Why Molecules Keep Vibrating: Elasticity, Inertia, and Friction

• When air molecules are set into vibration by a disturbance, such as a tuning fork being struck, they vibrate a tiny distance around their equilibrium positions, eventually coming to a stop.

• Once the molecules have been disturbed, three forces interact to keep them swinging back and forth for a while before they settle down again: elasticity, inertia, and friction.

• Elasticity is a restoring force; it refers to the property of an object to be able to spring back to its original size, form, location, and shape after being stretched, displaced, or deformed.

• All objects that have mass, including air molecules, possess a certain degree of elasticity. The amount of restoring force depends on the extent to which the object is displaced.

• Hooke's law states that the restoring force is proportional to the distance of displacement and acts in the opposite direction.

• Thus, the farther an object is displaced from its original location, the stronger the restoring force that pulls it back toward that position.

• Inertia refers to the resistance of any physical object to a change in its state of motion or rest.

• Friction is a force that opposes motion.

• It is easy to demonstrate Hooke's law with an elastic band.

• Stretch the elastic band to different lengths and feel the difference in the strength

of the recoil when you let go. The further you stretch it, the stronger is the recoil.

The vibration of the air molecules does not last indefinitely?

Because of the frictional resistance of the air, each time the molecules move back and forth around their equilibrium positions, they do so with slightly less amplitude.

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Amplitude refers to the maximum distance away from rest position that the molecules are displaced.

Amplitude is determined by the strength of the original disturbance: the stronger the disturbance, the greater will be the displacement of the molecules.

The decrease of amplitude is called damping.

Damping finally causes the molecules to settle down once again at their equilibrium positions.

At this point, no further changes in Pam Occur.

Sound Propagation

• As long as the vibration continues, the alternating increases and decreases in pressure continue in a wavelike motion that propagates (travels) from the source of the sound through the air in an ever-widening sphere. (see Figure 1.4).

• If an ear is in the path of some of these vibrating molecules, detects the fluctuations in pressure as the sound wave impinges upon the tympanic membrane (TM, ear- drum).

• At one instant in time, the TM detects high pressure corresponding to the arrival of a compression, followed by low pressure corresponding to the arrival of a rarefaction.

• The compression of air moves the TM inward slightly, whereas the rarefaction of air pulls the TM slightly outward.

• Thus, the TM vibrates inward and outward in response to the changes in air pressure arriving at the ear.

• The vibration of the tympanic membrane sets bones within the middle ear (ossicles) into vibration.

• The vibration of the tympanic membrane sets bones within the middle ear (ossicles) into vibration.

• The vibration of the ossicles in turn sets the fluid in the inner ear into vibration, resulting in the stimulation

of hair cells (nerve cells) in the inner ear.

• The triggering of the nerve cells generates a nerve impulse, which is conducted along the auditory

pathway to the appropriate areas of the nervous system and is then interpreted by the brain as sound.

• It is important to realize that sources of sound cause changes in air pressure not just in one direction but in

all directions.

Wave Motion of Sound

• A sound wave describes the alternating increases and decreases of pressure that move through a medium.

• Wave motion is characterized by minimal movement of individual molecules and potentially extensive propagation of the changes in pressure that are created by the molecular movement.

• Once molecules have been set into vibration, each molecule (or group of molecules) does not, itself, travel long distances to get to a listener's ear.

• The molecules are only temporarily disturbed from their rest position and travel a tiny distance around their equilibrium positions before their motion dies away and they return to their original position.

• It is the changes in air pressure caused by molecular vibration that travel toward the listener's ear.

Longitudinal Versus Transverse Waves

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Waves can be longitudinal or transverse (Figure 1.6).

Wave motion of water is transverse; that is, the individual molecules of water

move up and down at right angles to the direction that the wave is traveling.

We are all familiar with the phenomenon of water rippling outward in widening

circles when a stone is dropped into a pond.

These ripples are the waves, formed by water molecules moving up and down and transmitting the disturbance in all directions.

Sound waves are longitudinal.

In a longitudinal wave the individual molecules move parallel to the direction that the wave is traveling.

The area of compression around the vibrating source is followed by an area of rarefaction, followed by an- other area of compression, another of rarefaction, and so on, spreading outward in a sphere. The outermost area of the sphere is called the wave front.

• The farther the changes in air pressure travel from the source, the more damped they become because the area of the wave front is directly proportional to the of its distance from the source (called the inverse square law).

• This means square that as the sound wave propagates, the constant amount of energy in the wave is distributed over an exponentially increasing area.

• The sound's acoustic energy diminishes by a factor of 1/distance2.

Example: the energy of a sound wave that is twice the distance from the source is distributed over four times the area and therefore has one fourth of the intensity compared to the and the energy of a wave that is three times the distance of the source is distributed over nine times the area with a correspondingly decreased energy.

Mass/Spring System

• Discussion has focused on vibration using the example of a tuning fork.

• Vibration can also be described in terms of a mass/spring system (Figure 1.7).

• Such a system is comprised of a block of a certain mass attached to a spring.

