SOUND ,1 subjectively the sense impression of the See also:

• ORGAN

In this article, which covers the See also:

• SCIENCE (Lat. scientia, from scire; to learn, know)

Sound needs a Material See also:

• MEDIUM

That layer C A presses against and pushes forward the next layer and so on. Thus a push or a See also:

• COMPRESSION

In the figure the radius of the inner circle is 3 mm. and the radii of the circles drawn round it are 12, 16, 20, &c. If the figure thus drawn is spun round its centre in the right direction in its own See also:

• PLANE

Displacement See also:

• DIAGRAM

If this is done for every point we obtain a continuous curve APBQCRD, which represents the displacement at every point at the given instant, though by a length at right angles to the actual displacement and on an arbitrary See also:

• SCALE
• SCALE (1) A

See also:

• DISTRIBUTION (Lat. distribuere, to deal out)

Distribution of Velocity in a Wave.—If a wave travels on without alteration the travelling may be represented by pushing on the displacement curve. Let the wave AQBTC (fig. 5) travel to A'QB'TC' in a very short time. In that short time the displacement at H decreases from HP to HP' or by PP'. The motion of the particle is therefore backwards towards A. At J the displacement remains the same, or the particle is not moving. At K it increases by RR' forwards, or the motion is forwards towards B. At L the displacement backward decreases, or the motion is forward At M, as at J, there is no See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

4 and 5 in a general manner may easily be put into exact form. Let OX (fig. 6) be the direction R 0 M N of travel, and let x be the distance of any point M from a fixed point O. Let ON =x+dx. Let MP =y represent the forward displacement of the particle originally at M, and NQ=y+dy that of the particle originally at N. The layer of air originally of thickness dx now has thickness dx+dy, since N is displaced forwards dy more than M. The See also:

• VOLUME

But QR/MN = dy/dx. Then u/U = —dy/dx (2) This gives the velocity of any particle in terms of the displacement. Equating (I) and (2) u/U =w/E (3) which gives the particle velocity in terms of the pressure excess. Generally, if any See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

We may say here that the See also:

• ENERGY (from the Gr. ivipyela; iv, in, ipyov, work)

It is convenient to give this calculation before proceeding to describe the experimental determination of the velocity in air, in other gases and in water, since the calculation serves to some extent as a See also:

• GUIDE (in Mid. Eng. gyde, from the Fr. guide; the earlier French form was guie, English " guy," the d was due to the Italian form guida; the ultimate origin is probably Teutonic, the word being connected with the base seen in O. Eng. witan, to know)

For instance, if we have a wave with displacement curve of form See also:

• ABC

Now the See also:

• TRANSFER (from Lat. transferre, to bear across, carry over)

(4) If y is the displacement at A, and if E is the elasticity, substituting for a and u from (2) and (3) we get X—Edx+PU2 (I+dx) 2= poU2. But since the volume dx with density po has become volume dx+dy with density p d P (I +dx) =Pa' Then X—Edx+poU2 (I+dx) =poU2, or X (E—poU2)dy/dx. (5) If then we apply a pressure X given by (5) at every point, and move the medium with any uniform velocity U, the disturbance remains fixed in space. Or if we now keep the undisturbed parts of the medium fixed, the disturbance travels on with velocity U if we apply the pressure X at every point of the disturbance. If the velocity U is so chosen that E—poU2=o, then X=o, or the wave travels on through the See also:

• ACTION

He supposed that in air See also:

• BOYLE
• BOYLE, JOHN J
• BOYLE, ROBERT (1627-1691)

G. See also:

• STOKES, SIR G
• STOKES, SIR GEORGE GABRIEL, BART
• STOKES, WHITLEY (1830-1909)

In the momentum equation (4) we may now omit X and it becomes w+p(U—u)2=poU2. Let us seek a more exact value for w. If when P changes to P+w volume V changes to V —v then (P+w) (V v)?' = PVY, whence w=P (yV+y(y2 1) V2) = '- (I+y 2 I V )' We have U—u=U(i—a/U)=U(1—v/V), since u/U = —dy/dx = v/V. Also since p(V—v) =poV, or p=po/(I —v/V), then p(U—u)2= VpoU2(i —v/V). Substituting in the momentum equation, wee obtain yV (+y 21 V) +PoU2 (1 — v) =poU2, whence U2 = P-- P (1 +y 2 I U) . // If U='/(7P/po) is the velocity for small disturbances, we may put Uo for U in the small term on the right, and we have U=Uo(1+—F1 u) 4 Uo or U =Uo+'-h(y-1-i)u• This investigation is obviously not exact, for it assumes that the form is unchanged, i.e. that the momentum issuing from A (fig. 9) is equal to that entering at B, an assumption no longer tenable wtien the form changes. But for very small times the assumption may perhaps be made, and the result at least shows the way in which the velocity is affected by the addition of a small term depending on and changing sign with u. It implies that the different parts of a wave move on at different rates, so that its form must change. As we obtained the result on the supposition of unchanged form, we can of course only apply it for such short lengths and such short times that the part dealt with does not appreciably alter. We see at once that, where u=o, the velocity has its " normal " value, while where u is See also:

• POSITIVE (or PORTABLE) ORGAN

If the steepness gets very great our investigation ceases to apply, and neither experiment nor theory has yet shown what happens. Probably there is a breakdown of the wave somewhat Experiments, referred to later, have been made to find the amplitude of See also:

• SWING, DAVID (1830-1894)

Vautier (see below). W. W. Jacques (Phil. Mag., 1879, 7, p. 219) investigated the transmission of a report from a See also:

• CANNON (a word common to Romance languages, from the Lat. canna, a reed, tube, with the addition of the augmentative termination -on, -one)

The kinetic energy per cubic centimetre is ;put, where p is the density and a is the velocity of disturbance due to the passage of the wave. If V is the undisturbed volume of a small portion of the air at the undisturbed pressure P, and if it becomes V —v when the pressure increases to P+a, the See also:

• AVERAGE

(I2) The energy per cubic centimetre on the average is 2 p7r2U 2a2/X2 (13) and the energy passing per second through 1 sq. cm. perpendicular to the line of propagation is 2pa2U3a2/X2 (14) The Pressure of Sound Waves.—Sound waves, like light waves, exercise a small pressure against any surface upon which they impinge. The existence of this pressure has been demonstrated experimentally by W. Altberg (See also:

• ANN

The excess pressure on CD is therefore /1(W -1- pu2)dt. But the values of w+pu2 which occur successively during the second at AB exist simultaneously at the beginning of the second over the distance U behind AB. Or if the conditions along this distance U could be maintained constant, and we could travel back along it uniformly in one second, we should meet all the conditions actually arriving at AB and at the same intervals. If then dA is an See also:

• ELEMENT (Lat. elementum)

364). If the train of waves is reflected, the value of p at AB will be the sum of the values for the two trains, and will, on the average, be doubled. The pressure on CD will therefore be doubled. But the energy will also be doubled, so that (15) still gives the average excess of pressure- Experimental Determinations of the Velocity of Sound. An obvious method of determining the velocity of sound in air consists in starting some sound, say by firing a gun, and stationing an observer at some measured distance from the gun. The observer See also:

• MEASURES

He made no correction for temperature or humidity. But when the wind is steady its effect may be eliminated by reciprocal " observations, that is, by observations of the time of passage of sound in each direction over the measured distance. Let D be the distance, U the velocity of sound in still air, and w the velocity of the wind, supposed for simplicity to blow directly from one station to the other. Let Ti and To be the observed times of passage in the two directions. We have U+w=D/Ti and U — w = D/T2. Adding and dividing by l2 U=2 2 21+TZ/' If Ti and To are nearly equal, and if,T=2(T,+T2), this is very nearly. U= D/T. The reciprocal method was adopted in 1738 by,a See also:

• COMMISSION (from Lat. commissio, committere)

The theoretical investigation given above shows that if U is the velocity in air at t° C. then the velocity Uo at o° C. in the same air is independent of the barometric pressure and that Uo=U/(1-Fo•oo184t), whence Uo=332 met./sec. In 1822 a commission of the See also:

• BUREAU (a Fr. word from burel or bureau, a coarse cloth used for coverings)

