SOUND

  1. Nature of Sound.
    Sound requires some material to transmit it; sound cannot travel through empty space. This was first shown by Robert Boyle (1627-1691); a bell ringing in a evacuated jar could not be heard. Sound is example of mechanical wave motion. And since sound cannot be transmitted without a physical medium, it must consist of pressure or longitudinal waves. In air sound travels by alternately compressing and rarefied the air by moving the molecules back and forth in the direction of the wave as it passes a point. For example, sound waves are produced by vibrating the membrane of loud speaker back and forth in the direction the sound is to travel. This vibrating membrane sends out a compression wave by moving the air molecules back and forth. The sound is heard by setting a small diaphragm in the ear, called the eardrum, vibrating in sync with the sound compression wave and these vibrations are sensed by the ear and transmitted to the brain by the nerves between the ear and the brain. The sensitivity of the ear to sound is not the same for all people. The normal range is from about 20 to 20,000 cycles per second or Hertz (Hz). Waves higher than this are called ultrsonic waves. The sensitivity of the ear is not the same over the whole range; the ear is most sensitive at about 3000 Hz.

  2. Intensity and Loudness of Sound.
    All wave motion carry energy along with them. Sound waves are no exception. A loudspeaker sends out energy along with the sound it produces. The energy flows in the direction of propagation of the waves. The intensity of the wave is defined in terms of the energy carried by the wave. Imagine a surface erected perpendicular to the direction of propagation of the waves. The intensity I of the wave is defined to be the energy carried per second through an unit area of this surface by the wave. Since power is the time rate of transfer of energy, sound intensity is the power P of the sound energy passing through an unit area A of a surface erected perpendicular to the direction of propagation of the sound, that is,
    I = P/A.
    The unit of sound intensity I is typically measured in watts per square meter. The following table lists the representative sound intensities that the human ear can hear.

    Approximate Sound Intensities
    Type of Sound Intensity watt/m2 Intensity Level, dB
    Barely audible sound 10-12 0
    Rustle of leaves 10-11 10
    Average whisper 10-10 20
    Ordinary conversation 10-6 60
    Busy street traffic 10-5 70
    Power mower or subway 10-2 100
    Pain producing sound 1 120
    Jack hammer or riveter 101 130
    Jet aircraft 103 150
    As can be seen from this table that the loudness of a sound does not correspond directly to its intensity. Although the sound of ordinary conversation is louder than an average whisper, it is clearly not 10,000 times as loud; intensity clearly does not represent the loudness of the sound to the human ear. Thus there has been developed a way to approximate the loudness of sounds to the human ear. A intensity-level or loudness scale, called the decibel scale, has been defined to relate the intensity of sound to the way the human ear judges the loudness of sound. The intensity level (or loudness) is measured in decibels (dB) and is defined by
    B = 10 log (I/I0),
    where I is the intensity of the sound under consideration and I0 is a reference intensity which is usually taken to be 10-12 w/m2 and B is the intensity-level or loudness which is measured in decibels. The basic unit of loudness is the "bel" and is named for the inventor of the telephone, Alexander Graham Bell (1847-1922). The decibel is one tenth of the bel (the prefix "deci" is the metric system prefix for 10-1). On the decibel scale, the threshold of hearing (I = 10-12 w/m2) is
    B = 10 log (10-12/10-12) = 10 log (1) = 0 dB.
    At the other end of the scale, the threshold of pain (I = 1 w/m2) corresponds to an intensity-level or loudness of 120 dB.
    B = 10 log (1/10-12) = 10 log (1012) = 120 dB.
    Prolong exposure to high intensity sound at this level, such as a hard rock music, can cause serious damage to one's hearing. Ear plugs are recommended whenever the sound level exceeds 90 dB.

  3. The Speed of Sound.
    Sound waves are longitudinal waves, that is, the particles of the medium through which the wave travels move back and forth is the same direction that the wave is traveling. A simple system which can be a model for the propagation of longitudinal elastic waves is a number of identical point masses coupled together by springs to form a chain, each two masses connected by spring at each of its end. Each spring follows Hooke's law for small elongations or compressions. When any one mass is displaced longitudinal, the springs attached to it are stretched or compressed; these deformations produce a force not only on the displaced mass but also on the neighboring masses. Thus, neighboring masses are set in motion, and when the mass at the left end of chain is suddenly displaced to the right, a compressional disturbance is sent propagating to the right along the chain. Each mass undergoes, in turn, a longitudinal displacement to the right and then returns to its equilibrium position, as a longitudinal or copressional wave travels along the chain. Thus the masses oscillate about their equilibrium positions as the wave travels to the right along the chain.

