# Wave motion and Sound

In a general sense, anything that moves back and forth, to and fro, side to side, in and out, or up and down is vibrating. The to-and-fro vibratory motion is often called oscillatory motion.  A vibration is a wiggle in time. A wiggle in both space and time is a wave. A wave extends from one place to another. Light and sound are both vibrations that propagate throughout space as waves, but two very different kinds of waves. Sound is the propagation of vibrations through a material medium- a solid, liquid, or gas. If there is no medium to vibrate, then no sound is possible. Sound cannot travel in a vacuum. But light can, for, as we shall learn in later chapters, light is a vibration of electric and magnetic fields-a vibration of pure energy. Although light can pass through many materials, it needs none. This is evident when it propagates through the vacuum between the Sun and the Earth. The source of all waves – sound, light, or whatever-is something that is vibrating (or oscillating).

When you throw a stone into a lake or pool of water, circular waves form and move outward, as shown in the photos above. Waves will also travel along a cord that is stretched out flat on a table if you vibrate one end back and forth.  Water waves and waves on a cord are two common examples of mechanical waves, which propagate as oscillations of matter. Later we will discuss other kinds of waves, including electromagnetic waves and light.

If you have ever watched ocean waves moving toward shore before they break, you may have wondered if the waves were carrying water from far out at sea onto the beach. They don’t. Water waves move with a recognizable velocity. But each particle (or molecule) of the water itself merely oscillates about an equilibrium point. This is clearly demonstrated by observing leaves on a pond as waves move by. The leaves (or a cork) are not carried forward by the waves, but simply oscillate about an equilibrium point because this is the motion of the water itself.

Conceptual Example: Wave vs. particle velocity

Is the velocity of a wave moving along a cord the same as the velocity of a particle of the cord?

Answer: No. The two velocities are different, both in magnitude and direction. The wave on the cord moves to the right along the tabletop, but each piece of the cord only vibrates to and fro. (The cord clearly does not travel in the direction that the wave on it does.)

Waves can move over large distances, but the medium (the water or the cord) itself has only a limited movement, oscillating about an equilibrium point. Thus, although a wave is not matter, the wave pattern can travel in matter. A wave consists of oscillations that move without carrying matter with them.

Waves carry energy from one place to another. Energy is given to a water wave, for example, by a rock thrown into the water, or by wind far out at sea. The energy is transported by waves to the shore. The oscillating hand  transfers energy  to the cord, and that energy is transported down the cord and can be transferred to an object at the other end. All forms of traveling waves transport energy.

## Characteristics of Wave Motion

Let us look a little more closely at how a wave is formed and how it comes to “travel.” We first look at a single wave bump, or pulse. A single pulse can be formed on a cord by a quick up-and-down motion of the hand, Fig. 2. The hand pulls up on one end of the cord. Because the end section is attached to adjacent sections, these also feel an upward force and they too begin to move upward. As each succeeding section of cord moves upward, the wave crest moves outward along the cord. Meanwhile, the end section of cord has been returned to its original position by the hand. As each succeeding section of cord reaches its peak position, it too is pulled back down again by tension from the adjacent section of cord. Thus the source of a traveling wave pulse is a disturbance, and cohesive forces between adjacent sections of cord cause the pulse to travel. Waves in other media are created and propagate outward in a similar fashion. A dramatic example of a wave pulse is a tsunami or tidal wave that is created by an earthquake in the Earth’s crust under the ocean. The bang you hear when a door slams is a sound wave pulse.

A continuous or periodic wave, such as that shown in Fig. 1, has as its source a disturbance that is continuous and oscillating; that is, the source is a vibration or oscillation. In Fig. 15-1, a hand oscillates one end of the cord. Water waves may be produced by any vibrating object at the surface, such as your hand; or the water itself is made to vibrate when wind blows across it or a rock is thrown into it. A vibrating tuning fork or drum membrane gives rise to sound waves in air. And we will see later that oscillating electric charges give rise to light waves. Indeed, almost any vibrating object sends out waves.

The source of any wave, then, is a vibration. And it is a vibration that propagates outward and thus constitutes the wave. If the source vibrates sinusoidally in SHM, then the wave itself-if the medium is perfectly elastic-will have a sinusoidal shape both in space and in time. (1) In space: if you take a picture of the wave in space at a given instant of time, the wave will have the shape of a sine or cosine as a function of position. (2) In time: if you look at the motion of the medium at one place over a long period of time-for example, if you look between two closely spaced posts of a pier or out of a ship’s porthole as water waves pass by-the up-and-down motion of that small segment of water will be simple harmonic motion. The water moves up and down sinusoidally in time.