It is different, too, for different senses with the same observer, and different even for the same sense when the external stimuli differ in intensity. When the interval between a flash and a report is measured, the personal equations for the two arrivals are, in all See also:

• PROBABILITY (Lat. probabilis, probable or credible)

On the same band a tuning-fork electrically maintained and a seconds clock actuating another style wrote parallel records. The circuit was continued to the gun which served as a source, and stretched across its muzzle. When the gun was fired, the circuit was broken, and the break was recorded on the paper. The circuit was at once remade. When the wave travelled to the receiver it pushed back the disk from the contact-piece, and this break, too, was recorded. The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuning-fork record. The receiving apparatus had what we may term a personal equation, for the break of contact could only take See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

128o metres and 2445 metres, obtaining from the first U°=331.37 met./sec.; but the number of experiments over the longer distance was greater, and he appears to have put more confidence in the result from them, viz. Uo=33x.71 met./sec. In the Phil. Trans., 1872, 162, p. 1, is given an interesting determination made by E. J. See also:

• STONE
• STONE (0. Eng. shin; the word is common to Teutonic languages, cf. Ger. Stein, Du. steen, Dan. and Swed. sten; the root is also seen in Gr. aria, pebble)
• STONE, CHARLES POMEROY (1824-1887)
• STONE, EDWARD JAMES (1831-1897)
• STONE, FRANK (1800-1859)
• STONE, GEORGE (1708—1764)
• STONE, LUCY [BLACKWELL] (1818-1893)
• STONE, MARCUS (184o— )
• STONE, NICHOLAS (1586-1647)

This limit for a diameter 1.1 m. was Uo=33o•6 met./sec., while for a diameter o•ro8 it was U0= 324'25 met./sec. Regnault also set up a shorter length of pipes of diameter o•ro8 m. in a See also:

• COURT
• COURT (from the O. Fr. court, Late Lat. cortis, curtis, a popular form of class. Lat. cohors, gen. cohortis; the mod. Fr. form tour is due to the influence of the Lat. curia, the word used in medieval documents to translate " court " in the feudal sense)
• COURT, ANTOINE (1696-176o)

with a probable See also:

• ERROR (Lat. error, from errare, to wander, to err)

F. See also:

• STURM
• STURM, JULIUS (1816-1896)

When a wave of sound meets a surface separating two See also:

• MEDIA

He states that in the whispering See also:

• GALLERY (through Ital. galleria, from Med. Lat. galeria, of which the origin is unknown)

Rayleigh points out that this clinging of the sound to the surface of a concave wall does not depend on the exactness of the spherical form. He suggests that the propagation of See also:

• EARTHQUAKE

As with light the ratio involved in the second law is always equal to the ratio of the velocity of the wave in the first medium to the velocity in the second; in other words, the sines of the angles in question are directly proportional to the velocities. Hence sound rays, in passing from one medium into another, are See also:

• BENT
• BENT (E1. BENI)
• BENT, JAMES THEODORE (1852–1897)

Sondhauss (Pogg. Ann., 1852, 85. p. 378), who used a See also:

• COLLODION (from the Gr. &M a' , glue)

the upper part, moving faster, gains on the See also:

• LOWER

It is, of course, a matter of-common observation that the wind increases in velocity from the surface upwards. An excellent illustration of this increase was pointed out by F. See also:

• OSIER (through Fr. from Late Lat. osaria, auxaria, a bundle of osier or willow twigs)

14, where the wind is travelling from right to left. In the first case the waves are more likely to reach and be perceived by an observer level with the source, while in the second case they may go over his See also:

• HEAD (in 0. Eng. heafad; the word is common to Teutonic languages; cf. Dutch hoofd, Ger. Haupt, generally taken to be in origin connected with Lat. caput, Gr. KerbOvi7)
• HEAD, SIR EDMUND WALKER, BART
• HEAD, SIR FRANCIS BOND, BART

At the instant that the original wave reaches F the wave from E has travelled to a circle of radius very nearly equal to EF—not quite, as S is not quite in the plane of the rails. The wave from D has travelled to a circle of radius nearly equal to DF, that from C to a circle of radius nearly CF, and so on. As these " secondary waves " return to S their distance apart is nearly equal to twice the distance between the rails, and the observer then hears a note of wave-length nearly 2EF. But if an observer is stationed at S' the waves will be about half as far apart and will reach him with nearly twice the frequency, so that he hears a note about an octave higher. As he travels further round the frequency increases still more. The railings in fact do for sound what a diffraction grating does for light. Frequency and Pitch. Sounds may be divided into noises and musical notes. A mere See also:

• NOISE (a word of doubtful origin; O. Fr. nogse or nose; Prov. nausa, which points to Lat. nausea, sickness, as the origin; others take Lat. noxia, harm, as the source)

Various See also:

• INSTRUMENTS

Neither this instrument nor the next to be described is now used for exact work; they merely serve as illustrations of the law of pitch. The See also:

• SIREN

Hence the note produced with any given circle of holes rises in pitch as the disk revolves more rapidly; and if, the revolution of the disk being kept as steady as possible, the tube be passed rapidly across the circles of the first series, a series of notes is heard, which, if the lowest be denoted by C, form the sequence C, Ci, El, G,, C2, &c. In like manner, the first circle in which we have two sets of holes dividing the circumference, the one into say 8 parts, and the other into to, or in ratio 4: 5, the note produced is a compound one, such as would be obtained by striking on the piano two notes separated by the interval of a major third (t). Similar results are obtainable by means of the remaining perforations. A still simpler form of siren may be constituted with a good See also:

• SPINNING (from O. Eng. spinnen, to spin, cf. Ger. spinnen, &c., the Teut. root is seen, to draw out, cf. span, spider)

After 1/nth of a revolution, the two sets of perforations will again coincide, the lateral impulse of of rotation increased. This will go on continually as long as air is supplied to the cylinder, and the velocity of rotation of the upper plate will be accelerated up to a certain maximum, at which it may be maintained by keeping the force of the current constant. Now, it is evident that each coincidence of the perforations in the two plates is followed by a non-coincidence, during which the air-current is shut off, and that consequently, during each revolution of the upper plate, there occur n alternate passages and interceptions of the current. Hence arises the same number of successive impulses of the external air immediately in contact with the movable plate, which is thus thrown into a state of vibration at the rate of n for every revolution of the plate. .The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value. If, then, we can determine the number m of revolutions performed by the plate in every second, we shall at once have the number of vibrations per second corresponding to the audible note by multiplying m by n. For this purpose the axis is furnished at its upper part with a screw working into a toothed wheel, and See also:

• DRIVING (from " to drive," i.e. generally to propel, force along or in, a word common in various forms to the Teutonic languages)

See also:

• DOVE
• DOVE (Dutch duyve, Dan. due, Ice. dufa, Ger. Taube)

The motion of the plate may be simply produced by dropping it between two vertical grooves, the tuning-forks being properly fixed to a frame above. Greater accuracy may be attained with a revolving-See also:

• DRUM (early forms drome or dromme, a word common to many Teut. languages, cf. Dan. tromme, Ger. Trommel: the word is ultimately the same as " trumpet," and is probably onomatopoeic in origin; it appears late in Eng. about the middle of the 16th century)

See also:

• SCOTT, ALEXANDER (fl. 1550)
• SCOTT, DAVID (18o6-1849)
• SCOTT, MICHAEL (1789-1835)
• SCOTT, ROBERT (1811-1887)
• SCOTT, SIR GEORGE GILBERT (1811-1878)
• SCOTT, SIR WALTER, BART
• SCOTT, WILLIAM BELL (1811-1890)
• SCOTT, WINFIELD (1786-1866)

Instead of a mirror, the See also:

• OBJECTIVE, or OBJECT GLASS

If the rod is circular in section and perfectly uniform the end will describe a circle, ellipse or straight line; but, as the elasticity is usually not exactly the same in all directions, the figure usually changes and revolves. Various modifications of the kaleidophone have been made (Rayleigh, Sound, § 38). Koenig devised a clock in which a fork of frequency 64 takes the place of the pendulum (Wied. Ann., 188o, ix. 394). The motion of the fork is maintained by the clock acting through Koenlg's an escapement, and the dial registers both the number ,,Koenig'! of vibrations of the fork and the seconds, minutes and nig'! Clock. See also:

