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SOUND

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Originally appearing in Volume V25, Page 460 of the 1911 Encyclopedia Britannica.
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SOUND  ,1 subjectively the sense impression of the See also:

organ of 1" Sound " is an interesting example of the numerous homonymous words in the See also:English See also:language . In the sense in which it is treated in this See also:article it appears in See also:Middle English as soun, and comes through Fr. son from See also:Lat. sonus; the d is a See also:mere addition, as in the nautical See also:term " See also:bound " (outward, homeward bound) for the earlier " boun," to make ready, prepare . In the adjectival meaning, healthy, perfect, See also:complete, chiefly used of a deep undisturbed See also:sleep, or of a well-based See also:argument or See also:doctrine, or of a See also:person well trained in his profession, the word is in O . Eng. sund, and appears also in Ger. gesund, Du. gezond . It is probably cognate with the Lat. See also:saints, healthy, whence the Eng. sane, See also:insanity, sanitation, &c . Lastly, there is a See also:group of words which etymologists are inclined to treat as being all forms of the word which in O . Eng. is sund, meaning ' ' See also:swimming." These words are for (1) the swim-See also:bladder of a See also:fish; (2) a narrow stretch of See also:water between an inland See also:sea and the ocean, or between an See also:island and the mainland, &c., cf . Souxv, THE, below; (3) to test or measure the See also:depth of anything, particularly the depth of water in lakes or seas (see See also:SOUNDING, below) . As a substantive the term is used of a surgical See also:instrument for the exploration of a See also:wound, cavity, &c., a probe . In these senses the word has frequently been referred to Lat. sub unda, under the water; and Fr. sombre, gloomy, possibly from sub See also:umbra, beneath the shade, is given as a parallel . See also:hearing, and objectively the vibratory See also:motion which produces the sensation of sound . The physiological and psychical aspects of sound are treated in the article HEARING .

In this article, which covers the See also:

science of See also:Acoustics, we shall consider only the See also:physical aspect of sound, that is, the physical phenomena outside ourselves which excite our sense of hearing . We shall discuss the disturbance which is propagated from the source to the See also:ear, and which there produces sound, and the modes in which various See also:sources vibrate and give rise to the disturbance . Sound is due to Vibrations.—We may easily satisfy ourselves that, in every instance in which the sensation of sound is excited, the See also:body whence the sound proceeds must have been thrown, by a See also:blow or other means, into a See also:state of agitation or tremor, implying the existence of a vibratory motion, or motion to and fro, of the particles of which it consists . Thus, if a See also:common See also:glass-See also:jar be struck so as to yield an audible sound, the existence of a motion of this See also:kind may be See also:felt by the See also:finger lightly applied to the edge of the glass; and, on increasing the pressure so as to destroy this motion the sound forthwith ceases . Small pieces of See also:cork put in the jar will be found to See also:dance about during the continuance of the sound; water or See also:spirits of See also:wine poured into the glass will, under the same circumstances, exhibit a ruffled See also:surface . The experiment is usually performed, in a more striking manner, with a See also:bell-jar and a number of small See also:light wooden balls suspended by See also:silk strings to a fixed See also:frame above the jar, so as to be just in contact with the widest See also:part of the glass . On See also:drawing a See also:violin See also:bow across the edge, the pendulums are thrown off to a considerable distance, and falling back are again repelled, and so on . It is also in many cases possible to follow with the See also:eye the motions of the particles of the sounding body, as, for instance, in the See also:case of a violin See also:string or any string fixed at both ends, when the string will appear through the persistence of visual sensation to occupy at once all the positions which it successively assumes during its vibratory motion . Sound takes See also:Time to Travel.—If we See also:watch a See also:man breaking stones by the roadside some distance away, we can see the See also:hammer fall before we hear the blow . We see the See also:steam issuing from the See also:whistle of a distant See also:engine See also:long before we hear the sound . We see See also:lightning before we hear the See also:thunder which spreads out from the flash, and the more distant the flash the longer the See also:interval between the two . The well-known See also:rule of a mile for every five seconds between flash and peal gives a See also:fair estimate of the distance of the lightning .