Some of the important quantities used to describe a periodic sinusoidal wave are shown in Fig. 3. The high points on a wave are called crests; the low points, troughs. The amplitude, A, is the maximum height of a crest, or depth of a trough, relative to the normal (or equilibrium) level. The total swing from a crest to a trough is twice the amplitude. The distance between two successive crests is called the wavelength,  λ (the Greek letter lambda). The wavelength is also equal to the distance between any two successive identical points on the wave. The frequency, f, is the number of crests-or complete cycles-that pass a given point per unit time. The period, T, equals 1/ f and is the time elapsed between two successive crests passing by the same point in space.

The wave velocity, v, is the velocity at which wave crests (or any other part of the waveform) move forward. The wave velocity must be distinguished from the velocity of a particle of the medium itself.

A wave crest travels a distance of one wavelength, λ, in a time equal to one period, T. Thus the wave velocity is v = λ/T. Then, since l/T = f,

v = λ f

For example, suppose a wave has a wavelength of 5 m and a frequency of 3 Hz. Since three crests pass a given point per second, and the crests are 5 m apart, the first crest (or any other part of the wave) must travel a distance of 15 m during the 1 s. So the wave velocity is 15 m/s.

## Types of Waves: Transverse and Longitudinal

When a wave travels down a cord-say, from left to right as in Fig. 1-the particles of the cord vibrate up and down in a direction transverse (that is, perpendicular) to the motion of the wave itself. Such a wave is called a transverse wave (Fig. 4a). There exists another type of wave known as a longitudinal wave. In a longitudinal wave, the vibration of the particles of the medium is along the direction of the wave’s motion. Longitudinal waves are readily formed on a stretched spring or Slinky by alternately compressing and expanding one end. This is shown in Fig. 4b, and can be compared to the transverse wave in Fig. 4a.

Transverse wave:

Longitudinal wave:

A series of compressions and expansions propagate along the spring. The compressions
are those areas where the coils are momentarily close together. Expansions (sometimes called rarefactions) are regions where the coils are momentarily far apart. Compressions and expansions correspond to the crests and troughs of a transverse wave.

An important example of a longitudinal wave is a sound wave in air. A vibrating drumhead, for instance, alternately compresses and rarefies the air in contact with it,
producing a longitudinal wave that travels outward in the air, as shown in Fig. 5.

As in the case of transverse waves, each section of the medium in which a longitudinal wave passes oscillates over a very small distance, whereas the wave itself can travel large distances. Wavelength, frequency, and wave velocity all have meaning for a longitudinal wave. The wavelength is the distance between successive compressions (or between successive expansions), and frequency is the number of compressions that pass a given point per second. The wave velocity is the velocity with which each compression appears to move; it is equal to the product of wavelength and frequency, ν = λf.

A longitudinal wave can be represented graphically by plotting the density of air molecules versus position at a given instant, as shown in Fig. 6. Such a graphical representation makes it easy to illustrate what is happening. Note that the graph looks much like a transverse wave.

Both transverse and longitudinal waves are produced when an earthquake occurs. The transverse waves  that travel through the body of the Earth  are called S waves (S for Shear), and the longitudinal waves are called P waves (P for pressure) or  compression waves. Both longitudinal and transverse waves can travel through a solid  since the atoms or molecules can vibrate about their relatively fixed positions in any direction. But in a fluid, only longitudinal waves can propagate, because any transverse motion would experience no restoring force since a fluid is readily deformable. This fact was used by geophysicists to infer that a portion of the Earth’s core must be liquid: after an earthquake, longitudinal waves are detected diametrically across the Earth, but not transverse waves.

Besides these  two types of waves,  surface waves can travel along the boundary between two materials. A wave on water is actually a surface wave that moves on the boundary between water and air. Water waves involve a combination of both longitudinal and transverse motions. As a wave travels through the waver, the particles travel in clockwise circles. The radius of the circles decreases as the depth into the water increases.

Fig. 7  A water wave is an example of a surface wave, which is a combination of                              longitudinal and transverse motions.

## The Principle of Superposition

When two or more waves pass through the same region of space at the same time, it is found that for many waves the actual displacement is the vector (or algebraic) sum of the separate displacements. This is called the principle of superposition. It is valid for mechanical waves as long as the displacements are not too large and there is a linear relationship between the displacement and the restoring force of the oscillating medium. If the amplitude of a mechanical wave, for example, is so large that it goes beyond the elastic region of the medium, and Hooke’s law is no longer operative, the superposition principle is no longer accurate. For the most part, we will consider systems for which the superposition principle can be assumed to hold.