• HOURS, CANONICAL

(From Lord Rayleigh's Theory of Sound, by permission of See also:

• MACMILLAN

I), a cylinder is ruled with equi- M'Leodand distant See also:

• WHITE
• WHITE, ANDREW DICKSON (1832– )
• WHITE, GILBERT (1720–1793)
• WHITE, HENRY KIRKE (1785-1806)
• WHITE, HUGH LAWSON (1773-1840)
• WHITE, JOSEPH BLANCO (1775-1841)
• WHITE, RICHARD GRANT (1822-1885)
• WHITE, ROBERT (1645-1704)
• WHITE, SIR GEORGE STUART (1835– )
• WHITE, SIR THOMAS (1492-1567)
• WHITE, SIR WILLIAM ARTHUR (1824--1891)
• WHITE, SIR WILLIAM HENRY (1845– )
• WHITE, THOMAS (1628-1698)
• WHITE, THOMAS (c. 1550-1624)

The phonic wheel, invented independently by Paul La Cour and Lord Rayleigh (see Sound, i. § 68 c), consists of a wheel carrying Rayletgh's several soft-iron armatures fixed at equal distances round its circumference. The wheel rotates between Phonic Wheel. the poles of an electro-magnet, which is fed by an intermittent current such as that which is working an electrically maintained tuning-fork (see infra). If the wheel be driven at such rate that the armatures move one place on in about the See also:

• PERIOD
• PERIOD (Gr. irepioSos, a going or way round, circuit, aepi, round, and &Sis, way, road)

But if the fork has, say, rather greater frequency, the hole in the wheel comes round at the end of the two seconds before the bead has quite come into position, and the two flashes appear gradually to move back in the opposite way to the pendulum. Suppose that in N beats of the clock the flashes have moved exactly one place back. Then the first flash in the new position is viewed by the 8Nth passage of the opening, and the second flash in the original position of the first is viewed when the pendulum has made exactly N beats and by the (8 N + I)th passage of the hole. Then the wheel makes 8 N + 1 revolutions in N clock beats, and the fork makes 32 N + 4 vibrations in the same time. If the clock is going exactly right, this gives a frequency for the fork of 32 + 4/N. If the fork has rather less frequency than 32 then the flashes appear to move forward and the frequency will be 32 -4/N. In Rayleigh's experiment the 32 fork was made to drive electrically one of frequency about 128, and somewhat as with the phonic wheel, the frequency was controlled so as to be exactly four times that of the 32 fork. A standard 128 fork could then be compared either optically or by beats with the electrically driven fork. Scheibler's Tonometer.—When two tones are sounded together with frequencies not very different, " beats " or swellings-out of the sound are 'heard of frequency equal to the difference of frequencies of the two tones (see below). Johann Heinrich Scheibler (1777—1838) tuned two forks to an exact octave, and then prepared a number of others dividing the octave into such small steps that the beats between each and the next could be counted easily. Let the forks be numbered o, 1, 2, . N.

If the frequency of o is .n, that of N is 211. Suppose that No. I makes m1 beats with No. o, that No. 2 makes m2 beats with No. 1, and so on, then the frequencies are n, n+mi, n+mi+m2, ..., n+--mi+m2+ . . . + MN. Since n+ml-hm2+ . . . + mN--2f, n=m1+m2+ . . . +mN, and it follows that when n is known, the frequency of every fork in the range may be determined.

Any other fork within this octave can then have its frequency determined by finding the two between which it lies. Suppose, for instance, it makes 3 beats with No. to, it might have frequency either 3 above or below that of No. lo. But if it lies above No. to it will beat less often with No. 11 than with No. 9; if below No. to less often with No. 9 than with No. II. Suppose it lies between No. to and No. 11 its frequency is that of No. 10+3. Manometric Flames.—This is a device due to Koenig (Phil. Mag., 1873, 45) and represented diagrammatically in fig.

19. f is a flame from a pinhole burner, fed through a cavity C, one side of which is closed by a membrane m; on the other side of the membrane is another cavity C', which is put into connexion with a source of sound, as, for instance, a Helmholtz resonator excited by a fork of the same frequency. The membrane vibrates, and alternately checks and increases the gas supply, and the flame jumps up and down with the frequency of the source. It then appears elongated. To show its intermittent See also:

• CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

All experiments in frequency show that two notes, forming a definite musical interval, have their frequencies always in the same ratio wherever in the musical scale the two notes are situated. In the scale of C ,the intervals from the key-note, the frequency ratios with the key-note, the successive frequency ratios and the successive intervals are as follows: If we pass through two intervals in succession, as, for instance, if we ascend through a See also:

• FOURTH

i[ See also:

• RAS

The construction would be complicated and the playing exceedingly difficult. The same string or pipe and the same key have therefore to serve for what should be slightly different notes. A See also:

• COMPROMISE (pronounced compromize; through Fr. from Lat. compromittere)

Here it is sufficient to say that the French standard is a' =435 with c" practically 522, and that in See also:

• ENGLAND
• ENGLAND, THE CHURCH OF

Koenig constructed a series of bars forming a harmonicon, the frequency of each See also:

• BAR
• BAR (0. Fr. barre, Late Lat. barra, origin unknown)
• BAR, CONFEDERATION OF
• BAR, FRANCOIS DE (1538-1606)
• BAR, THE

It draws off from any wave it has sent backward and so spreads the succeeding waves over a longer distance than if it had remained at rest. Hence the forward waves are shorter and the backward waves are longer. Since U =n X where U is the velocity of sound, a the wave-length, and n the frequency, it follows that the forward frequency is greater than the backward frequency. The more general case of motion of source, medium and receiver Note C D E F G A B C Interval with C second major fourth fifth major seventh octave third See also:

• SIXTH

20. If all were still, the n waves emitted by S in one second would spread over a length U. But through the wind velocity the first wave is carried to a distance U + w from S, while through the motion of the source the last wave is a distance u from S. Then the n waves occupy a space U + w — u. Now turning to the receiver, let us consider what length is occupied by the waves which pass him in one second. If he were at rest, it would be the waves in length U + w, for the wave passing him at the beginning of a second would be so far distant at the end of the second. But through his motion v in the second, he receives only the waves in distance U + w— v. Since there are n waves in distance U + w — u the number he actually receives is n(U'+ w — v)/(U + w — u). If the velocities of source and receiver are equal then the frequency is not affected by their motion or by the wind. But if their velocities are different, the frequency of the waves received is affected both by these velocities and by that of the wind. The change in pitch through motion of the source may be illustrated by putting a pitch-pipe in one end of a few feet of rubber tubing and blowing through the other end while the tubing is whirled round the head. An observer in the plane of the motion can easily hear a change in the pitch as the pitch-pipe moves to and from him.

Musical Quality or Timbre.—Though a musical note has definite pitch or frequency, notes of the same pitch emitted by different instruments have quite different quality or timbre. The three characteristics of a longitudinal periodic disturbance are its amplitude, the length after which it repeats itself, and its form, which may be represented by the shape of the displacement curve. Now the amplitude evidently corresponds to the loudness, and the length of period corresponds to the pitch or frequency. Hence we must put down the quality or timbre as depending on the form. The simplest form of wave, so far as our sensation goes—that is, the one giving rise to a pure tone—is, we have every See also:

• REASON (Lat. ratio, through French raison)

AC is the shortest distance after which the curve begins to repeat itself ; this length X is termed the wave-length. The maximum height of the curve HM =a is the amplitude. If w.e transfer 0 to A, e=o, and the curve may be represented by y=a sin V x. If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e=OA at any time t will be Ut, so that the value of y may be represented as y=a sin (x—U1). (16) The velocity perpendicular to the axis of any point on the curve at a fixed distance x from 0 is dy = --3---cos—(x—Ut). dt — X X The See also:

• ACCELERATION (from Lat. accelerare, to hasten, celer, quick)

The See also:

• CHIEF (from Fr. chef, head, Lat. caput)

(21) where the terms may be See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