Sound needs a Material See also:

Medium to Travel Through.—In See also:order that the ear may be affected by a sounding body there must be continuous See also:matter reaching all the way from the body to the ear . This can be shown by suspending an electric bell in the See also:receiver of an See also:air-See also:pump, the wires conveying the current passing through an air-tight cork closing the hole at the See also:top of the receiver . These wires See also:form a material channel from the bell to the outside air, but if they are See also:fine the sound which they carry is hardly appreciable . If while the air within the receiver is at atmospheric pressure the bell is set ringing continuously, the sound is very audible . But as the air is withdrawn by the pump the sound decreases, and when the exhaustion is high the bell is almost inaudible . Usually air is the medium through which sound travels, but it can travel through solids or liquids . Thus in the air-pump experiment, before exhaustion it travels through the glass of the receiver and the See also:base See also:plate . We may easily realise its trans-See also:mission through a solid by putting the ear against a table and scratching the See also:wood at some distance, and through a liquid by keeping both ears under water in a See also:bath and tapping the See also:side of the bath . Sound is a Disturbance of the See also:Wave Kind.—As sound arises in See also:general from vibrating bodies, as it takes time to travel, and as the medium which carries it does not on the whole travel for-See also:ward, but subsides into its See also:original position when the sound has passed, we are forced to conclude that the disturbance is of the wave kind, We can at once gather some See also:idea of the nature of sound waves in air by considering how they are produced by a bell . Let AB (fig . I) be a small portion of a bell which vibrates to and fro from CD to EF and back . As AB moves from CD to EF it pushes forward the layer of air in contact with it .

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

compression of the air is transmitted onwards in the direction OX . As AB returns from EF towards CD the layer of air next to it follows it as if it D were pulled back by AB . Really, FIG . I . of course, it is pressed into the space made for it by the See also:rest of the air, and flowing into this space it is extended . It makes See also:room for the next layer of air to move back and to be extended and so on, and an See also:extension of the air is transmitted onwards following the compression which has already gone out . As AB again moves from CD towards EF another compression or push is sent out, as it returns from EF towards CD another extension or pull, and so on . Thus waves are propagated along OX, each wave consisting of one push and one pull, one wave emanating from each complete vibration to and fro of the source AB . Crova's Disk.—We may obtain an excellent See also:representation of the motion of the layers of air in a See also:train of sound waves by means of a See also:device due to Crova and known as " Crova's disk." A small circle, say 2 or 3 mm. See also:radius, is See also:drawn on a card as in fig . 2, and See also:round this circle equidistant points, say 8 or 12, are taken . From these points as centres, circles are drawn in See also:succession, each with radius greater than the last by a fixed amount, say 4 or 5 mm .

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 waves appear to travel out from the centre along any radius . If a second card with a narrow slit in it is held in front of the first, the slit See also:running from the centre outwards, the wave motion is still more evident . If the figure be photographed as a See also:lantern slide which is mounted so as to turn round, the wave motion is excellently shown on the See also:screen, the compressions and extensions being represented by the crowding in and opening out of the lines . Another See also:illustration is afforded by a long See also:spiral of See also:wire with coils, say 2 in. in See also:diameter and z in. apart . It may be hung up by threads so as to See also:lie horizontally . If one end is sharply pressed in, a compression can be seen running along the See also:spring . The Disturbance in Sound Waves is See also:Longitudinal.—The motion of a particle of air is, as represented in these illustrations, to and fro in the direction of See also:propagation, i.e. the disturbance 0 x B F is " longitudinal." There is no " transverse " disturbance, that is, there is in air no motion across the See also:line of propagation, for such motion could only be propagated from one layer to the next by the " viscous " resistance to relative motion, and would See also:die away at a very See also:short distance from the source . But trans-See also:verse disturbances may be propagated as waves in solids . For instance, if a rope is fixed at one end and held in the See also:hand at the other end, a transverse jerk by the hand will travel as a trans-verse wave along the rope . In liquids sound waves are longitudinal as they are in air . But the waves on the surface of a liquid, which are not of the sound kind, are both longitudinal and transverse, the See also:compound nature being easily seen in watching the motion of a floating particle .

Displacement See also:

Diagram.—We can represent waves of longitudinal displacement by a See also:curve, and this enables us to draw very important conclusions in a very See also:simple way . Let a train of waves be passing from See also:left to right in the direction See also:ABCD (fig . 3) . At every point p R mMc Q FIG . 3 . let a line be drawn perpendicular to AD and proportional to the displacement of the particle which was at the point before the disturbance began . Thus let the particle which was at L be at 1, to the right or forwards, at a given instant . Draw LP upward and some convenient multiple of Ll . Let the particle which was at M originally be at m at the given instant, being displaced to the left or backwards . Draw MQ downwards, the same multiple of :See also:elm . Let N be displaced forward to n . Draw NR the same multiple of Nn and upwards .

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 . At the points ABCD there is no displacement, and the line AD through these points is called the See also:axis . Forward displacement is represented by height above the axis, backward displacement by depth below it . In See also:ordinary sound waves the displacement is very See also:minute, perhaps of the order io-6 cm., so that we multiply it perhaps by See also:ioo,000 in forming the displacement curve . Wave Length and Frequency.—If the waves are continuous and each of the same shape they form a " train," and the displacement curve repeats itself . The shortest distance in which this repetition occurs is called the wave-length . It is usually denoted by X . In fig . 3, AC =X . If the source makes n vibrations in one second it is said to have " frequency " n . It sends out n waves in each second . If each wave travels out. from the source with velocity U the n waves emitted in one secnd must occupy a length U and therefore U = na .