One result of the superposition principle is that if two waves pass through the same region of space, they continue to move independently of one another. You may have noticed, for example, that the ripples on the surface of water (two-dimensional waves) that form from two rocks striking the water at different places will pass through each other.

Figure 8 shows an example of the superposition principle. In this case there are three waves present, on a stretched string, each of different amplitude and frequency. At any time, such as at the instant shown, the actual amplitude at any position x is the algebraic sum of the amplitude of the three waves at that position. The actual wave is not a simple sinusoidal wave and is called a composite (or complex) wave. (Amplitudes are exaggerated in Fig. 8).

It can be shown that any complex wave can be considered as being composed of many simple sinusoidal waves of different amplitudes, wavelengths, and frequencies. This is known as Fourier’s theorem.  Although we will not go into the details here, we see the importance of considering sinusoidal waves (and simple harmonic motion): because any other wave shape can be considered a sum of such pure sinusoidal waves.

### Superposition: Is Schrödinger’s Cat Dead or Alive? *

One especially mind-boggling principle of the quantum realm is that of superposition. This concept suggests that a quantum particle can be in multiple states at once, completely indistinguishable, until it is observed. The idea is so strange that most people can’t wrap their heads around the principle. That bizarre concept leads to another rule of quantum mechanics – when we measure or observe a particular property, the quantum object has to choose one state of the many probable states in order for us to observe it. This property of superposition boggled scientists of the 1920s, and still confuses many people today.

Schrödinger’s cat is a thought experiment devised by Austrian physicist Erwin Schrodinger in 1935.  The scenario presents a cat that may be simultaneously both alive and dead – a state known as a quantum superposition.  Schrödinger coined the term Verschränkung (entanglement) in the course of developing the thought experiment.

A cat is placed in a steel box along with a Geiger counter (Note: Geiger counters measure radioactivity) , a flask of poison, a hammer, and a radioactive substance. When the radioactive substance decays, the Geiger detects it and triggers the hammer to release the poison, which subsequently kills the cat. The radioactive decay is a random process, and there is no way to predict when it will happen. Physicists say the atom exists in a state known as a superposition—both decayed and not decayed at the same time. Until the box is opened, an observer doesn’t know whether the cat is alive or dead—because the cat’s fate is tied to whether or not the atom has decayed and the cat would, as Schrödinger put it, be “living and dead … in equal parts” until it is observed.

The state of the cat has the schematic form:

|Cat>$(\frac{1}{2})^{1/2}$|Alive >  +  $(\frac{1}{2})^{1/2}$|Dead>

In this state, called an equal superposition, there are equal probabilities of measuring either |Alive > or |Dead>  as

The cat is neither alive nor dead, but rather a linear combination of the two, until a
measurement occurs—until, say, you peek in the window to check. At that moment
your observation forces the cat to “take a stand”: dead or alive. And if you find
him to be dead, then it’s really you who killed him, by looking in the window.

Schrodinger regarded this as patent nonsense, and most physicists would agree with him. There is something absurd about the very idea of a macroscopic object being in a linear combination of two palpably different states. An electron can be in a linear combination of spin up and spin down, but a cat simply cannot be in a linear combination of alive and dead. How are we to reconcile this with the orthodox interpretation of quantum mechanics?

The most widely accepted answer is that the triggering of the Geiger counter constitutes the “measurement,” in the sense of the statistical interpretation, not the intervention of a human observer. It is the essence of a measurement that some macroscopic system is affected (the Geiger counter, in this instance). The measurement occurs at the moment when the microscopic system (described by the laws of quantum mechanics) interacts with the macroscopic system (described by the laws of classical mechanics) in such a way as to leave a permanent record. The macroscopic system itself is not permitted to occupy a linear combination of distinct states.

Probably this is not an entirely satisfactory resolution, but at least it avoids the stultifying solipsism of Wigner and others, who persuaded themselves that it is the involvement of human consciousness that constitutes a measurement in quantum mechanics. Part of the problem is the word “measurement” itself, which certainly carries a suggestion of human participation. Heisenberg proposed the word “event,” which might be preferable. But “measurement” is so ingrained by now that we’re stuck with it. And, in the end, no manipulation of the terminology can completely exorcise this mysterious ghost.

