Sound waves

Sound

Sound travels as distinct waves with two physical characteristics: amplitude, which is volume or loudness; and frequency, which is pitch. Amplitude is the height of a sound wave. Frequency represents how often a sound wave goes through a full cycle. As a sound wave travels across distance, the amplitude decreases. This is why we can't hear a quiet whisper across a room.

The Nature of a Sound Wave

Sound is a Mechanical Wave


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 the 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 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

Sound is a Longitudinal Wave

Sound waves are longitudinal waves because particles of the medium through which the sound is transported vibrate parallel to the direction which the sound moves. 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 move 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 interaction; thus, 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 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 is a longitudinal wave. 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.

Sound is a Pressure Wave

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 be the human ear or a man-made instrument, is 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. Since the fluctuations in pressure as detected by the detector occur at periodic and regular time intervals, a plot of pressure vs. time would appear as a sine curve. The crests of the sine curve correspond to compressions; the troughs correspond to rarefactions; and the "zero point" corresponds to the pressure which 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 and the pressure-time fluctuations which it creates.



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 which has crests and troughs. Sound is indeed a longitudinal wave with compressions and rarefactions. As sound passes through a medium, the particles of that medium do not vibrate in a transverse manner. Do not be misled - sound is a longitudinal wave.

Interference and Beats

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 a previous unit of The Physics Classroom, if two crests 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 a crest with twice the amplitude of the two interfering crests. This type of interference is known as constructive interference. If a crest and a trough 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.



But how can sound waves which do not possess crests and troughs 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. 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 in a previous unit, locations along the medium where constructive interference continually occurs are known as anti-nodes.

The Doppler Effect and Shock Waves

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 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 which it produces. The diagram at the right depicts snapshots in time of a variety of wavefronts produced by an aircraft which is moving at the same speed as sound. The circular lines represent compressional wavefronts of the sound waves. Notice that these circles are bunched up at the front of the aircraft. This phenomenon is known as a shock wave. 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 of the waves which it produces. The diagram at the right depicts snapshots in time of a variety of wavefronts produced by an aircraft which 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.

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 inte***ce 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 boundary behaviors by which a sound wave could behave: 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.

Reflection, Refraction, and Diffraction

Like any wave, a sound wave doesn't just stop when it reaches the end of the medium or when it encounters an obstacle in its path. Rather, a sound wave will undergo certain behaviors when it encounters the end of the medium or an obstacle. Possible behaviors include reflection off the obstacle, diffraction around the obstacle, and transmission (accompanied by refraction) into the obstacle or new medium .
When a wave reaches the boundary between one medium another medium, a portion of the wave undergoes reflection and a portion of the wave undergoes transmission across the boundary. The reflected wave may or may not undergo a phase change (i.e., be inverted) depending on the relative densities of the two media. It was also mentioned that the amount of reflection is dependent upon the dissimilarity of the two medium. For this reason, acousticly minded builders of auditoriums and concert halls avoid the use of hard, smooth materials in the construction of their inside halls. A hard material such as concrete is as dissimilar as can be to the air through which the sound moves; subsequently, most of the sound wave is reflected by the walls and little is absorbed.

Reflection of sound waves off of su***ces can lead to one of two phenomenon - an echo or a reverberation. A reverberation often occurs in a small room with height, width, and length dimensions of approximately 17 meters or less. Why the magical 17 meters? The effect of a particular sound wave upon the brain endures for more than a tiny fraction of a second; the human brain keeps a sound in memory for up to 0.1 seconds. If a reflected sound wave reaches the ear within 0.1 seconds of the initial sound, then it seems to the person that the sound is prolonged. The reception of multiple reflections off of walls and ceilings within 0.1 seconds of each other causes reverberations - the prolonging of a sound. Since sound waves travel at about 340 m/s at room temperature, it will take approximately 0.1 s for a sound to travel the length of a 17 meter room and back, thus causing a reverberation (t = v/d = (340 m/s)/(34 m) = 0.1 s). This is why reverberations is common in rooms with dimensions of approximately 17 meters or less. Perhaps you have observed reverberations when talking in an empty room, when honking the horn while driving through a highway tunnel or underpass, or when singing in the shower. In auditoriums and concert halls, reverberations occasionally occur and lead to the displeasing garbling of a sound.

