Sound

"If a tree falls in the forest, and there is no one to hear it, will there be a sound?... This is a very old philosophical dilemma which relies on using the word "sound" for two different purposes. One use is as a description of a particular type of physical disturbance: 'Sound is an organized movement of molecules caused by a vibrating body in some medium - water, air, rock or whatever' [Olson, 1967]. The other is as a description of a sensation: 'Sound is the auditory sensation produced through the ear by the alteration ... in pressure, particle displacement, or particle velocity which is propagated in an elastic medium' [Olson 1967]. Both [of] these definitions are correct, they differ only in the first being a cause and the second being an effect" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/sound/sound1.html).

"Sound is the movement of air particles created by a vibrating source. Air particles are in constant random motion, exerting very small pressure variations around the steady-state atmospheric pressure. Each particle is subject to both an inertial force (due to the elasticity of the medium). When an object - a sound source - is set into vibration, each air particle moves to and fro about it's average position along an axis parallel to the direction in which the wave propagates. Air particles themselves do not move very far, they simply transfer pressure changes by what is referred to as sound propagation. This constitutes what we call a 'sound wave' which moves away from the sound source at a velocity determined by the medium. The velocity of propagation of a sound wave in air is about 344 meters per second, while in water it is 1437 meters per second" (http://www.neurophys.wisc.edu:80/~ychen/textbase/s1-p2.html).

"Sound originates when a body moves back and forth rapidly enough to send a coursing wave through the medium in which it is vibrating. A simple form of sound waves is produced by an explosion of a small balloon of compressed air. By bursting the balloon, potential energy (energy of position) is converted to kinetic energy (energy of motion)" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/sound/sound1.html).

"Sound waves move out spherically from a point source of sound, and as they do so they become less intense. Sound pressure is inversely proportional to distance from the source as long as the sound does not encounter obstacles, like the head and external ears for example. Obstacles, which create a change in the medium, impede or resist the propagation of sound. When a sound wave encounters an obstacle or change in medium, a portion of the sound wave is reflected from the surface. That portion of a sound wave not reflected from an obstacle is absorbed and continues to be propagated through the new medium. Reflectance of a sound is at the heart of our understanding of the action of the middle ear, whose purpose is to overcome the impedance mismatch at the interface of air and fluid of the inner ear. Reflected sound may encounter the original sound wave and, depending on the relative timing of the two, they may either reinforce or cancel one another. Sound waves may also be diffracted, which means that, depending on the frequency of the sound, they are able to wrap around small or medium-size objects. Reflectance and diffraction are two principal ways that sound waves are altered by the head" (http://www.neurophys.wisc.edu:80/~ychen/textbase/s1-p4.html).

"Any motion that is repeated is called periodic motion. Examples of periodic motion are the moon's orbit of the earth, a beating heart, the whirring tail of a frightened rattlesnake, the wings of a hummingbird, and the movement of the valve on a revolving bicycle tire. A vibrating body in contact with the atmosphere will produce sound waves. One simple example is a vibrating piston" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/sound/sound2.html).

"Most sound generators produce recurrent waves which are generally similar to each other. These waves are propagated at a definite velocity. This velocity depends on the medium of propagation.

One cycle of a sound wave in air consists of one compression of the air together with the subsequent rarefaction that occurs. The air molecules are forced together (compression or compaction) and then subsequently (in accordance with the 2nd law of thermodynamics) they immediately begin returning to their equilibrium state. The equilibrium state of the air molecules is the state in which they were before the compression under observation occurred. (Always taking into account that other disturbances of the atmosphere may have been occurring simultaneously with this compression.) In doing so they acquire momentum and thus become compressed again and so on" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/sound/sound3.html).

"The energy flow associated with a sound wave is the total mechanical energy (potential and kinetic energies associated with elastic oscillations of the medium) that is transferred during each second through a surface of unit area (1m) - expressed in Joule/m/sec or Watt/m. This is commonly called the intensity of the wave..." (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/intensity/Intensity2.html).

"Sound originates when a vibrating source disturbs the air in a quasi-periodic fashion and sends a sound wave traveling through it. Another way of saying this is that a sound source radiates acoustical energy which is transferred to the medium (air) and this energy propagates through the air in the form of a sound wave.

Our ears are not very sensitive at all to the total acoustical energy which reaches them. They are, however, sensitive to the rate at which the energy arrives. This rate is what determines loudness. The rate at which an instrument radiates acoustical energy is the instrument's acoustic power output. (Power is the rate of doing work.) The unit of power is the watt" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/intensity/Intensity1.htmlDefinition of Intensity).

"The perceived loudness of a sound, is however not directly proportional to its intensity. A built-in safety mechanism cuts down the sensitivity of the ear as intensity increases. When the intensity of a sound is doubled, the perceived loudness increases by only about 23%. Also, our perception of intensity varies with frequency - as we shall see later. Keeping frequency constant at say 1 KHz (1000 Hz)" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/loud/Loud1.html).