### Dispersion*

When the restoring force is not precisely proportional to the displacement for mechanical waves in some continuous medium, the speed of sinusoidal waves depends on the frequency. The variation of speed with frequency is called dispersion. The different sinusoidal waves that compose a complex wave will travel with slightly different speeds in such a case. Consequently, a complex wave will change shape as it travels if the medium is “dispersive.” A pure sine wave will not change shape under these conditions, however, except by the influence of friction or dissipative forces. If there is no dispersion (or friction), even a complex linear wave does not change shape.

## Reflection and Transmission

When a wave strikes an obstacle, or comes to the end of the medium in which it is
traveling, at least a part of the wave is reflected. You have probably seen water waves reflect off a rock or the side of a swimming pool. And you may have heard a shout reflected from a distant cliff-which we call an “echo.”

Partial Transmittance and Partial Reflectance: A wave experiences partial transmittance and partial reflectance when the medium through which it travels suddenly changes.

A wave pulse traveling down a cord is reflected as shown in Fig. 9. The reflected pulse returns inverted as in Fig. 9a if the end of the cord is fixed; it returns right side up if the end is free as in Fig. 9b. When the end is fixed to a support, as in Fig. 9a, the pulse reaching that fixed end exerts a force (upward) on the support. The support exerts an equal but opposite force downward on the cord (Newton’s third law). This downward force on the cord is what “generates” the inverted reflected pulse.

Consider next a pulse that travels down a cord which consists of a light section and a heavy section, as shown in Fig. 10. When the wave pulse reaches the boundary between the two sections, part of the pulse is reflected and part is transmitted, as shown. The heavier the second section of the cord, the less the energy that is transmitted. (When the second section is a wall or rigid support, very little is transmitted and most is reflected, as in Fig. 10a.) For a periodic wave, the frequency of the transmitted wave does not change across the boundary since the boundary point oscillates at that frequency. Thus if the transmitted wave has a lower speed, its wavelength is also less (λ = ν/f).

For a two- or three-dimensional wave, such as a water wave, we are concerned with wave fronts, by which we mean all the points along the wave forming the wave crest (what we usually refer to simply as a “wave” at the seashore). A line drawn in the direction of motion, perpendicular to the wave front, is called a ray, as shown in Fig. 11. Wave fronts far from the source have lost almost all their curvature (Fig. 11b) and are nearly straight, as ocean waves often are; they are then called plane waves.

For reflection of a two- or three-dimensional plane wave, as shown in Fig. 12,  the angle that the incoming or incident wave makes with the reflecting surface is equal to the angle made by the reflected wave. This is the law of reflectionthe angle of reflection equals the angle of incidence.

The “angle of incidence” is defined as the angle  θi the incident ray makes with the perpendicular to the reflecting surface (or the wave front makes with a tangent to the surface), and the “angle of reflection” is the corresponding angle  θr  for the reflected wave.

## Interference

Interference refers to what happens when two waves pass through the same region of space at the same time. Consider, for example, the two wave pulses on a cord traveling toward each other as shown in Fig. 13. In Fig. 13a the two pulses have the same amplitude, but one is a crest and the other a trough; in Fig. 13b they are both crests. In both cases, the waves meet and pass right by each other. However, in the region where they overlap, the resultant displacement is the algebraic sum of their separate displacements (a crest is considered positive and a trough negative). This is another example of the principle of superposition. In Fig. 13a, the two waves have opposite displacements at the instant they pass one another, and they add to zero. The result is called destructive interference. In Fig. 13b, at the instant the two pulses overlap, they produce a resultant displacement that is greater than the displacement of either separate pulse, and the result is constructive interference.

When two rocks are thrown into a pond simultaneously, the two sets of circular waves interfere with one another as shown in Fig. 14a. In some areas of overlap, crests of one wave repeatedly meet crests of the other (and troughs meet troughs), Fig. 14b. Constructive interference is occuring at these points, and the water continuously oscillates up and down with greater amplitude than either wave separately. In other areas, destructive interference occurs where the water does not move up and down at all over time. This is where crests of one wave meet troughs of the other, and vice versa. Figure 14a shows the displacement of two identical waves graphically as a function of time, as well as their sum, for the case of constructive interference. For constructive interference (Fig. 14a), the two waves are in phase. At points where destructive interference occurs (Fig. 14b) crests of one wave repeatedly meet troughs of the other wave and the two waves are out of phase by one-half wavelength or 180°. The crests of one wave occur a half wavelength behind the crests of the other wave. The relative phase of the two water waves in Fig. 14 in most areas is intermediate between these two extremes, resulting in partially destructive interference, as illustrated in Fig. 15c. If the amplitudes of two interfering waves are not equal, fully destructive interference (as in Fig. 15b) does not occur.