23) and is then let go. Let H S' :L> H L (17) (18) Then, as we shall prove later, the vibrations of the string may be FIG. 23. us imagine it to form half a wave-length of the extended train ZGAHBKC, on an indefinitely extended stretched string, the values of y at equal distances from A (or from B) being equal and opposite. represented by the travelling of two trains in opposite directions each with velocity tension =mass per unit length each half the height of the train represented in fig. 23. For the superposition of these trains will give a stationary wave between A II and B. Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths a, IX, IX, &c., where a =2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve. Since the velocity is the same for all disturbances they all travel at the same speed, and the two trains will always remain of the same form. If then we resolve AHBKC into harmonics by Fourier's theorem, we may follow the motion of the separate harmonics, and their superposition will give the form of the string at any instant. Further, the same harmonics with the same amplitude will always be present. We see, then, that the conditions for the application of Fourier's theorem are See also:

• EQUIVALENT

In many vibrating systems this does not hold, and then Fourier's theorem is no longer an appropriate resolution. But where it is appropriate, the disturbance sent out into the air contains the same harmonic series as the source. The question now arises whether the sensation produced by a periodic disturbance can be analysed in See also:

• CORRESPONDENCE
• CORRESPONDENCE (from med. scholastic Lat. correspondentia, corrcspondere, compounded of Lat. cum, with, and respond ere, to answer; cf. Fr. correspondance)

Vibrations thus excited are termed forced vibrations, and their amplitude is greater the more nearly the period of the applied force approaches that of the system when vibrating freely. The mathematical investigation of forced vibrations (Rayleigh,Sound, i. § 46) shows that, if there were nodissipation of energy, the vibration would increase indefinitely when the periods coincided. But there is always leakage of energy either through See also:

• FRICTION (from Lat. fricare, to rub)

If p=n the amplitude becomes infinite. This is the case of " resonance." The amplitude does not, of course, become infinite in practice. There is always loss of energy by dissi- pation in the vibrating machinery and by radiation into the medium, and the amplitude only increases until this loss is balanced by the gain from the work done by the applied force. According to Helmholtz, the ear probably contains within it a series of resonators, with small intervals between the periods of the successive members, while the series extends over the The Ear as whole range of audible pitch. We need not here enter a Fourier into the question of the structure constituting these Analyser. resonators. Each of them is supposed to have its own natural frequency, and to be set into vibration when the ear receives a train of waves of that frequency. The vibration in some way arouses the sensation of the corresponding tone. But the same resonator will be appreciably though less affected by waves of frequency differing slightly from its own. Thus Helmholtz from certain observations (Sound, ii. § 388) thought that if the intensity of response by a given resonator in the ear to its own tone is taken as 1, then its response to an equally loud tone a semitone different may be taken as about i o. According to this theory, then, when a pure tone is received the auditory apparatus corresponding to that tone is most excited, but the apparatus on each side of it is also excited, though by a rapidly diminishing amount, as the interval increases. If the sensations corresponding io these neighbouring elements are thus aroused, we have no such perception as a pure tone, and what we regard as a pure tone is the mean of a group of sensations.

The sensitiveness of the ear in judging of a given tone must then correspond to the accuracy with which it can See also:

• JUDGE (Lat. judex, Fr. juge)

The amount radiated out in the form of sound waves is deduced, and hence the energy of the stream at any distance is known. In the other, the waves produce a measurable effect on a vibrating system of the same frequency, and the amplitude in the waves can be deduced. The first may be illustrated by Lord Rayleigh's experiments to determine the amplitude of vibration in waves only just audible (Sound, ii. § 384). He used two kinds of experiment, Mlnlmum but it will be sufficient here to indicate the second. A Am fork of frequency 256 was used as the source. The energy Amplitude ie of this fork with a given amplitude of vibration could Sounds. be calculated from its dimensions and elasticity, and the amplitude was observed by measuring with a microscope the line into which the See also:

• IMAGE (Lat. imago, perhaps from the same root as imitari, copy, imitate)

Assuming this energy to be propagated in hemispherical waves, it is easy to find the quantity per second going through I sq. cm. at the distance of the listener, and thence from the energy in a wave, found above, to determine the amplitude. The result was an amplitude of 1.27 X Io cm. Other forks gave results not very different. In a later series of experiments Lord Rayleigh (Phil. Mag., 1907, 14, p. 596) found that the least energy stream required to excite sensation did not vary greatly between frequencies of 512 and 256, (22) but that the stream required increased rapidly as the frequency was reduced below 256. The second method may be illustrated by the experiments of M. Wien (Wied. Ann., 1889, See also:

• XXXVI

When the plate vibrated the mirror was vibrated about the fixed edge, and the image of a reflected slit was broadened out into a band, the broadening giving the amplitude of vibration of the plate. From subsidiary experiments (for which the original memoir must be consulted) the pressure See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

131) compared the indications of Wien's resonator See also:

• MANOMETER (Gr. µavos, thin or loose; µerpov, a measure)

21. If we measure t from an instant at which the two trains exactly coincide, then as U for the other train has the opposite sign, its displacement is represented by ya=a sin (x+Ut). (25) The sum of the disturbance is obtained by adding (24) and (25) 'r y=y1+y2=2a cos 2Ut sin 2 ;rx, (26) At any given instant t this is a sine curve of amplitude 2a cos (27r/A) Ut, and of wave-length A, and with nodes at x=o, ,A, A, ... , that is, there is no displacement at these nodes whatever the value of t, and between them the displacement is always a sine curve, but of amplitude varying between +2a and -2a. The ordinate of the curve changes sign as we pass through a node, so that successive sections are moving always in opposite directions and have opposite displacements. Each section then vibrates, and its amplitude goes through all its values in time given by 27rUT/A =27r, or T=A/U, and the frequency is U/A. We may represent such a train of " stationary waves " by fig. 25, where the curves give the two extreme amplitudes. The points A, 13, C, D are termed " nodes," and the points half-way between them " loops." The general character of these results may be obtained by a graphic construction. Let fig. 26 (I) represent a wave-length of each train when they are coincident. It is sufficient to take a single wave-length.

The dotted curve represents the superposition, which simply doubles each ordinate. Divide the wave-length into,say, eight equal parts as marked. Then move one train marked (I) IA to the right, and the other train (II) iX to the left, introducing new parts of each train at one end, and segding out old parts at the other. Then we get fig. 26 (2), the dotted curve representing the resultant with amplitude IN2 that of (1). Another See also:

• MOVEMENT

Elementary Theory of Pipes.—The longitudinal vibration of air in cylindrical pipes is made use of in various wind instruments. We shall deduce the modes of vibration of the air column in a cylindrical pipe from the consideration that the air in motion within the pipe forms some part of a system of stationary waves, one train being formed by the exciter of the disturbance, and the other being formed by the reflection of the train at the end of the pipe. In order to justify the use of stationary waves we must show that two such trains can move in opposite directions over the same ground without modifying each other so long as the displacement in either is small. For this it is necessary that the total force on an element due to the sum of the displacements should be equal to the sum of the forces due to the two displacements considered separately. The medium then acts for the second train just as if it were undisturbed by the first. It is sufficient then to show that the excess of pressure at any point is the sum of the excesses due to either train separately. If w is the total pressure excess, and if y is the total displacement at x, then =E Xchange of volume =original volume = —Edy/dx. If y, and y2 are the two separate displacements and if y=y,+ya, then w= — E (dyi/dx + dya/dx) =See also:

• COL
• COL (Fr. for " neck," Lat. collum)

At the nodes A, B, C, D, E there is no displacement, but there are maximum volume and pressure changes. Consider, for instance, the point B. When the displacement is represented by AHBKC the particles on each side of B are displaced towards it The reader will be able to make out the simultaneous motions and pressures at various points. It is obvious that the nodes are alternately in compression and extension, or vice versa, and that for IX on each side of a node the motion is either to it on both sides or from it on both sides. The first mode of vibration gives the " fundamental tone," and the succeeding modes are termed " overtones." The whole series forms the series of See also:

• ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

The flame there is much Using this result Rayleigh found the correction for an unflanged open end by sounding two pipes nearly in unison, each provided with a flange, and counting the beats. Then the flange was removed from one and the beats were again counted. The change in virtual length by removal of the flange was thus found, and the open end correction for the unflanged pipe was o.6 R. This correction has also been found by See also:

• DAVID
• DAVID (a Hebrew name meaning probably beloved 1)
• DAVID, FELICIEN (1810—1876)
• DAVID, GERARD [GHEERAERT DAVIT], (?=1523)
• DAVID, PIERRE JEAN (1789–1856)
• DAVID, ST (Dewi, Sant)