But reflection of sound waves in auditoriums and concert halls do not always lead to displeasing results, especially if the reflections are designed right. Smooth walls have a tendency to direct sound waves in a specific direction. Subsequently the use of smooth walls in an auditorium will cause spectators to receive a large amount of sound from one location along the wall; there would be only one possible path by which sound waves could travel from the speakers to the listener. The auditorium would not seem to be as lively and full of sound. Rough walls tend to diffuse sound, reflecting it in a variety of directions. This allows a spectator to perceive sounds from every part of the room, *** it seem lively and full. For this reason, auditorium and concert hall designers prefer construction materials which are rough rather than smooth.



Reflection of sound waves also lead to echoes. Echoes are different than reverberations. Echoes occur when a reflected sound wave reaches the ear more than 0.1 seconds after the original sound wave was heard. If the elapsed time between the arrival of the two sound waves is more than 0.1 seconds, then the sensation of the first sound will have died out . In this case, the arrival of the second sound wave will be perceived as a second sound rather than the prolonging of the first sound. There will be an echo instead of a reverberation.

Reflection of sound waves off of su***ces is also effected by the shape of the su***ce. Reflection of sound waves off of curved su***ces leads to a more interesting phenomenon. Curved su***ces with a parabolic shape have the habit of focusing sound waves to a point. Sound waves reflecting off of parabolic su***ces concentrate all their energy to a single point in space; at that point, the sound is amplified. Perhaps you have seen a museum exhibit which utilizes a parabolic-shaped disk to collect a large amount of sound and focus it at a focal point. If you place your ear at the focal point, you can hear even the faintest whisper of a friend standing across the room. Parabolic-shaped satellite disks use this same principle of reflection to gather large amounts of electromagnetic waves and focus it at a point (where the receptor is located). Scientists have recently discovered some evidence which seem to reveal that the bull moose utilizes his antlers as a satellite disk to gather and focus sound. Finally, scientists have long believed that owls are equipped with spherically-shaped facial disks which can be maneuvered in order to gather and reflect sound towards their ears.

Diffraction involves a change in direction of waves as they pass through an opening or around a barrier in their path. The diffraction of water waves was discussed in Unit 10 of The Physics Classroom. In that unit, we saw that water waves have the ability to travel around corners, around obstacles and through openings. The amount of diffraction (the sharpness of the bending) increases with increasing wavelength and decreases with decreasing wavelength. In fact, when the wavelength of the waves are smaller than the obstacle or opening, no noticeable diffraction occurs.

Diffraction of sound waves is commonly observed; we notice sound diffracting around corners or through door openings, allowing us to hear others who are speaking to us from adjacent rooms. Many forest-dwelling birds take advantage of the diffractive ability of long-wavelength sound waves. Owls for instance are able to communicate across long distances due to the fact that their long-wavelength hoots are able to diffract around forest trees and carry farther than the short-wavelength tweets of song birds. Low-pitched (high wavelength) sounds always carry further than high pitched (low wavelength) sounds.



Scientists have recently learned that elephants emit infrasonic waves of very low frequency to communicate over long distances to each other. Elephants typically migrate in large herds which may sometimes become separated from each other by distances of several miles. Researchers who have observed elephant migrations from the air have been impressed and puzzled by the ability of elephants at the beginning and the end of these herds to make extremely synchronized movements. The matriarch at the front of the heard might make a turn to the right which is immediately followed by elephants at the end of the herd *** the same turn to the right. These synchronized movements occur despite the fact that the elephants' vision of each other is blocked by dense vegetation. Only recently have they learned that the synchronized movements are preceded by infrasonic communication. While low wavelength light waves are unable to diffract around the dense vegetation, the high wavelength sounds produced by the elephants have sufficient diffractive ability to communicate long distances.