## Standing Waves; Resonance

If you shake one end of a cord and the other end is kept fixed, a continuous wave will travel down to the fixed end and be reflected back, inverted, as we saw in Fig. 16a. As you continue to vibrate the cord, waves will travel in both directions, and the wave traveling along the cord, away from your hand, will interfere with the reflected wave coming back. Usually there will be quite a jumble. But if you vibrate the cord at just the right frequency, the two traveling waves will interfere in such a way that a large-amplitude standing wave will be produced, Fig. 16. It is called a “standing wave” because it does not appear to be traveling. The cord simply appears to have segments that oscillate up and down in a fixed pattern. The points of destructive interference, where the cord remains still at all times, are called nodes. Points of constructive interference, where the cord oscillates with maximum amplitude, are called antinodes. The nodes and antinodes remain in fixed positions for a particular frequency.

Standing waves can occur at more than one frequency. The lowest frequency of vibration that produces a standing wave gives rise to the pattern shown in Fig. 16a. The standing waves shown in Figs. 16b and 16c are produced at precisely twice and three times the lowest frequency, respectively, assuming the tension in the cord is the same. The cord can also vibrate with four loops (four antinodes) at four times the lowest frequency, and so on.

The frequencies at which standing waves are produced are the natural frequencies or resonant frequencies of the cord, and the different standing wave patterns shown in Fig. 16 are different “resonant modes of vibration.” A standing wave on a cord is the result of the interference of two waves traveling in opposite directions. A standing wave can also be considered a vibrating object at resonance. Standing waves represent the same phenomenon as the resonance of a vibrating spring or pendulum. However, a spring or pendulum has only one resonant frequency, whereas the cord has an infinite number of resonant frequencies, each of which is a whole-number multiple of the lowest resonant frequency.

Consider a string stretched between two supports that is plucked like a guitar or violin string, Fig. 17a. Waves of a great variety of frequencies will travel in both directions along the string, will be reflected at the ends, and will travel back in the opposite direction. Most of these waves interfere with each other and quickly die out. However, those waves that correspond to the resonant frequencies of the string will persist. The ends of the string, since they are fixed, will be nodes. There may be other nodes as well. Some of the possible resonant modes of vibration (standing waves) are shown in Fig. 17b. Generally, the motion will be a combination of these different resonant modes, but only those frequencies that correspond to a resonant frequency will be present.

To determine the resonant frequencies, we first note that the wavelengths of the standing waves bear a simple relationship to the length L of the string. The lowest
frequency, called the fundamental frequency, corresponds to one antinode (or loop).
And as can be seen in Fig. 17b, the whole length corresponds to one-half wavelength. Thus  L = λ1/2, where λ1 stands for the wavelength of the fundamental frequency. The other natural frequencies are called overtones; for a vibrating string they are whole-number (integral) multiples of the fundamental, and then are also called harmonics, with the fundamental being referred to as the first harmonic. The next mode of vibration after the fundamental has two loops and is called the second harmonic (or first overtone), Fig. 17b. The length of the string L at the second harmonic corresponds to one complete wavelength: L = λ2 . For the third harmonic, L=(3/2)λ3  and so on. In general, we can write

L = n.λn/2     where n = 1,2,3, ….

The integer n labels the number of the harmonic: n = 1 for the fundamental, n = 2 for the second harmonic, and so on. Thus

λn = 2L/n,     n = 1,2 , 3,….      [string fixed at both ends]

A standing wave does appear to be standing in place (and a traveling wave appears to move). The term “standing” wave is also meaningful from the point of view of energy. Since the string is at rest at the nodes, no energy flows past these points. Hence the energy is not transmitted down the string but “stands” in place in the string.

Standing waves are produced not only on strings, but on any object that is struck, such as a drum membrane or an object made of metal or wood. The resonant frequencies depend on the dimensions of the object, just as for a string they depend on its length. Large objects have lower resonant frequencies than small objects. All musical instruments, from stringed instruments to wind instruments (in which a column of air vibrates as a standing wave) to drums and other percussion instruments, depend on standing waves to produce their particular musical sounds.

## Sound

Most sounds are waves produced by the vibrations of material objects. In a piano, a violin, and a guitar, the sound is produced by the vibrating strings; in a saxophone, by a vibrating reed; in a flute, by a fluttering column of air at the mouthpiece. Your voice results from the vibration of your vocal chords.

