole number mathematical relationship between them; these are said to produce a rich sound. A tuba tends to vibrate at a set of frequencies which are mathematically related by whole number ratios; it produces a rich tone. Still other objects will vibrate at a set of multiple frequencies which have no simple mathematical relationship between them. These objects are not musical at all and the sounds which they create could be described as noise. When a meter stick or pencil is dropped on the floor, it vibrates with a number of frequencies, producing a complex sound wave which is clanky and noisy.

23. Forced Vibrations
If you were to take a guitar string and stretch it to a given length and a given tightness and have a friend pluck it, you would hear a noise; but the noise would not even be close in comparison to the loudness produced by an acoustic guitar. On the other hand, if the string is attached to the sound box of the guitar, the vibrating string is capable of forcing the sound box into vibrating at that same natural frequency. The sound box in turn forces air particles inside the box into vibrational motion at the same natural frequency as the string. The entire system (string, guitar, and enclosed air) begins vibrating and forces surrounding air particles into vibrational motion. The tendency of one object to force another adjoining or interconnected object into vibrational motion is referred to as a forced vibration. In the case of the guitar string mounted to the sound box, the fact that the surface area of the sound box is greater than the surface area of the string, means that more surrounding air particles will be forced into vibration. This causes an increase in the amplitude and thus loudness of the sound.

24. This same principle of a forced vibration is often demonstrated in a Physics classroom using a tuning fork. If the tuning fork is held in your hand and hit with a rubber mallet, a sound is produced as the tines of the tuning fork set surrounding air particles into vibrational motion. The sound produced by the tuning fork is barely audible to students in the back rows of the room. However, if the tuning fork is set upon the whiteboard panel or the glass panel of the overhead projector, the panel begins vibrating at the same natural frequency of the tuning fork. The tuning fork forces surrounding glass (or vinyl) particles into vibrational motion. The vibrating whiteboard or overhead projector panel in turn forces surrounding air particles into vibrational motion and the result is an increase in the amplitude and thus loudness of the sound. This principle of forced vibration explains why demonstration tuning forks are mounted on a sound box, why a commercial music box mechanism is mounted on a sounding board, why a guitar utilizes a sound box,
and why a piano string is attached to a sounding
board. A louder sound is always produced when
an accompanying object of greater surface area
is forced into vibration at the same natural frequency.

25. Resonance
Now consider a related situation which resembles another common Physics demonstration. Suppose that a tuning fork is mounted on a sound box and set upon the table; and suppose a second tuning fork/sound box system having the same natural frequency (say 256 Hz) is placed on the table near the first system. Neither of the tuning forks is vibrating. Suppose the first tuning fork is struck with a rubber mallet and the tines begin vibrating at its natural frequency Hz. These vibrations set its sound box and the air inside the sound box vibrating at the same natural frequency of 256 Hz. Surrounding air particles are set into vibrational motion at the same natural frequency of 256 Hz and every student in the classroom hears the sound. Then the tines of the tuning fork are grabbed to prevent their vibration and remarkably the sound of 256 Hz is still being heard. Only now the sound is being
produced by the second tuning fork - the
one which wasn't hit with the mallet. Amazing!!
The demonstration is often repeated to
assure that the same surprising results are
observed. They are! What is happening?

26. In this demonstration, one tuning fork forces another tuning fork into vibrational motion at the same natural frequency. The two forks are connected by the surrounding air particles. As the air particles surrounding the first fork (and its connected sound box) begin vibrating, the pressure waves which it creates begin to impinge at a periodic and regular rate of 256 Hz upon the second tuning fork (and its connected sound box). The energy carried by this sound wave through the air is tuned to the frequency of the second tuning fork. Since the incoming sound waves share the same natural frequency as the second tuning fork, the tuning fork easily begins vibrating at its natural frequency. This is an example of resonance - when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion.
The result of resonance is always a large vibration. Regardless of the vibrating system, if resonance occurs, a large vibration results.

27. Interference
Wave interference is the phenomenon which occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape which results from the net effect of the two individual waves upon the particles of the medium. As mentioned in the last chapter, 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.

28. 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.
But how can sound waves which do not possess upward and downward displacements interfere constructively and destructively? Sound is a pressure wave which consists of compressions 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.

29. 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.

30. Two Source Sound 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 cancel each 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 in the last chapter when talking about standing waves, 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 on the next page.

31. 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 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.

32. Destructive interference of sound waves becomes an important issue in the design of concert halls and auditoriums. The rooms 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.
The destructive interference of sound waves can also be used advantageously in noise reduction systems. Ear phones have been produced which can be used by factory and construction workers to reduce the noise levels on their jobs. Such ear phones capture sound from the environment and use computer technology to produce a second sound wave which one-half cycle out of phase. The combination of these two sound waves within the headset will result in destructive interference and thus reduce a worker's exposure to loud noise.

33. Beats
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 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).

34. 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 which varies in volume at a regular rate.
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 previous diagrams 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.