• The spring is fixed at one point and is free to move at the point of attachment to the

block.

• When not disturbed, the system is in equilibrium with no net force acting on it.

• If the block is pulled and then released, the spring with the attached block will recoil

toward the rest position due to inertia.

• As the block and spring continue to overshoot rest position, the elastic recoil forces increase and pull the block and spring back toward equilibrium.

• As with the example of the tuning fork, this back-and-forth motion around rest position gradually loses amplitude and dies out.

Simple Harmonic Motion

• The vibration of the tuning fork and the movement of the mass/spring system occur in simple harmonic motion (SHM).

• Simple harmonic motion occurs when the restoring force acting on an object (i.e., elasticity) is directly proportional to the displacement of the object from its equilibrium position but in the opposite direction (i.e., toward its equilibrium position).

• Simple harmonic motion is characterized by a pattern of acceleration through the equilibrium position and deceleration at the endpoints of the movement.

• The example of a swing illustrates this motion

Frequency, Period, Wavelength, Velocity, and Amplitude

• As an object vibrates it does so at a certain rate, and the vibrations have a certain amplitude.

• The resulting wave travels with a certain velocity through a medium. Figure 1.9 illustrates the related concepts of frequency, period, wavelength, and amplitude.

Frequency

• Frequency One back-and-forth movement of the tuning fork tine or molecule around the equilibrium position makes up one cycle of vibration.

• One cycle of vibration occurs when the tine or molecule moves to a maximum distance away from its original spot, back toward rest position, moves to a maximal point in the site direction, and then moves back again to the equilibrium position.

• For sound, cycles of vibration are typically described in terms of pressure changes rather than in terms of individual movements of molecules.

• Acoustically, a cycle of vibration consists of an increase in pressure from Pam (compression), a decrease in pressure back to P am ́ further decrease in pressure below Pam (rarefaction), and a return to baseline Pam .

• Cycles of vibration are measured in terms of time in seconds.

• The tines of a tuning fork might vibrate at the rate of 100 cycles per second (cps), causing the surrounding air molecules to vibrate at a rate of 100 cps as well.

• If this vibration eventually reaches a listener, that person's eardrum will vibrate at 100 cycles per second.

• The number of cycles per second is called frequency, and the unit of measurement of frequency is hertz (abbreviated Hz).

• Thus, a tuning fork vibrating at 100 cps has a frequency of 100 Hz.

• The sound wave produced by the tuning fork also has a frequency of 100 Hz, and the

eardrum in the path of this 100 Hz sound wave would be set into vibration at 100 Hz.

• Frequency can also be expressed in terms of kilohertz (kHz):

• 1000 Hz equals 1 kHz, and 2500 Hz equals 2.5 kHz.

Period

• Period is the time it takes for one cycle of vibration to occur.

Example: a wave with a frequency of 100 Hz has 100 cycles that occur in 1 second.

• Assuming that each cycle in the wave lasts for the same amount of time, it is clear that each must

take 1/100 second (0.01 s).

• Period is symbolized as t.

• There is a reciprocal relationship between frequency and period. This relationship is expressed by

the formula F = 1/t, where F is frequency, and t equals period.

• If one knows the frequency of a wave, its period can be calculated by putting a 1 over the

frequency; if the period of the wave is known, its frequency is determined by putting a 1 over the

period. Sound waves can be periodic or aperiodic.

• Aperiodic wave is a wave in which every cycle takes the same amount of time to occur as every

other cycle.

• Such a wave would have a musical tone and a specific pitch.

• Example, the vibrating string of a guitar or violin produces a periodic sound

wave with a musical tone and pitch.

• However, not all sound waves have cycles lasting the same amount of time.

• A wave in which individual cycles do not take the same amount of time to occur

is called aperiodic wave (without a period). Why?

• Because this type of sound does not have a specific period, by definition, it cannot have a specific frequency, or a specific pitch.

• Perceptually, such a wave sounds like noise.

Wavelength

• Sound waves travel through space.

• The measurement of the distance traveled by a sound wave is its wavelength (λ).

• Wavelength refers to the distance in meters or centimeters covered by one complete

cycle of pressure change-that is, the distance covered by the wave from any starting point

to the same point on the next cycle.

• Frequency, period, and wavelength are closely related.

• The higher the frequency (the more cycles per second), the shorter in duration is the

period and the shorter is the wavelength.

• The lower the frequency (the fewer cycles per second), the longer in duration is the

period and the longer is the wavelength.

Velocity

• How fast a wave moves (its velocity) depends on the density and elastic properties of the medium through which it is moving.

• Because water is denser than air, sound travels about four times as quickly through water as it does through air and even faster than that in some solids, such as steel.

• The speed of sound in air is around 331 m/s at 0°C (32°F), compared to 1461 m/s in water at 19°C.

• In a steel rod, sound may travel at a speed of 5000 m/s (Dull, Metcalfe, & Williams, 1960).

• The speed of sound in liquids and solids is not affected much by temperature.