27 between two loops. The fundamental mode is that in, which H and K represent the ends of the pipe. In this case HK=2Xi=1, and the frequency is ni=UN =U/21. There is a node in the middle. In the next mode H and L represent the ends and HL=See also:

• A2(z)

If the frequency is 256 the corrected length for the fundamental is about (33,000/2)(256) (I +.001841) at O. The pipe will also resound to forks of frequencies 512, 768, 1024 and so on. An open " flue " organ pipe is of this type. The wind rushing through the slit S (fig. 30) maintains the vibration in a way to be discussed later, and the opening 0 makes the lower end a loop. The modes of vibration in an open organ pipe may be exhibited by means of Koenig's manometric flames (Phil. Mag., 1873, vol. 45). The pipe is provided with manometric flames at its middle point, and at one-See also:

• QUARTER (through Fr. from Lat. quartarius, fourth part)

The tangent to the displacement curve is always parallel to the axis, that is, for a small distance the successive particles are always equally displaced, and therefore always occupy the same volume. This means that at the loops while the motion is greatest there are no pressure changes. We have now to select such portion of this system as will suit the conditions imposed by any actual pipe. There are three distinct types, which we will consider in succession. I. Pipe Closed at One End, Open at the Other.—At the closed end there is no motion, for the pressure always constrains the air to remain in contact with the end. The closed end is therefore a node. At the open end, as a first approximation to be corrected later, there are no pressure changes, for any tendency to excess can be relieved by immediate expansion into the outer air, and any tendency to defect can be filled up by an inrush from the outer air. The open end is therefore a loop. It is to be noted that the exciter of the vibrations is in general at the open end, and that the two trains forming the stationary system consist of the direct waves from the exciter travelling into the tube, and the waves reflected back from the closed end. In fig. 27 we may have the length AH occupying the tube.

In is case AH = IXI =l, the length of the tube, and the frequency Li = U/Xi = U/4l. But we may also have a shorter wave-length X2 such that the length AK occupies the tube. In this case AK= IX2=1, and the frequency n2=U/X2=3U/41. With a still shorter wave-length )u we may have the length AL occupying the tube and AL =1X3=1, and the frequency ni = U/Xi = 5U/41, and so on, as we take succeeding loops for the open end. In fig. 28 are represented the stationary wave systems of the first four modes, and any of the succeeding ones are easily drawn.[PIPES condition for a loop at the open end, that of no pressure variation, cannot be exactly fulfilled. This would require that the air outside should have no mass in order that it should at once move out and relieve the air at the end of the pipe from any excess of pressure, or at once move in and fill up any defect. There are variations, therefore, at the open end, and these are such that the loop may be regarded as situated a short distance outside the end of the pipe. It may be noted that in practice there is another reason for pressure variation at the end of the pipe. The stationary wave method regards the vibration in the pipe as due to a series of waves travelling to the end and being there reflected back down the pipe. But the reflection is not complete, for some of the energy comes out as waves; hence the direct and reflected trains are quite equal, and cannot neutralize each other at the loop. The position of the loop has not yet been calculated for an ordinary open pipe, but Lord Rayleigh has shown (Sound, ii.

§ 307) that for a cylindrical tube of radius R, provided with a flat extended flange, the loop may be regarded as about o•82 R, in advance of the end. That is, the length of the pipe must be increased by 0.82 R before applying Bernoulli's theory. This is termed the " end correction." F affected by the nodal pressure changes, while the other two vibrate only slightly. If, however, the pipe is blown strongly, the fundamental See also:

• DIES, CHRISTOPH ALBERT (1755-1822)

31, a loosely- S fitting cork or card piston being fixed on one end of the sounder, which is inserted within the dust-tube. The other end of the sounder is stroked outwards with a See also:

• DAMP

If yi is known this gives 72. Kundt and Warburg applied the method to find y for mercury vapour (Pogg. Ann., 1876, 157, p. 356), using a double form of the apparatus in which there are two dust- tubes worked by the same sounding rod. This rod is supported at a and ; of its length where it enters the two dust-tubes, as represented diagrammatically in fig. 32. It is stroked in the middle so as to excite its second mode of vibration. The method ensures that the two frequencies shall be exactly the same. In the mercury experiment the sounding rod was sealed into the dust-tube, which was exhausted of air, and contained only come mercury and some See also:

• QUARTZ

From the temperature, P2/p2 was known, and y2/y1 could then be found. Taking 71=1.41, 72 was determined to be 1.66. Lord Rayleigh and Sir William See also:

• RAMSAY, ALLAN (1686-1758)
• RAMSAY, ALLAN (1713-1784)
• RAMSAY, ANDREW MICHAEL (1686-1743)
• RAMSAY, DAVID (1749—1815)
• RAMSAY, SIR WILLIAM (1852– )
• RAMSAY, SIR WILLIAM MITCHELL (185r– )

163), A. J. I. Topler andAmplttudeot L. Boltzmann (Pogg. Ann.,. vol. 141, or Rayleigh, vibration. Sound, ii. § 422a), and E. Mach (Optisch-akustischen Versuche, 1873). Mach's method is perhaps the most direct. The pipe was fixed in a horizontal position, and along the top wall ran a See also:

• PLATINUM [symbol Pt, atomic weight 145.0 (0=16)]

When the wire was heated by an electric current a fine line of vapour descended from each drop. The pipe was closed at the centre by a membrane which prevented a through See also:

• DRAUGHT (from the common Teutonic word " to draw "; cf. Ger. 7'racht, load; the pronunciation led to the variant form " draft," now confined to certain specific meanings)

C is a constant, equal to the coefficient of viscosity in Helmholtz's theory, but less simple in Kirchhoff's theory. Experiments on the velocity in pipes were carried out by H. Schneebeli (Pogg. Ann., 1869, 136, p. 296) and by T. J. Seebeck (Pogg. Ann., 187o, 139, p. 104) which accorded with this result as far as R is concerned, but the diminution of velocity was found to be more nearly proportional to Nom. Kundt also obtained results in general agreement with the formula (Rayleigh, Sound, ii. § 26o). He used his dust-tube method.

Elementary Theory of the Transverse Vibration of Musical Strings. We shall first investigate the velocity with which a disturbance travels along a string of mass m per unit length when it is stretched with a constant tension T, the same at all points. We shall then show that on certain limitations two trains of disturbance may be superposed so that stationary waves may be formed, and thence we shall deduce the modes of vibration as with pipes. Let AB (fig. 33) represent the string with the ends AB fixed. Let a disturbance once set going travel along unchanged in form from A to B with velocity U. Then move AB from right to left with this velocity, and the disturbance remains fixed in space. Take a point P in the disturbed part, and a point Q which the disturbance has not yet reached. Since the conditions in the region PQ remain always the same, the momentum perpendicular to AB entering the region at Q is equal to the momentum perpendicular to AB leaving the region at P. But, since the motion at Q is along AB, there is no momentum there perpendicular to AB. So also there is on the whole none in that direction leaving at P. Let the tangent at P make angle ¢ with AB.

The velocity of the string at P parallel to PM is U sin, , and the mass of string passing P is mU per second, so that mU2 sin . is carried out per second. But the tension at P is T, parallel to the tangent, and T sin fp parallel to PM, and through this —T sin . is the momentum passing out at P per second. Since the resultant is zero, mU2 sin <p—T sin tp =o, or U2 =Tim. Now keep AB fixed, and the disturbance travels with velocity U. We might make this investigation more general by introducing a force X as in the investigation for air, but it hardly appears necessary. To form stationary waves two equal trains must be able to travel in opposite directions with equal velocities, and to be superposed. We must show then that the force called out by the sum of the disturbances is equal to the sum of the forces called out by each train separately. In order that the velocity shall remain unchanged the tension T must remain the same. This implies that the disturbance is so small that the length is not appreciably altered. The component of T A Q --- (2/2l)J(T/m). In the third mode acting parallel to the axis or straight string is Tdx/ds, and when the disturbance is sufficiently small the curve of displacement is so nearly parallel to the axis that dx/ds1, and this component is T. The component of T perpendicular to the axis is Tdy/ds=Tdy/dx.