    The essential conditions for the existence of longitudinal waves are that the medium, in the case, the chain of masses and springs, must possess inertia and be elastically deformable. Clearly, if there were no coupling between the masses (the springs were infinitely weak), then displacing one mass would not effect the neighboring masses, and the wave would not be produced or propagated. Or, on the other hand, if the medium perfectly rigid (if the springs were infinitely rigid), then displacing one mass would cause all the other masses to be displaced simultaneously. Again, a wave traveling at finite speed would not be produced. With an elastic medium, however, when an external force initially deforms and sets in motion only a portion of the medium, this deformed protion through its coupling with adjoining portions, sets them in motion. Suppose that left end of chain is moved longitudinally in simple harmonic motion, instead of a single pulse. All the masses will eventually be set in simple harmonic motion with same amplitude and frequency as that of the source, each oscillating mass having a phase lag relative to the source which is proportional to its distance from it.

    To represent this motion mathematically, let the coordinate x give the equilibrium location of each mass in the chain relative to, say, the left end (the origin of coordinate system). And let the coordinate y represent the longitudinal displacement of a point mass from its equilibrium position along its direction of propagation. Let the waveform travel to the right at the wave speed v. In general, this wave speed is equal to the square root of the ratio of the elastic property to inertial property. All types of mechanical waves follow a relation of this type. In a transverse mechanical wave in a string, the wave speed v is equal to square root of the ratio of the string's tension to the linear density of the string. In the longitudinal mechanical wave of sound the wave speed is equal to the square root of the ratio of the bulk modulus B to mass density ρ (mass per unit volume).
    The mathematical expression for this longitudinal wave is
    y = A sin (kx - ωt),
    where A is the maximum longitudinal amplitude,
    k is the propagation constant,
    and ω is the angular frequency.

    Let the waveform travel to the right at the wave speed v. In general, this wave speed is equal to the square root of the ratio of the elastic property to inertial property. Since elastic modulus expresses the elastic property of the medium and the density of the medium expresses quantitatively the inertial property, the wave speed is equal to the square root of the ratio of the elastic modulus of the medium to its density. All types of mechanical waves follow a relation of this type. In a transverse mechanical wave in a string, the wave speed v is equal to the square root of the ratio of the strings's tension FT to the linear density μ (mass per unit length) of the string. That is,
    v = √(FT/μ). (3)
    In the longitudinal mechanical wave of sound through a gas or liquid, the wave speed is equal to the square root of the ratio of the bluk modulus Bto the mass density ρ (mass per unit volume). That is, v = √(B/ρ). (4)
    Similarly, the speed of propagation of a longitudinal pressure wave in a solid is equal to the square root of the ratio of the Young's modulus Y of the solid rod to its mass density ρ. That is,
    v = √(Y/ρ). (5)

    The Speed of Sound in a Real Gas.
    According to the kinetic-molecular theory, the molecules of the gas are on the average very far apart compared with the size of the molecules. It is only through occasional random collisions between the molecules that one portion of the gas knows, so to speak, what is going on elsewhere in the gas. This would imply that speed of sound in real gas cannot be exceed the average speed of the molecules of the gas; in fact the two speed are nearly equal. According to Kinetic-molecular theory the average speed of molecular motion is related to the absolute temperature T of the gas by the following equation:
    (1/2)m(vrms)2 = (3/2)kT, (6)
    where
    vrms is the root-mean-square velocity,
    m is the mass of the gas molecule,
    k is Boltzmann's constant (k = 1.38 × 10-23 J/K),
    and T is the absolute temperature in degrees Kelvin.
    Now the speed of sound in a real gas is directly proportional to rms velocity of the gas molecules; that is, in mathematical symbols,
    v = √[(γ/3) × vrms], (7)
    where γ is the ratio of the specific heat of the gas at constant pressure to the specific heat at constant volume; it is largest for a monoatomic gas and is nearly unity for polyatomic gases. Solving equation (6) for vrms and substituting into equation (7), we get
    v = √(γkT/m). (8)
    This equation tells us that the speed of sound in a real gas is directly proportional to the square root of the absolute temperature of the gas and for gases that have the same molecular structure is inversely proportional to the square root of the molecuar weight of the gas. Note that the speed of sound is independent of the pressure of the gas. The speed of sound of various gases are listed in the following table.

    Substance Temperature, deg C. Speed, m/s
    Gases

    Carbon dioxide 0 259
    Oxygen 0 316
    Air 0 331
    Air 20 343
    Nitrogen 0 334
    Helium 0 965
    Liquids

    Mercury 25 1450
    Water 25 1498
    Seawater 25 1531
    Solids

    Rubber - 1800
    Lead - 2100
    Lucite - 2700
    Gold - 3000
    Iron - 5000-6000
    Glass - 5000-6000
    Granite - 6000
    In liquids, and especially in solids, the effects of local pressure fluctuations are transmitted far more quickly than in gases because of the displacement of an atom caused by its neighboring atoms through their mutual interatomic forces and not as result of their occasional random collisions.

To continue, click here.