In each of these cases, the original vibration stimulates the vibration of something larger or more massive, such as the sounding board of a stringed instrument, the air column within a reed or wind instrument, or the air in the throat and mouth of a singer. This vibrating material then sends a disturbance through the surrounding medium, usually air, in the form of longitudinal waves. Under ordinary conditions, the frequency of the vibrating source and the frequency of the sound waves produced are the same.

We describe our subjective impression about the frequency of sound by the word pitch. Frequency corresponds to pitch: A high-pitched sound from a piccolo has a high frequency of vibration, while a low-pitched sound from a foghorn has a low frequency of vibration. The ear of a young person can normally hear pitches corresponding to the range of frequencies between about 20 and 20,000 hertz. This frequency range is called the audible range.  As we grow older; the limits of this human hearing range shrink, especially at the high-frequency end. Sound waves with frequencies below 20 hertz are infrasonic, and those with frequencies above 20,000 hertz are called ultrasonic. We cannot hear infrasonic and ultrasonic sound waves.

The speed of sound is different in different materials. In air at O°C and 1 atm, sound travels at a speed of 331 m/s. The speed depends on the elastic modulus, B, and the density, p, of the material. Thus for helium, whose density is much less than that of air but whose elastic modulus is not greatly different, the speed is about three times as great as in air. In liquids and solids, which are much less compressible and therefore have much greater elastic moduli, the speed is larger still. The speed of sound in various materials is given in Table 1. The values depend somewhat on temperature, but this is significant mainly for gases. For example, in air at normal (ambient) temperatures, the speed increases approximately 0.60 m/s for each Celsius degree increase in temperature:

v ≈ (331 + 0.60 T) m/s,       [speed of sound in air]

where T is the temperature in °C.

## Intensity of Sound: Decibels

Loudness is a sensation in the consciousness of a human being and is related to a physically measurable quantity, the intensity of the wave. Intensity is defined as the energy transported by a wave per unit time across a unit area perpendicular to the energy flow.  Intensity has units of power per unit area, or $watts/meter^{2} (W/m^{2})$:

$I=\frac{E}{St}$

The human ear can detect sounds with an intensity as low as $10^{- 12} W/m^{2}$ and as high as  $1 W/m^{2}$ (and even higher, although above this it is painful). This is an incredibly wide range of intensity, spanning a factor of $10^{12}$ from lowest to highest. Presumably because of this wide range, what we perceive as loudness is not directly proportional to the intensity. To produce a sound that sounds about twice as loud requires a sound wave that has about 10 times the intensity. This is roughly valid at any sound level for frequencies near the middle of the audible range. For example, a sound wave of intensity $10^{- 2} W/m^{2}$ sounds to an average human being like it is about twice as loud as one whose intensity is $10^{- 3}W/m^{2}$ , and four times as loud as  $10^{- 4} W/m^{2}$.

### Sound Level

Because of this relationship between the subjective sensation of loudness and the physically measurable quantity “intensity,” sound intensity levels are usually specified on a logarithmic scale. The unit on this scale is a bel, after the inventor Alexander Graham Bell, or much more commonly, the decibel (dB), which is $\frac{1}{10}$ bel (10 dB = 1 bel). The sound level, β, of any sound is defined in terms of its intensity, I, as

β( in dB) = $10 \log \frac{I}{I_0}$

where $I_0$ is the intensity of a chosen reference level, and the logarithm is to the base 10. $I_0$ is usually taken as the minimum intensity audible to a good ear-the “threshold of hearing,” which is $I_0 = 10^{- 12} W/m^{2}$ Thus, for example, the sound level of a sound whose intensity $I = 1.0 x 10^{- 10} W/m^{2}$ will be

β = $10 \log (\frac{10^{- 10} W/m^{2}}{10^{- 12} W/m^{2}})$ =10 log 100= 20 dB

since log 100 is equal to 2. Notice that the sound level at the threshold of hearing is 0 dB (since log 1 = 0 ). Notice too that an increase in intensity by a factor of 10 corresponds to a sound level increase of 10 dB. An increase in intensity by a factor of 100 corresponds to a sound level increase of 20 dB. Thus a 50-dB sound is 100 times more intense than a 30-dB sound, and so on.

Intensities and sound levels for a number of common sounds are listed in Table 1.