Now if yi and y2 are the displacements due to the two trains separately, and y=yi+y2, the two separate forces are Tdy,/dx and Tdy2/dx, while that due to y is Tdy/dx. But since y=yi+y2, Tdy/dx=Tdy1/dx+Tdy2/dx, or the condition for superposition holds when the displacement is so small that we may put dx/ds=l. Evidently this comes to neglecting 02. Let two trains of equal waves moving in opposite directions along such a string of indefi- nite length form the stationary system of fig. 27. Since the nodes are always at rest we may represent the vibration of a given string by the length between any two nodes. The fundamental mode is that in which A and B represent the ends pf the string. In this case AB = ZXi =l the length, and the frequency ni = U/X1 = U/21 = (1/21)11(T/m). The middle of the string is a loop. In the next mode A and C represent the ends and AC = X2 =1 and 112 = U/X2 = 2U/21= A and D represent the ends and AD=IX3=l and n3=U/X3=3U/21= (3/A11 (T /m) and so on. In fig. 34 the stationary wave systems of the first four modes are represented.

The complete series of harmonics The experimental demonstration of these results is easily made by the sonometer or See also:

• MONOCHORD (Gr. µovb op6ov, rcavwv µovvucos)

When the harmonic is sounded the riders at the loops are thrown off, while those at the nodes remain seated. Not only may the fundamental and its harmonics be obtained seoarately, but they are also to be heard simultaneously, particularly the earlier ones, which are usually more prominent than those higher in the series. A practised ear easily discerns the coexistence of these various tones when a See also:

• PIANOFORTE (Ital. piano, soft, and forte, loud)

If the length of the thread be kept invariable, a certain tension will give but one ventral segment; the fundamental note of the thread is then of the same pitch as the note of the body to which it is attached. By reducing the tension to one quarter of its previous amount, the number of ventral segments will be seen to be increased to two, indicating that the first harmonic of the thread is now in unison with the solid, and consequently that its fundamental is an octave lower than it was with the former tension; thus confirming the law that n varies as 'T. In like manner, on further lowering the tension to one ninth, three ventral segments will be formed, and so on. The law that, caeteris paribus, n varies inversely as the thickness may be tested by forming a string of four lengths of the single thread used before, and consequently of double the thickness of the latter, when, for the same length and tension, the compound thread will exhibit double the number of ventral segments presented by the single thread. The other laws admit cf similar illustration. Longitudinal Vibrations of Wires and Rods. Subject to a See also:

• LIMITATION, STATUTES OF

Then the disturbance remains fixed in space. Let A be a point in A dx the disturbance, and B a point in the undisturbed part. The material between A and B, though continually changing, is always in the same condition, and therefore the momentum within AB is constant. Hence the amount carried out at A is equal to that carried in at B. Now momentum is transferred in two ways, viz. by the force acting between contiguous portions of a body and by the transfer of moving matter. At B there is only the latter kind, and since the transfer of matter is po 'oU, where po is the undisturbed density and &o is the undisturbed cross-section, since its velocity is U the passage of momentum per second is po&oUo2. At A, if the velocity of the disturbance relative to undisturbed parts of the rod is u from left to right, the velocity relative to A is U—u. If p is the density at A, and Co the cross-section, then the momentum carried past A is p&(U —u)2. But if y is the displacement at A, dy/dx is the extension at A, and the force acting is a pull across A equal to Y&ody/dx, where Y is Young's modulus of elasticity. Then we have Y&ody/dx +p& (U — u)2 = po&oU2. (27) Now u/U = —dy/dx, (28) for the particle at A moves over dy backwards, while the disturbance moyes over U. Also since dx has been stretched to dx+dy p& (dx+dy) = po&odx or p&(I +dy/dx) = po&o• Substituting from (28) in (27) (( Tx) dy dy Y&°dx + pWU2 (i + Tx) 2 = po&0U2, and substituting from (29) in (30) dy) Y&odx + po&oU2 (1 dx/ whence Y&o = po&oU2, or U2 =Y/p, (32) where now p is the normal density of the rod.

The velocity with which the rod must travel in order that the disturbance may be fixed in space is therefore U = al (Y/p), or, if the rod is kept fixed, this is the velocity with which the disturbance travels. This investigation is subject to the limitation that the diameter of the cross-section must be small compared with the wave-length. When the rod extends or contracts longitudinally it contracts or - U B = po&oU2, (29) (30) (31) extends radially and in the ratio a, known as See also:

• POISSON, SIMEON DENIS (1781-1840)

For an iron wire Y/p is about 1012/4, so that for a frequency of 500 in a wire fixed at both ends a length about 5 metres is required. If the wire is stretched across a room and stroked in the middle with a damp cloth the fundamental is easily obtained, and the first harmonic can be brought out by stroking it at a quarter the length from one end. A glass or brass rod free at both ends may be held by the hand in the middle and excited by stroking one end outwards with a damp cloth. If it is clamped at one-quarter and three-quarters of the length from the ends, and is stroked in the middle, the first harmonic sounds. Young's modulus may be obtained for the material of a rod by clamping it in the middle and obtaining the frequency of the fundamental when Y=412n2p. The value thus obtained is generally appreciably greater than that obtained by a statical method in which the rod is pulled out by an applied tension. Rods of different materials may be used as sounders in a Kundt's dust tube, and their Young's moduli may be compared, since: length of rod velocity in rod = velocity in air X distance between dust-heaps. Torsional Vibrations of Rods and Wires.—The velocity of propagation of a torsional disturbance along a wire of circular section may be found by the transfer of momentum method, remembering that we must now replace linear momentum by angular momentum. Let the disturbance be supposed to travel unchanged in form from left to right with velocity U. Now suppose that the wire or rod is moved from right to left with velocity U. The disturbance is then fixed in space. Let A be a point in the disturbance and B a point in the undisturbed portion.

The condition of the matter between A and B remains constant, though fresh matter keeps coming in at B and an equal quantity leaves at A. Hence the angular momentum of the part between A and B remains constant, or as much enters at B as leaves at A. But at B there is no torsion, and no torsion couple of one part of the wire on the next. So that no angular momentum enters at B, and therefore on the whole none leaves at A. The transfer of angular momentum through A is of two kinds—first, that due to the passage of rotating matter, and, secondly, that due to the couple with which matter to the right of A acts upon matter to the left of A. The mass of matter moving through A per second is paa2U, where a is the radius of the wire and p is its density. If 0 is the angle of twist, the angular velocity is do/dt. The radius of gyration of the section is za2. Hence the angular momentum conveyed per second outwards is Zp:ra4Ud0/dt. The couple due to the twist of a wire of length 1 through $ is G = Znaa4 /l, and we may put 0/1=do/dx. Since no angular momentum goes out on the whole inira4d0/dx-1-paa"Ud8/dt =o. (33) But the condition of unchanged form requires that the matter shall twist through (dO/dx)dx while it is travelling over dx, i.e. in time dx/U.

Transverse Vibrations of Bars or Rods.—When a bar or rod is of considerable cross-section, a transversal disturbance calls into play forces due to the strain of the material much more important than the forces due to any tension which is ordinarily applied. The velocity of a disturbance along such a bar, and its modes of vibration, depend therefore on the elastic properties of the material and the dimensions of the bar. We cannot investigate the vibrations in an elementary manner. A full discussion will be found in Rayleigh's Sound, vol. i. ch. 8. We shall only give a few results. The cases interesting in sound are those in which (1) the bar is free at both ends, and (2) it is clamped at one end and free at the other. For a bar free at both ends the fundamental mode of vibration has two nodes, each 0.224 of the length from the end. The next mode has a node in the middle and two others each 0.132 from the end. The third mode has four nodes 0.094 and 0.357 from each end, and so on. The frequencies are nearly in the ratios 32:52:72• ... Such bars are used in the harmonicon.

