SOUND

8. Harmonic Sound Waves.
   If the source of a longitudinal wave, such as a vibrating diaphragm, oscillates with simple harmonic motion, then the resulting disturbance will be harmonic. One can produce a one-dimensional harmonic sound wave in a long, narrow tube containing a gas, by means of a vibrating piston at one end. A compressed layer is formed at times when the piston is being pushed into the tube. This compressed region, called a condensation, moves down the tube as a pulse, continuously compressing the layers in front of it. When the piston is pulled back, the gas in front of the withdrawing piston expands and the pressure and density in this region fall below their equilibrium values. These low pressure regions, called rarefactions, are also propagated down the tube, following the condensations. Both of these pressure regions move with a speed equal to the speed of sound in that medium. The distance between two successive condensations (or two successive rarefactions) is the wavelength, λ, of the sound. As these waves travel down the tube, any small volume of the medium moves with simple harmonic motion parallel to the direction of the wave. Let s(x, t) be the displacement of a small volume measured from its equilibrium position. This harmonic displacement function can be expressed as
   s(x, t) = S cos (kx - ωt), (1)
   where S is the maximum displacement from its equilibrium position and is called the displacement amplitude, k is the propagation constant called the wave number, and ω is the angular frequency of the piston. Note that the displacement s is along the x-axis, the direction of the sound wave, which of course means that the wave is longitudinal. The variation is the pressure of the gas, p, measured from its equilibrium value is also harmonic and maybe expressed as
   p = P sin (kx - ωt), (2)
   where P is the maximum change in pressure from its equilibrium value and is called the pressure amplitude. This pressure amplitude P is directly proportional to the displacement amplitude, S, which can be expressed by the following equation:
   ΔP = ρvωS,
   where ωS is the maximum longitudinal velocity of the medium in front of the piston.
   Note that a sound wave may be considered as either a displacement wave or a pressure wave. A comparison of the equation for displacement wave with the equation for the pressure wave shows that the pressure wave is 90 degrees out of phase with the displacement wave. That is, the pressure variation is maximum when the displacement is zero, whereas the displacement is a maximum when the pressure variation is zero. Since the pressure is directly proportional to the density ρ, the variation in density from the equilibrium value follows a relationship similar to the equation (2) for pressure variation.

9. Longitudal Standing Waves.
   Standing longitudal waves can be set up in tube of air, such as in an organ pipe, as the result of interference between longitudinal waves traveling in the opposite directions. The phase relation between the incident wave and the reflected wave depends upon whether the end of pipe is open or closed. The is analogous to the phase relations between incident and reflected transverse waves at the ends of a string. The closed end of an air column in a pipe is a displacement node, just as the fixed end of a vibrating string is a displacement node. As a result, at the closed end of a tube of air, the reflected wave is 180 degrees out of phase with the incident wave. Furthermore, since the pressure wave is 90 degrees out of phase with the displacement wave, the closed end of an column of air corresponds to a pressure antinode (that is, a point of maximum pressure variation).
   If the air column is open to the atmosphere, then the air molecules have complete freedom of motion. Therefore, the wave reflected from the open end is nearly in phase with the incident wave, when the tube's diameter is small relative to the wavelength of the sound. Consequently, the open end of an air column is approximately a displacement antinode and a pressure node. Strictly speaking, the open end of an air column is not exactly an antinode. When a condensation reaches an open end, it does not reach full expansion until it passes somewhat beyond the end. For a thin-walled tube of a circular cross section, the end correction is about 0.6R, where R is the radius of the tube. Hence, the effective length of the tube is somewhat longer than the true length L.
   By directing air against an edge at the left end of the pipe, a longitudinal standing waves are formed and pipe resonates at its natural frequencies. All modes of vibration are excited simultaneously (although not with the same amplitude). Note that the ends are displacement antinodes (approximately). In the fundamental mode, the wavelength is twice the length of the pipe, and hence the frequency of the fundamental, f₁, is given by
   v/2L.
   where v is the speed of sound in air.
   Similarly, the frequencies of the overtones are
   2f₁, 3f₁,....
   Thus, in a pipe open at both ends, the natural frequencies of vibration form a harmonic series, that is, the overtones are integral multiples of the fundamental frequency, f₁. Since all harmonics are present, the natural frequencies can be expressed as
   f_n = n(v/2L). (n=1,2,3,...)
   where v is the speed of sound in air.
   If the pipe is closed at one end and open at the other, the closed end is a displacement node. In this case, the wavelength for the fundamental mode is four times the length of tube. Hence, the fundamental, f₁, is equal to v/4L, and the frequencies of the overtones are equal to 3f₁, 5f₁, .... That is, in a pipe closed at one end only odd harmonics are present, and these are given by
   f_n = n(v/4L), (n=1,3,5,...)
   where v is the speed of sound in air.