Table 1

## Sound sources; Musical Sounds

Most of the sounds we hear are noises. The impact of a falling object, the slamming of a door, the roaring of a motorcycle, and most of the sounds from traffic in city streets are noises. Noise corresponds to an irregular vibration of the eardrum produced by some irregular vibration in our surroundings. The sound of music has a different character, having periodic tones-or musical “notes.” Although noise doesn’t have these characteristics, the line that separates music and noise is thin and subjective. To some contemporary composers, it is nonexistent.

Some people consider contemporary music (e.g. chalga) and music from other cultures to be noise. Differentiating these types of music from noise becomes a problem of aesthetics. However, differentiating traditional music-that is, Western classical music and most types of popular music-from noise presents no problem. A person with total hearing loss could distinguish between these by using an oscilloscope.

Musicians usually speak of musical tones in terms of three principal characteristics:
pitch, loudness, and quality. Rapid vibrations of the sound source (high frequency) produce a high note, whereas slow vibrations (low frequency) produce a low note.

Different musical notes are obtained by changing the frequency of the vibrating sound source. This is usually done by altering the size, the tightness, or the mass of the vibrating object. A guitarist or violinist, for example, adjusts the tightness, or tension, of the strings of the instrument when tuning them, Then different notes can be played by altering the length of each string by “stopping” it with the fingers.

In wind instruments, the length of the vibrating air column can be altered (trombone and trumpet) or holes in the side of the tube can be opened and closed in various combinations (saxophone, clarinet, flute) to change the pitch of the note produced.

We have no trouble distinguishing between the tone from a piano and a tone of the same pitch from a clarinet. Each of these tones has a characteristic sound that differs in quality, or timbre. Most musical sounds are composed of a superposition of many tones differing in frequency. The various tones are called partial tones, or simply partials. The quality of a tone is determined by the presence and relative intensity of the various partials. The sound produced by a certain tone from the piano and a clarinet of the same pitch have different qualities that the ear recognizes because their partials are different.

Conventional musical instruments can be grouped into one of three classes: those in which the sound is produced by vibrating strings, those in which the sound is produced by vibrating air columns, and those in which the sound is produced by percussion-as with the vibrating of a two-dimensional surface.

### Categories of Sound Waves

Sound waves fall into three categories covering different ranges of frequencies. Audible waves are longitudinal waves that lie within the range of sensitivity of the human ear, approximately 20 to 20 000 Hz. Infrasonic waves are longitudinal waves with frequencies below the audible range. Earthquake waves are an example. Ultrasonic waves are longitudinal waves with frequencies above the audible range for humans and are produced by certain types of whistles. Animals such as dogs can hear the waves emitted by these whistles.

### Applications of Ultrasound

Ultrasonic waves are sound waves with frequencies greater than 20 kHz. Because of their high frequency and corresponding short wavelengths, ultrasonic waves can
be used to produce images of small objects and are currently in wide use in medical
applications, both as a diagnostic tool and in certain treatments. Internal organs can be examined via the images produced by the reflection and absorption of ultrasonic waves. Although ultrasonic waves are far safer than x-rays, their images don’t always have as much detail. Certain organs, however, such as the liver and the spleen, are invisible to x-rays but can be imaged with ultrasonic waves.

Medical workers can measure the speed of the blood flow in the body with a device called an ultrasonic flow meter, which makes use of the Doppler effect. The flow speed is found by comparing the frequency of the waves scattered by the flowing blood with the incident frequency.

Physicians commonly use ultrasonic waves to observe fetuses. This technique presents far less risk than do x-rays, which deposit more energy in cells and can produce birth defects. First the abdomen of the mother is coated with a liquid, such as mineral oil. If this were not done, most of the incident ultrasonic waves from the piezoelectric source would be reflected at the boundary between the air and the mother’s skin. Mineral oil has a density similar to that of skin, and a very small fraction of the incident ultrasonic wave is reflected. The ultrasound energy is emitted in pulses rather than as a continuous wave, so the same crystal can be used as a detector as well as a transmitter. An image of the fetus is obtained by using an array of transducers placed on the abdomen. The reflected sound waves picked up by the transducers are converted to an electric signal,
which is used to form an image on a fluorescent screen. Difficulties such as the likelihood of spontaneous abortion or of breech birth are easily detected with this technique. Fetal abnormalities such as spina bifida and water on the brain are also readily observed.

Ultrasound is also used to break up kidney stones that are otherwise too large to
pass. Previously, invasive surgery was more often required.