When one end is clamped and the other is free the clamped end is always a node. The fundamental mode has that node only. The next mode has a second node 0.226 from the free end; the next, nodes at 0.132 and 0.5 from the free end, and so on. The frequencies are nearly in the ratios 1:6.25:17.5. Such bars are used in musical boxes and as free reeds in organ pipes. The most important example of this type is the tuning-fork, which may be regarded as consisting of two parallel bars clamped together at the base. The first overtone has frequency 6.25 that of the fundamental, and is not in the harmonic series. If the fork be mounted on a resonance box or held in front of a cavity resounding to the fundamental and not to the first overtone, the fundamental is brought out in great purity. Vibrations of Plates.—These are for the most part interesting rather from the point of view of elasticity than of sound. We shall not See also:

• ATTEMPT (Lat. adtemptare, attentare, to try)

Chladni (1756-1827). As usually arranged, a thin metal plate is screwed on to the top of a See also:

• FIRM

But with larger plates, which alone will furnish the more complicated figures, a clamp-screw must be used for fixing the plate, and, at the same time, one or more other nodal points ought to be touched with the fingers while the bow is being applied. In this way, any of the possible configurations may be easily produced. By similar methods, a circular plate may be made to exhibit nodal lines dividing the surface by diametral lines into four or a greater, but always even, number of sectors, an odd number being Incompatible with the general law of stationary waves that the parts of a body adjoining a nodal line on either side must always vibrate oppositely to each other. Another class of figures consists of circular nodal lines along with diametral lines (fig. 39). Then d L — dxor =—Udx. Substituting in (33) we get U2 =snip. (34) If we now keep the wire at rest the disturbance travels along it with velocity U= (n/p), and it depends on the rigidity and density of the wire and not upon its radius. It is easy to deduce the modes of vibration from stationary waves as in the previous cases. If a rod is clamped at one end and free at the other, the fundamental frequency is (1/1)J (nip). For iron n'p is of the order See also:

• ROLL (O. Fr. rolle, roulle, mod. role, Lat. rotulus, dim. of rota, wheel)
• ROLL, ALFRED PHILIPPE (1846– )

The torsional vibrations of a wire are excited when it is bowed. If small paper rings are put on a monochord wire they rotate through these vibrations when the wire is bowed. Circular nodal lines unaccompanied by intersecting lines cannot so that any deviation from the straight See also:

• RUSH
• RUSH, BENJAMIN (1745–1813)
• RUSH, RICHARD (1780–1859)

4o, so as to rise out of a See also:

• VESSEL (O. Fr. vaissel, from a rare Lat. vascellum, dim. of vas, vase, urn)

Sensitive Flames and Jets.—When a flame is just not flaring, any one of a certain range of notes sounded near it may make it flare while the note is sounding. This was first noticed by See also:

• JOHN
• JOHN (1167–1216)
• JOHN (1290-c. 1320)
• JOHN (1296-1346)
• JOHN (1371–1419)
• JOHN (1468-1532)
• JOHN (1801-1873)
• JOHN (Heb. llni')
• JOHN (ZAPOLYA) (1487-1540)
• JOHN, 4TH MARQUESS OF TWEEDDALE (c. 1695-1762)
• JOHN, DON (1545-1578)
• JOHN, DON (1629–1679)
• JOHN, GOSPEL OF ST
• JOHN, or HAYS (1513–1571)
• JOHN, THE APOSTLE
• JOHN, THE EPISTLES OF

§ 7 seq.) describes a number of beautiful experiments with jets at higher pressure than ordinary, say to in. of water, issuing from a pinhole steatite burner. The flame may be 16 in. high, and on receiving a suitably high sound it suddenly drops down and roars. The sensitive point is at the orifice. Lord Rayleigh (Sound, ii. § 370), using as a source a " See also:

• BIRD

If a tube be placed over such a flame it makes an excellent singing tube. The flame of an incandescent gas See also:

• MANTLE

In order that the vibration may be maintained, a periodic force must be applied either to aid the internal restoring force on the return See also:

• JOURNEY (through O. Fr. jornee or journee, mod. Fr. journee, from med. Lat. diurnata, Lat. diurnus, of or belonging to dies, day)

In the organ pipe--as in the common whistle—a thin sheet of air is forced through a narrow slit at the bottom of the embouchure and impinges against the top edge, which is made very organ Plpe. sharp. The disturbance made at the commencement of the blowing will no doubt set the air in the pipe vibrating in its own natural period, just as any irregular air disturbance will set a suspended body swinging in its natural period, but we are to consider how the vibration is maintained when once set going. When the motion due to the vibration is up along the pipe from the embouchure, the air moves into the pipe from the outside, and carries the sheet-like stream in with it to the inside of the sharp edge. This stream does work on the air, aiding the motion. When the motion is reversed and the air moves out of the pipe at the embouchure, the sheet is deflected on to the outer side of the sharp edge, and no work is done against it by the air in the pipe. Hence the stream of air does work during half the vibration and this is not abstracted during the other half, and so it goes on increasing the motion until the supply of energy in blowing is equal to the loss by friction and sound. The maintenance of the vibration of the air in the singing tube has been explained by Lord Rayleigh (Sound, vol. ii. § 322 h) as due Singing to the way in which the heat is communicated to the Tube. vibrating air. When the air in a pipe open at both ends is vibrating in its simplest mode, the air is alternately moving into and out from the centre. During the quarter swing ending with greatest nodal pressure, the kinetic energy is changed to potential energy manifested in the increase of pressure. This becomes again kinetic in the second quarter swing, then in the third quarter it is changed to potential energy again, but now manifested in the decrease of pressure. In the last quarter it is again turned to the kinetic form.

Now suppose that at the end of the first quarter swing, at the instant of greatest pressure, heat is suddenly given to the air. The pressure is further increased and the potential energy is also increased. There will be more kinetic energy formed in the return journey and the vibration tends to grow. But if the heat is given at the instant of greatest rare-See also:

• FACTION (through the French, from Lat. factio, a company of persons combined for action, facere, to do; from the other French derivative facon comes " fashion ")

But when the See also:

• PASTE (O. Fr. paste, modern pate, Late Lat. pasta, whence also in Span., Port. and Ital., from Gr. 1r&vrrl or 1raara, barley porridge, or salted pottage, ir&ao'ety, to sprinkle with salt)

The two waves, therefore, being in opposite phases, neutralize one another, and the result is a faint sound. On turning round the upper chest into any inter-mediate position, the intensity of the sound will increase up to a maximum, which occurs when the air in both chests is being admitted and cut off contemporaneously. If two organ pipes in unison are mounted side by side on a wind-chest with their ends close together, and are blown for a very short time, they sound. But if the blowing is continued, usually in less than a second the sound dies away to a small fraction of that due to either alone. Yet the air within the pipes is vibrating more vigorously than ever, but in opposite phases in the two pipes. This may be shown by furnishing the pipes with manometric flames placed in the same vertical line. When the flames are viewed in a revolving mirror and the pipes are blown, each image of one flame lies between two images of the other. The essential fact, as pointed out by Lord Rayleigh (Scientific Papers, i. 409), is not the common wind chest, but the nearness of the open ends, so that the outrush from one pipe can supply the inrush to the other, and the converse. If, the two pipes are slightly out of tune when sounded separately together they sound a common note which may be higher than that due to either alone. Lord Rayleigh (loc. cit.) points out that this Thus we learn that two musical notes, of the same pitch, conveyed to the ear through the air, will produce the effect of a single note of the same pitch, but of increased loudness, if they are in the same phase, but may affect the ear very slightly, if at all, when in opposite phases. If the difference of phase be varied gradually from zero to!A, the resulting sound will 2 gradually decrease from a maximum to a minimum.