"A musical instrument is a transformer. It is supplied with power by a performer or electricity. Much of this power is used to overcome friction/resistance and is "wasted" in other ways. Only a small proportion is transferred into musical sound. A pianist may use energy at the rate of 200 watts in a very loud passage without more than 0.4 watt being radiated as sound.

1 watt is the power required to raise 0.45 Kg 215 mm/sec (approx.) A man doing hard continuous labor develops about 100 watts - the power required to keep a 100W electric light bulb burning. It would take thus approximately 2 million people in conversation to keep a 50 Watt electric light bulb burning" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/intensity/Intensity1.htmlDefinition of Intensity).

"The psychological magnitude loudness is associated with a given SPL. Judgments of whether two sine tones sound equally loud show fairly low dispersion among different individuals. Judgments on "how much" louder one tone is than another require previous conditioning or training and yield results that fluctuate greatly from individual to individual and from occasion to occasion.

Tones of the same SPL but with different frequencies are in general judged as having different loudness. SPL is thus not a good measure of loudness, if we inter compare tones of different frequency. Experiments have been performed to establish curves of equal loudness, taking the SPL at 1 KHz as a reference quantity" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/loud/Loud3.html).

"These size-of-domain difficulties are avoided if we use a logarithmic scale based on a ratio of intensities. As it happens, we are normally more interested in comparing intensities than dealing with absolute values. This comparison is achieved by observing their ratio. For example, if the ratio of the intensities of two sounds is 3, one intensity is thrice that of the other. Instead of specifying the power density (intensity) itself it is customary to specify how many times the actual power is greater than the power of a reference-level soundwave. This reference level is 10 Watt/m. This is about the weakest sound that humans can hear" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/intensity/Intensity3.htmlComparison of Sound Intensity Level (SL)).

"The fundamental physical descriptors of a sound are its frequency and its intensity. Frequency and intensity translate into the psychological attributes of pitch (frequency) and loudness (intensity), respectively" (http://www.neurophys.wisc.edu:80/~ychen/textbase/s1-p4.html).

"Frequency is the number of complete waves or oscillations or cycles of a periodic quantity occurring in unit time (usually 1 second). Note the difference between Frequency and Pitch. Frequency is a measure of the rate of disturbance whilst pitch is what our heads do with this phenomenon. In defining frequency, note the fundamental reliance on the concept of time" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/sound/sound3.html).

"A useful stimulus for testing auditory function is a pure tone for which the sound pressure is a sine wave when plotted against time. The frequency of a pure tone is the number of cycles or complete oscillations of condensation and rarefaction in one second. The unit of measure of frequency is Hertz (Hz). Thus, a pure tone that goes through 1000 cycles per second is said to have a frequency of 1000 Hz, or 1 kHz (kiloHertz). The period is the time required for one complete cycle, or the time that elapses between two successive condensation or rarefaction peaks. The wave length is the distance between two successive peaks on the wave" (http://www.neurophys.wisc.edu:80/~ychen/textbase/s1-p5.html).

The wavelength of a sound wave is the distance the sound travels to complete one cycle...A sound wave travels with a definite velocity. The actual velocity depends on the medium through which the wave is traveling. Now we know that the frequency of a sound wave is the number of cycles that pass an observation point per second. Thus, the velocity of propagation of a sound wave is its wavelength times its frequency. The frequency of a wave is independent of the waves' medium, however, the wavelength will depend on the wave velocity in the medium through which it is traveling" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/sound/sound3.html).

Hermann von Helmholtz (1821-1894) [invented] resonators - hollow glass spheres that have two short tubular necks, diametrically opposite one another. One opening was put to the ear, the other directed at the sound source. By using a series of these Helmholtz was able to estimate the strengths of the harmonics of a periodic sound. He also found the frequencies of inharmonic partials of bells and gongs. Resonances have a noticeable effect on the timbre of musical sounds. For brass instruments this is controlled by the length and shape of tubing, and how the player constricts the lips. For woodwind instruments, their tubular structures are resonators whose resonant frequencies are controlled by opening or closing various holes. The vocal tract has several resonances that emphasize various ranges of frequency in the sound produced by the vibration of the vocal cords. By changing the shape of the vocal tract, the frequencies of these resonances or formats determine which vowel sound is produced. The resonances of the soundboard of a violin greatly affect the timbre. The suppression of some partials is important for the musical quality of the violin tone" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/spectra/spectra2.html).