10. Musical Scales and Harmony.
    Nearly all music is based on a basic set of notes or tones called a scale, in which the notes bear a definite relationship to one another. Many scales have been used throughout history. The simplest one in Western music is the diatonic scale, which consists of the eight familiar notes "do-re-mi-fa-so-la-ti-do." Each note is named by a letter of the alphabet from A to G, and each corresponds to a particular pitch or frequency. The highest "do" has twice the frequency of the lower "do." The difference in pitch between each tone on diatonic scale is called a whole interval, except for those between mi and fa and between ti and do, which are half-intervals. The ratio of the successive frequencies for whole intervals is 9/8 or 10/9 and for half-intervals it is 16/15. The pitch of the first do can be chosen arbitarily, but the pitch of the remaining notes then conform to the regular seqauence indicated in the following table; in this table the initial do is taken as middle C at 264 Hz, and this is the C major scale.

    Diatonic C major scale:

    Note	Letter name	Frequency (Hz)	Frequency ratio	Interval
    do	C	264
    			9/8	whole
    re	D	297
    			10/9	whole
    mi	E	330
    			16/15	half
    fa	F	352
    			9/8	whole
    sol	G	396
    			10/9	whole
    la	A	440
    			9/8	whole
    ti	B	495
    			16/15	half
    do	C	528

    The interval from middle C to the next C, which is called "C above middle C," C', is called an octave from the Latin word for "eight," because there are eight notes counting both ends (but only seven intervals). C' has twice the frequency of C. This is always the case as you go up the scale. Each C has twice the frequency of the preceding C, each G has twice the frequency of the preceding G, as so on. A diatonic scale can start on any note, and the one beginning on D, is called the D major scale; the following table shows that scale.

    Diatonic D major scale:

    Note	Letter name	Frequency (Hz)	Frequency ratio	Interval
    do	D	297
    			9/8	whole
    re	E	334
    			10/9	whole
    mi	F#	371
    			16/15	half
    fa	G	396
    			9/8	whole
    sol	A	445
    			10/9	whole
    la	B	495
    			9/8	whole
    ti	C#	557
    			16/15	half
    do	D	594

    As can be seen, the spacing of half and whole intervals is the same. Some of the notes must thus be raised or lowered a half interval; when raised a half-interval, it is called a sharp (#) and when lowered, a flat (b). Sharps and flats are also used in the "minor scales," in which the spacing arrangement of intervals is slighly different than for a major scale. A scale that includes all sharps and flats is a chromatic scale. The equally tempered chromatic scale is given in the following table:

    Equally tempered chromatic scale:

    Note	Letter name	Frequency (Hz)	Frequency ratio	Interval
    do	C	262
    			1.06	half
    	C# or Db	277
    			1.06	half
    re	D	297
    			1.06	half
    	D# or Eb	311
    			1.06	half
    mi	E	330
    			1.06	half
    fa	F	349
    			1.06	half
    	F# or Gb	370
    			1.06	half
    sol	G	392
    			1.06	half
    	G# or Ab	415
    			1.06	half
    la	A	440
    			1.06	half
    	A# or Bb	466
    			1.06	half
    ti	B	494
    			1.06	half
    do	C	524