Bats use high frequency (low wavelength) ultrasonic waves in order to enhance their ability to hunt. The typical prey of a bat is the moth - an object not much larger than a couple of centimeters. Bats use ultrasonic echolocation methods to detect the presence of bats in the air. But why ultrasound? The answer lies in the physics of diffraction. As the wavelength of a wave becomes smaller than the obstacle which it encounters, the wave is no longer able to diffract around the obstacle, instead the wave reflects off the obstacle. Bats use ultrasonic waves with wavelengths smaller than the dimensions of their prey. These sound waves will encounter the prey, and instead of diffracting around the prey, will reflect off the prey and allow the bat to hunt by means of echolocation. The wavelength of a 50 000 Hz sound wave in air (speed of approximately 340 m/s) can be calculated as follows

wavelength = speed/frequency
wavelength = (340 m/s)/(50 000 Hz)

wavelength = 0.0068 m

The wavelength of the 50 000 Hz sound wave (typical for a bat) is approximately 0.7 centimeters, smaller than the dimensions of a typical moth.

Refraction of waves involves a change in the direction of waves as they pass from one medium to another. Refraction, or bending of the path of the waves, is accompanied by a change in speed and wavelength of the waves. So if the medium (and its properties) are changed, the speed of the waves are changed. Thus waves passing from one medium to another will undergo refraction. Refraction of sound waves is most evident in situations in which the sound wave passes through a medium with gradually varying properties. For example, sound waves are known to refract when traveling over water. Even though the sound wave is not exactly changing media, it is traveling through a medium with varying properties; thus, the wave will encounter refraction and change its direction. Since water has a moderating effect upon the temperature of air, the air directly above the water tends to be cooler than the air far above the water. Sound waves travel slower in cooler air than they do in warmer air. For this reason, the portion of the wavefront directly above the water is slowed down, while the portion of the wavefronts far above the water speeds ahead. Subsequently, the direction of the wave changes, refracting downwards towards the water. This is depicted in the diagram at the right.

wave equation

We will use sound propagation in a gas such as air as an example.
Sound waves are elastic waves that propagate in a fluid or solid.
In fluids long wavelength sound waves consist of an alternating pattern of rarefaction and compression.
In a solid transverse waves can also propagate.
In ordinary sound the changes in pressure tend to be very small.
The intensity of a sound wave is often measured in decibels



The reference pressure is



and Pe is the root mean square pressure amplitude of the peak excess pressure). Even at the pain level of 120 dB the peak excess pressure will be
which is small compared to the ambient pressure

The mechanism for a sound wave is that gas motion generates a change in the density, which causes a change in pressure. There will then be an unbalanced force which accelerates the gas and causes the cycle to repeat.

We write for the pressure and density

where the subscript a stands for average while e stands for excess.

The relationship between changes in density and pressure depends on the properties of the medium in which the sound waves propagate. We will assume that the processes are fast enough that they can be considered to be adiabatic, i.e. without any heat transport. Since the amplitudes Pe andare small compared to the ambient conditions we assume that they are proportional to each other

with

Let us consider a column of cross-section A.

When the air is at rest in equilibrium this column extends from x to x+dx. We assume that a sound wave is traveling in the x-direction and that at some instant the left end of the column is displaced an amountwhich is small compared to the wavelength of the sound wave.

Conservation of mass givesor

giving

The column is subject to a net force
causing an acceleration

We get the wave equation

For an ideal gas assuming an adiabatic process

with or

Differentiating we find
where m is the mass of a molecule andthe molecular weight. We finally get
It is interesting to compare the speed of sound with typical molecular speeds. From the thermodynamics of ideal gases we have for the rms speed/wavequation/waveequation/img25.gif[/img]
It is interesting to compare the speed of sound with typical molecular speeds. From the thermodynamics of ideal gases we have for the rms speed

for air we see that the rms speed and the sound speed are quite comparable. By substituting into the wave equation we see that it admits solutions in the form of traveling waves where
It can be shown (see Powers sect 3.1)that the general solution to the wave equations can be written

The same equation that we derived for sound waves also holds for a vibrating string (see Powers section 3.1)
The speed of the wave is then

where T is the tension andis the mass/unit length of the string.

Often one is not interested in the details of the solution but only in knowing the frequencies which can be excited. One is then led to an eigenvalue problem which is similar to what we did for the heat equation.

The Speed of a Wave