"The French Mathematician François Marie Charles Fourier (1772-1837) invented a type of mathematical analysis by which it can be proved that any periodic wave can be represented as a sum of sine waves having the appropriate amplitude, frequency and phase. The function that sums all the harmonics into a sound wave is called a Fourier integral. Furthermore for harmonic spectra, the frequencies of the component waves are related in a simple way: they are whole number multiples of a single frequency...A square (or pulsed wave with a mark to space ratio of 1:1) requires the sum of an infinite number of sine components whose frequencies are odd whole number multiples of the fundamental... and whose amplitudes decrease in proportion to the inverse of the harmonic number (1, 1/3, 1/5, 1/7, ...) and the proper phases. Most periodic sound waves consist of both odd and even frequency components although closed organ pipes and some wind instruments (eg clarinets) do have predominantly odd frequency components. A triangular wave also has only odd numbered frequency components. Their amplitudes are different to those of a square wave however, reducing to the inverse of the square of the harmonic number (1, 1/9, 1/25, 1/49 ...) A Fourier representation of a complex wave of finite duration (as musical sounds are) requires an infinite number of different harmonics. Trying to represent actual sounds as sums of true sine waves, which persist from an infinite past into an infinite future, is a mathematical artifice. Consider the nearly periodic sounds produced by musical instruments. A sum of harmonically related sine waves doesn't correctly represent such a sound, because the sound starts, persists a while and dies away.

'Noisy' sounds such as the hiss of escaping air or the "sh" or "s" sounds of speech can be represented as the sum of sine waves (a Fourier integral) that have slightly different frequencies. When the sound is repeated the waveforms won't be exactly the same. The power of the sound in any narrow range of frequencies will be about the same, but the amplitudes and phases of the individual frequency components won't be identical. Nevertheless the two different "sh" sounds will sound the same; we will hear them as being identical. It may however, for certain purposes, be adequate to use fewer components" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/spectra/spectra3.html).

 

"Most sound generators produce recurrent waves which are generally similar to each other. These waves are propagated at a definite velocity. This velocity depends on the medium of propagation. One cycle of a sound wave in air, consists of one compression of the air together with the subsequent rarefaction that occurs. The air molecules are forced together (compression or compaction) and then subsequently (in accordance with the 2nd law of thermodynamics) they immediately begin returning to their equilibrium state. The equilibrium state of the air molecules is the state in which they were before the compression under observation occurred. (Always taking into account that other disturbances of the atmosphere may have been occurring simultaneously with this compression.) In doing so they acquire momentum and thus become compressed again and so on. Frequency is the number of complete waves or oscillations or cycles of a periodic quantity occurring in unit time (usually 1 second). Note the difference between Frequency and Pitch. Frequency is a measure of the rate of disturbance whilst pitch is what our heads do with this phenomenon. In defining frequency, note the fundamental reliance on the concept of time. The wavelength of a sound wave is the distance the sound travels to complete one cycle. The symbol used to denote wavelength is the Greek letter lambda (). The preceding examples have shown that a sound wave travels with a definite finite velocity. The actual velocity depends on the medium through which the wave is travelling. In fact the following can be observed to be true:

V light wave >> V radio wave< V sound wave >V water wave >> V earth's rotation

Since a wave advances a distance of one wavelength in a time interval of one period, it follows that the velocity of a wave is given by

but since T = 1/f, we can write

Now we know that the frequency of a sound wave is the number of cycles that pass an

observation point per second. Thus the velocity of propagation of a sound wave is its wavelength times its frequency.

v = f or

where v = velocity of propagation, in centimetres per second = wavelength, in centimetres

f = frequency, in Hz.

The frequency of a wave is independent of the waves' medium, however, the wavelength will

depend on the wave velocity in the medium through which it is travelling" (http://online.anu.edu.au/ITA/ACAT/drw/PPofM/sound/sound1.htmlFrequency of a Sound Wave).

The sounds that we hear from vibrating objects are complex in the sense that they contain many different frequencies. This is due to the complex way that objects vibrate. A "note" (say Middle C) played on a piano sounds different from the same "note" played on a saxophone. In both cases there are different frequencies present above the common fundamental note sounded. These different frequencies are part of what enables you to distinguish between different instruments. (Their difference in timbre.) Spectra can be classified as being of one of two types:

 

(1) Harmonic, in which the spectral components (different frequencies) are

mostly whole number multiples of the lowest, and most often loudest,

frequency. The lowest tone is called the fundamental and the higher (i.e. in

frequency) spectral components, the tones over the fundamental, are called

overtones, or harmonics.

 

(2) Inharmonic, in which the criteria in (1) are not met i.e in which the spectral

components are mostly NOT whole number multiples of the lowest frequency,

which is often NOT the loudest tone. Many percussion instruments fall into this

category.

 

The harmonic series is a series of frequencies that are whole number multiples of a fundamental. For example, taking the tone Middle C whose fundamental frequency is approximately 261 Hz,the harmonic series on this frequency is:

Hs = f , 2f , 3f , 4f , ......... nf and where f = 261 Hz.,

= 261 Hz , 522 Hz , 783 Hz , 1044 Hz , ................... n x 261 Hz.

This is expressed in musical notation thus:

Note that these notes are 'representing' the tones. The frequencies of the actual tones are at

varying distances from the frequencies of these notes depending on the tuning system used. In

particular the 7th