    On this scale F# and Gb, for example, are taken to have the same frequency. Originally, the frequencies of these two notes (and other like pairs) were slightly different, and string soloists often play them that way today. However, taking the sharp and corresponding flat to be the same frequency greatly reduces the number of keys need on a piano and other fixed note instruments. Note also that the note A does not have precisely the same frequency on the C major and D major scales. This true of many notes when all the different when all the different diatonic scales are considered. The equally tempered chromatic scale, in the interests of reducing the number of keys on fixed-note instruments, ignores these slight differences and thus does not reproduce precisely a diatonic scale. The equally tempered scale is made up of 12 equally spaced intervals, so each note has a frequency that is 1.059 times that of the previous bite, This particular number arises because (1.059)¹² = 2, as required for an octave.

    Many other scales have been used in the West and in other cultures. One that is used in Africa and the Orient is the pentatonic scale, based on just five tones.

    Harmony is an important aspect of music. Two notes are said to be harmonious, if they sound pleasant when sounded together and are not harsh. In general, people find notes harmonious if the ratio of their frequencies equals the ratio of two small whole numbers. Thus a ratio of 2/1, which is an octave, sounds very harmonious. A ratio of 3/2 (such as G = 396 Hz and C = 264 Hz) is also very harmonious, it is called a "fifth," because G is five notes above C (counting both ends). Ratios of 4/3 and 5/4 are also harmonious, but a little less so, whereas the ratios of larger numbers such as 9/8 or 10/9 are quite dissonant; this can be checked by striking two successive notes (say C and D) on the piano at the same time.

    The ancient Greeks first found the relation between harmoniuos notes and the ratio of small whole numbers. They did not, however, relate it to frequency but rather to the length of strings. If the ratio of the lengths of two equally taut strings was 2/1 or 3/2, the sound was harmonious. In modern times it is clearly seen how this discovery relates to frequency, because of the modern understanding of standing waves.

    Tone Quality or Timbre. Whenever we hear a musical sound, we are aware of its loudness, its pitch, and also of a third aspect called "tone quality." For example, when a clarinet and an oboe play one after the other the same note with the same loudness and same pitch (say middle C), there is a clear difference in the musical sound. The sound of clarinet will not confused with the sound of an oboe. This difference is a difference of tone quality; the technical term is timbre or tone color.
    Tone quality, like loudness and pitch, is related to a physically measurable quantities. Tone quality of a sound depends upon the presence of overtones or harmonics, the number of them and their amplitude. When a note is played on a musical instrument, the fundamental frequency as well as the harmonic frequencies are present simultaneously. This superposition of the fundamental and its harmonics give a composite wave form characteristic of the instrument. Not only is the presence of the harmonics but also the relative amplitude of the harmonics is what gives to each instrument its distinctive sound or tone quality. Usually the fundamental has the greatest amplitude and its frequency is what is heard as the pitch of the note.
    The tone quality of sound is also affected by the manner in which an instrument is played. Plucking a violin string, for example, makes a very different sound than pulling a bow across it. Also the pitch of a note can be affected by the way the instrument is played. In some instruments like the flute the pitch of the note can be determined by the way the stream of air is blown into the mouthpiece. When the stream of air is blown directly into the mouthpiece instead across it, the fundamental is hardly excited at all and the first harmonic, at twice of the frequency, predominates. The pitch, then, is an octave higher.

    The sound spectrum of an instrument is largely determined by the shape of the instrument and the material from which it is made, as well as the way it is played. Often the way the instrument is constructed can affect the quality of the sound that the instrument produces. This is particularly true of string instruments which have a sound box; the sounding box not only amplifies the sound, but, because it has resonant frequencies of its own, determines which harmonics will be emphasized.