• In gases such as air, however, temperature plays an important part in how fast the sound travels.

• The warmer the air, the more quickly sound gates, with the speed increasing at the rate of about

0.6 m/s/°C.

• The relationship among the wave speed (v), the wavelength (λ), and the frequency (F) is expressed as:

(frequency equals velocity of sound divided by wavelength) or

(wavelength equals velocity of sound divided by frequency).

• This relationship is important in calculating certain aspects of vowel production.

• Amplitude represents the amount of displacement from an equilibrium position

over time and corresponds perceptually to the loudness of the sound.

Visually Depicting Sound Waves:

• Waveforms Sound waves are invisible and intangible; the pressure changes are miniscule and cannot be seen.

• This could be a problem in understanding and working with sound, which is fleeting and insubstantial.

• A waveform is a graph with time on the horizontal axis and amplitude on the vertical axis.

• The amplitude represents the amount of whatever is being graphed.

• The vertical axis of the resulting waveform would represent the distance that the tine vibrated around its rest position, and the horizontal axis would represent the time that the

tine was moving.

• When dealing with sound, what is represented on the waveform are the changes in air pressure that result from molecular motion.

• So, an acoustic wave- form shows the amplitude of air pressure changes over time.

• If a line is drawn at the midlevel of the graph, it would represent normal

• When the line goes above baseline, it represents an increase in pressure (compression), and the height of the line at any point represents the magnitude of

pressure increase.

• Similarly, when the line goes below baseline, it represents a decrease in pressure

(rarefaction), and the depth of the line at any point represents the magni- or baseline pressure. amplitude of pressure decrease.

• A waveform is useful in showing many different aspects of a sound.

• By counting the peaks in the waveform, the frequency of the wave can be calculated.

• By measuring the time of each cycle, the period of the wave can be determined.

• Because the vertical axis measures the magnitude of pressure changes, it is easy to

visualize the relative amplitude of the wave and whether or not the wave is damping.

• Shape is another aspect of sound that is visible from a waveform.

• A smoothly varying shape, a sinusoid, indicates that the wave is a pure tone vibrating in

SHM.

• If all the cycles in the wave repeat themselves in a predictable fashion, the wave is

periodic.

• If the cycles look different and take different amounts of time to occur, the wave is

aperiodic.

• If the amplitude of the wave is decreasing over time, then the sound is damping.

• If the cycles in the wave repeat themselves in a regular but the shape of the wave is not sinusoidal (the cycles look more irregular way, in shape), then the wave is depicting a periodic complex sound.

• Pure Tones and Complex Sounds

• Pure Tones Sounds may be composed of one or more frequencies.

• A sound of a single frequency is called a pure tone.

• This type of sound is generated when an object vibrates in simple harmonic motion

(SHM) and is graphed as a sinusoidal wave (see Figures 1.8 and 1.10).

• Perceptually, a pure tone can be matched to a specific pitch on a musical scale and has a

thin quality. •

Complex Sounds

• Complex sounds are characterized by waves that consist of two or more frequencies

• Complex sounds occur when waves of different frequencies combine (inter- fere) with each other (see Figures 1.10 and 1.11).

• The interference results in a more complex vibration of the air molecules. Corresponding to the more complex molecular movement in complex sound waves, when acted upon by the wave the tympanic membrane also vibrates in a more complex manner. .

Periodic Versus Aperiodic Complex Waves

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Complexsoundsmaybeperiodicoraperiodic.

Periodic complex sounds consist of a series of frequencies that are systematically related to each other.

The lowest frequency of the sound is the fundamental frequency (F0), and the frequencies above the fundamental are called harmonic frequencies, or just harmonics. The harmonics in a complex periodic sound are whole-number multiples of the fundamental frequency.

A complex periodic wave has a musical tone and a specific pitch, and it sounds richer and more resonant than a pure tone wave. In fact, the more harmonics in a sound wave, the more resonant it will sound, and vice versa.

• Most musical instruments, as well as the human voice, produce sounds that are periodic and complex.

• The harmonics in a complex periodic sound can be identified through a process called Fourier analysis.

• Jean-Baptiste Fourier (1768–1830) was a French mathematician who showed that any complex wave can be represented by the sum of its component frequencies as well as their amplitudes and phases.

• Fourier analysis is a highly complex mathematical procedure, which is performed these days by computer.

• Aperiodic complex sounds also consist of two or more frequencies, but the frequencies are not systematically related to each other.

• Rather, a broad range of frequencies make up the sound.

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An periodic complex sound could contain all frequencies between 100 and 5000 Hz. Another aperiodic sound might include frequencies from 2000 to 4000 Hz.

Such waves sound like noise, with no musical tone and no specific pitch.

Examples include the sound of steam escaping from a radiator or the sound of applause. There are two kinds of aperiodic complex sounds, differentiated on the basis of their duration.

Continuous sounds are able to be prolonged, whereas transient sounds are extremely brief in duration.

Steam hissing out from the radiator is continuous, whereas the sound made by a person hitting his or her hand on a desk is transient.