Among the many experimental confirmations which may be adduced of these proportions we will mention the following: Take a circular plate, such as is available for the See also:

• PRODUCTION (Lat. productionem, from producere, to pro-duce)

But there is a slight delay in ignition, partly due to See also:

• EXPULSION (Lat. expulsio, from expellere)

At this moment, therefore, opposite displacements will be produced of the air particles by the two notes, and the sound due to them will be at a minimum. This will be followed by an increase of intensity until the lapse of another sixth of a second, when the less rapidly vibrating note will have lost another half-vibration relatively to the other, or one vibration reckoning from the original period of time, and the two component vibrations will again conspire and reproduce a maximum effect. Thus, an interval of one-third of a second elapses between two successive See also:

• MAXIMA

If one prong of each fork be furnished with a small See also:

• PLAIN (O. Fr. plain, from Lat. plenum)

We are then led to conclude that beats are the physical See also:

• FOUNDATION (Lat. fundatio, from fundare, to found)

The two tones are now dissonant, and, as we have seen, about the middle of the scale the maximum dissonance is when there are between 30 and 40 beats per second. If the pitch is raised still further the dissonance lessens, and when there are about 130 beats per second the, interval is consonant. If all tones were pure, dissonance at this part of the scale would not occur if the interval were more than a third. But we have to remember that with strings, pipes and instruments generaIly the fundamental tone is accompanied by overtones, called also " upper partials," and beating within the dissonance range may occur between these overtones. Thus, suppose a fundamental 256 has present with it overtone harmonics 512, 768, 1024, 1280, &c., and that we sound with it the major seventh with fundamental 480, and having harmonics 960, 1440, &c. The two sets may be arranged thus c 256 512 768 1024 1280 h 480 960 1440, and we see that the fundamental of the second will beat 32 times per second with the first overtone of the first, giving dissonance. The first overtone of the second will beat 64 times per second with the third of the first, and at such height in the scale this frequency will be unpleasant. The very marked dissonance of the major seventh is thus explained. We can see, too, at once how the octave is such a smooth consonance. Let the two tones with their harmonic overtones be 256 512 768 1024 1280 1536 512 1024 1536. The fundamental and overtones of the second all coincide with overtones of the first. Take as a further example the fifth with harmonic overtones as under 256 512 768 1024 1280 1536 384 768 1152 1536.

The fundamental and overtones of the second either coincide with or fall midway between overtones in the first, and there is no approach to a dissonant frequency of beats, and the concord is perfect. But obviously in either the octave or the fifth, if the tuning is imperfect, beats occur all along the line wherever the tones should coincide with perfect tuning. Thus it is easy to detect a want of tuning in these intervals. The harshness of deep notes on instruments See also:

• RICH, BARNABE (c. 1540-1617)
• RICH, CLAUDIUS JAMES (1787-1821)
• RICH, JOHN (1692-1761)
• RICH, PENELOPE, LADY (c. 1562–1607)
• RICH, RICHARD, 1ST BARON RICH (1490?-1567)

Thus, take the major seventh with frequencies 256 and 480. There will be a tone frequency 480—256=224, and this will be very dissonant with 256. The tone of the frequency of the beats was discovered by Georg Andreas Sorge in 1740, and independently a few years later by Giuseppe See also:

• TARTINI, GIUSEPPE (1692–1770)

The reader is referred to the full discussion by Helmholtz. We shall conclude by a brief account of the ways in which combination tones may be produced. There appears to be no doubt that they are produced, and the only question is whether the theory accounts sufficiently for the intensity of the tones actually heard. Combination tones may be produced in three ways: (I) In the neighbourhood of the source; (2) in the receiving mechanism of the ear; (3) in the medium conveying the waves. 1. We may illustrate the first method by taking a case discussed by Helmholtz (Sensations of Tone, app. xvi.) where the two sources are reeds or pipes blown from the same wind-chest. Let us suppose that with constant excess of pressure, p, in the wind-chest, the amplitude produced is proportional to the pressure, so that the two tones issuing may be represented by pa sin 27rn]t and pb sin 27rn21. Now as each source lets out the wind periodically it affects the pressure in the chest so that we cannot regard this as constant, but may take it as better represented by p+Xa sin (27rnit+e)+µb sin (27rnzt+f). Then the issuing disturbance will be {p+Xa sin (21rnit+e)+µb sin (27rn2t-{-f)} (a sin 27rnit+b sin 27rnit} =pa sin 27rnit+pb sin 27rn2t a2~ a2X + 2 cos e— 2 cos (47rnit+e) z + 2 cos f -2 cos (47rn21+f) a2)` abX + cos {27r(ni—n2)t+e}- - cos {27r(ni+n2)t+e} + aµ cos {27r(ni—n2)t—f}—a'~ cos {27r(ni+n2)t+f} 2 2 Thus, accompanying the two original pure tones there are (I) the octave of each; (2) a tone of frequency (n —n2); (3) a tone of frequency (ni+n2). The second is termed by Helmholtz the difference lone, and the third the summation tone. The amplitudes of these tones are proportional to the products of a and b multiplied by X or µ. These combination tones will in turn react on the pressure and produce new combination tones with the original tones, or with each other, and such tones may be termed of the second, third, &c., order.

It is evident that we may have tones of frequency hnl kn2 hni—kn2 hni+kn2, where h and k are any integers. But inasmuch as the successive orders are proportional to X a2 X3, or µ µ2 µ3, and X and are small, they are of rapidly decreasing importance, and it is not certain that any beyond those in equation (35) correspond to our actual sensations. The combination tones thus produced in the source should have a physical existence in the air, and the amplitudes of those represented in (35) should be of the same order. The conditions assumed in this investigation are probably nearly realized in a harmonium and in a double siren of the form used by Helmholtz, and in these cases there can be no doubt that actual objective tones are produced, for they may be detected by the aid of resonators of the frequency of the tone sought for. If the tones had no existence outside the ear then resonators would not increase their loudness. There is not much difficulty in detecting the difference tone by a resonator if it is held, say, close to the reeds of a harmonium, and Helmholtz succeeded in detecting the summation tone by the aid of a resonator. Further, Rucker and Edser, using a siren as source, have succeeded in making a fork of the appropriate pitch See also:

• RESPOND

500, or Rayleigh, Sound, § 386) that sounds of considerable intensity when heard by themselves are liable to be completely obliterated by graver sounds of sufficient force goes far to explain this, for the summation tones are of course always accompanied by such graver sounds. 2. The second mode of production of combination tones, by the mechanism of the receiver, is discussed by Helmholtz (Sensations of Tone, App. xii.) and Rayleigh (Sound, i. § 68). It depends on the restoring force due to the displacement of the receiver not being accurately proportional to the displacement. This want of proportionality will have a periodicity, that of the impinging waves, and so will produce vibrations just as does the variation of pressure in the case last investigated. We may see how this occurs by supposing that the restoring force of the receiving mechanism is represented by Tx-3-µz2, where x is the displacement and µx2 is very small. Let an external force F act on the system, and for simplicity suppose its period is so great compared with that of the mechanism that we may take it as practically in See also:

• EQUILIBRIUM (from the Lat. aequus, equal, and libra, a balance)

Probably in most cases the combination tones which we hear are thus made, and possibly, too, the tones detected by Koenig, and by him named " beat-tones." He found that if two tones of frequencies p and q are sounded, and if q lies between Np and (N +1)p, then a tone of frequency either (N+1)p-g, or of frequency q—Np, is heard. The difficulty in Helmholtz's theory is to account for the audibility of such beat tones when they are of a higher order than the first. Rucker and Edser quite failed to detect their external existence, so that apparently they are not produced in the source. If we are to assume that the tones received by the ear are pure and free from partials, the loudness of the beat-tones would appear to show that Helmholtz's theory is not a complete account. 3. The third mode of production of combination tones, the production in the medium itself, follows from the varying velocity of different parts of the wave, as investigated at the beginning of this article. It is easily shown that after a time we shall have to superpose on the original displacement a displacement proportional to the square of the particle velocity, and this will intro-duce just the same set of combination tones. But probably in practice there is not a sufficient interval between source and hearer for these tones to grow into any importance, and they can at most be only a small addition to those formed in the source or the ear. (1899). The standard See also:

• TREATISE

J. See also:

• ELLIS (originally SHARPE), ALEXANDER JOHN (1814-189o)
• ELLIS, GEORGE (1753-1815)
• ELLIS, ROBINSON (1834- )
• ELLIS, SIR HENRY (1777-1869)
• ELLIS, WILLIAM (1794 – 1872)

John Tyndall, Sound (5th ed., 1893), originally delivered as lectures, treats the subject descriptively, and is illustrated by a large number of excellent experiments. Good general accounts are given in J. L. G. Violle, Cours de physique, tome i1., " Acoustique "; A. See also:

• WINKELMANN, EDUARD (1838-1896)

Wullner, Lehrbuch der Experimenlalphysik (1907), 6th ed., vol. i.; and C. L. See also:

• BARNES, ALBERT (1798–1870)
• BARNES, BARNABE (1569 ?–16o9)
• BARNES, JOSHUA (1654-1712)
• BARNES, ROBERT (1495-1540)
• BARNES, SIR EDWARD (1776-1838)
• BARNES, THOMAS (1785-1841)
• BARNES, W
• BARNES, WILLIAM (1800-1886)