## Doppler Effect

You may have noticed that you hear the pitch of the siren on a speeding fire truck drop abruptly as it passes you. Or you may have noticed the change in pitch of a blaring horn on a fast-moving car as it passes by you. The pitch of the engine noise of a race car changes as the car passes an observer. When a source of sound is moving toward an observer, the pitch the observer hears is higher than when the source is at rest; and when the source is traveling away from the observer, the pitch is lower. This phenomenon is known as the Doppler effect, named for the Austrian physicist Christian Doppler (1803–1853), who discovered it. Although the Doppler effect is most often associated with sound, it’s common to all waves, including light.

Consider the siren of a fire truck at rest, which is emitting sound of a particular frequency in all directions. The sound waves are moving at the speed of sound in air, $v_s$, which is independent of the velocity of the source or observer.

If our source, the fire truck, is moving, the siren emits sound at the same frequency as it does at rest. But the sound wavefronts it emits forward, in front of it, are closer together than when the fire truck is at rest. This is because the fire truck, as it moves, is “chasing” the previously emitted wavefronts, and emits each crest closer to the previous one. Thus an observer on the sidewalk in front of the truck will detect more wave crests passing per second, so the frequency heard is higher. The wavefronts emitted behind the truck, on the other hand, are farther apart than when the truck is at rest because the truck is speeding away from them. Hence, fewer wave crests per second pass by an observer behind the moving truck  and the perceived pitch is lower.

### Doppler Effect for Light

The Doppler effect occurs for other types of waves as well. Light and other types of electromagnetic waves (such as radar) exhibit the Doppler effect. Another important application is to astronomy, where the velocities of distant galaxies can be determined from the Doppler shift. Light from distant galaxies is shifted toward lower frequencies, indicating that the galaxies are moving away from us. This is called the redshift since red has the lowest frequency of visible light. The greater the frequency shift, the greater the velocity of recession. It is found that the farther the galaxies are from us, the faster they move away. This observation is the basis for the idea that the universe is expanding, and is one basis for the idea that the universe began as a great explosion, affectionately called the “Big Bang”

### Shock Waves and the Sonic Boom

An object such as an airplane traveling faster than the speed of sound is said to have a supersonic speed. Such a speed is often given as a Mach number, which is defined as the ratio of the speed of the object to the speed of sound in the surrounding medium. For example, a plane traveling 600 m/s high in the atmosphere, where the speed of sound is only 300 m/s, has a speed of Mach 2.

When a source of sound moves at subsonic speeds (less than the speed of sound), the pitch of the sound is altered as we have seen (the Doppler effect); see also Fig. 19a and b. But if a source of sound moves faster than the speed of sound, a more dramatic effect known as a shock wave occurs. In this case, the source is actually “outrunning” the waves it produces. As shown in Fig. 19c, when the source is traveling at the speed of sound, the wave fronts it emits in the forward direction “pile up” directly in front of it. When the object moves faster, at a supersonic speed, the wave fronts pile up on one another along the sides, as shown in Fig. 19d. The different wave crests overlap one another and form a single very large crest which is the shock wave. Behind this very large crest there is usually a very large trough. A shock wave is essentially the result of constructive interference of a large number of wave fronts. A shock wave in air is analogous to the bow wave of a boat traveling faster than the speed of the water waves it
produces, Fig. 20.

When an airplane travels at supersonic speeds, the noise it makes and its disturbance of the air form into a shock wave containing a tremendous amount of sound energy. When the shock wave passes a listener, it is heard as a loud sonic boom. A sonic boom lasts only a fraction of a second, but the energy it contains is often sufficient to break windows and cause other damage. Actually, a sonic boom is made up of two or more booms since major shock waves can form at the front and the rear of the aircraft, as well as at the wings, etc. Bow waves of a boat are also multiple, as can be seen in Fig. 20.

When an aircraft approaches the speed of sound, it encounters a barrier of sound waves in front of it (see Fig. 19c). To exceed the speed of sound, the aircraft needs extra thrust to pass through this “sound barrier.” This is called “breaking the sound barrier.” Once a supersonic speed is attained, this barrier no longer impedes the motion. It is sometimes erroneously thought that a sonic boom is produced only at the moment an aircraft is breaking through the sound barrier. Actually, a shock wave follows the aircraft at all times it is traveling at supersonic speeds. A series of observers on the ground will each hear a loud “boom” as the shock wave passes. The shock wave consists of a cone whose apex is at the aircraft. The angle of this cone, θ (see Fig. 19d), is given by

sin θ = u/v

where v is the velocity of the object (the aircraft) and u is the velocity of
sound in the medium.