See also:

Distribution of Compression and Extension in a Wave.—Let fig . 4 be the displacement diagram of a wave travelling from left to right . At A the air occupies its original position, while at See also:Hit is displaced towards the right or away from A since HP is above the axis . Between A and H, then, and about H, it is extended . At J the displacement is forward, but since the curve at Q is parallel to the axis the displacement is approximately the same for all the points See also:close to J, and the air is neither extended nor compressed, but merely displaced bodily a distance represented by JQ . At B there is no displacement, but at K there is displacement towards B represented by KR, i.e. there is compression . At L there is also displacement towards B and again compression . At M, as at J, there is neither extension nor compression . At N the displacement is away from C and there is extension . The dotted curve represents the distribution of compression by height above the axis, and of extension by depth below it . Or we may take it as representing the pressure—excess over the normal pressure in compression, defect from it in extension . The figure shows that when the curve of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal .

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, and at N it is easily seen that the motion is backward . The distribution of velocity then is represented by the dotted curve and is forward when the curve is above the axis and backward when it is below . Comparing See also:figs . 4 and 5 it is seen that the velocity is forward in compression and backward in extension . The Relations between Displacement, Compression and Velocity.—The relations shown by figs .

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 dx, then, has increased to dx+dy or volume i has increased to I+dy/dx and the increase of volume I is dy/dx . Let E be the bulk modulus of See also:elasticity, defined as increase of pressure = decrease of volume per unit volume where the pressure increase is so small that this ratio is See also:constant, w the small increase of pressure, and — (dy/dx) the volume decrease, then E=w/(—dy/dx) or w /E= —dy/dx (I) This gives the relation between pressure excess and displacement . To find the relation of the velocity to displacement and pressure we shall See also:express the fact that the wave travels on carrying all its conditions with it, so that the displacement now at M will arrive at N while the wave travels over MN . Let U be the velocity of the wave and let u be the velocity of the particle originally at N . Let MN=dx=Udt . In the time dt which the wave takes to travel over MN the particle displacement at N changes by QR, and QR = —udt, so that QR/MN = —u/U .

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 q in the wave is carried forward unchanged with velocity U, the change of at a given point in time dl is equal to the change of 4 as we go back along the curve a distance dx=Udt at the beginning of dt . ds/s _ Ids dx— Udt . The Characteristics of Sound Waves Corresponding to Loudness, See also:Pitch and Quality.—Sounds differ from each other only in the three respects of loudness, pitch and quality . The loudness of the sound brought by a train of waves of given wave-length depends on the extent of the to and fro excursion of the air particles . This is obvious if we consider that the greater the vibration of the source the greater is the excursion of the air in the issuing waves, and the louder is the sound heard . See also:Half the See also:total excursion is called the See also:amplitude . Thus in fig . 4 QJ is the amplitude . Methods of measuring the amplitude in sound waves in air have been devised and will be described later .

We may say here that the See also:

energy or the intensity 'of the sound of given wave-length is proportional to the square of the amplitude . The pitch of a sound, the See also:note which we assign to it, depends on the number of waves received by the ear per second . This is generally equal to the number of waves issuing from the source per second, and therefore equal to its frequency of vibration . Experiments, which will be described most conveniently when Then we discuss methods of determining the frequencies of sources, prove conclusively that for a given note the frequency is the same whatever the source of that note, and that the ratio of the frequencies of two notes forming a given musical interval is the same in whatever part of the musical range the two notes are situated . Here it is sufficient to say that the frequencies of a note, its See also:major third, its fifth and its See also:octave, are in the ratios of 4: 5: 6: 8 . The quality or timbre of sound, i.e. that which differentiates a note sounded on one instrument from the same note on another instrument, depends neither on amplitude nor on frequency or wave-length . We can only conclude that it depends on wave form, a conclusion fully See also:borne out by investigation . The displacement curve of the waves from a tuning-See also:fork on its resonance See also:box, or from the human See also:voice sounding oo, are nearly smooth and symmetrical, as in fig . 7a . That for the air waves from a violin are probably nearly as in fig . 7b . Calculation of the Velocity of Sound Waves in Air.—The velocity with which waves of longitudinal disturbance travel in air or in any other fluid can be calculated from the resistance to compression and extension and the See also:density of the fluid .

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 conducting and interpreting the observations . The waves from a source surrounded by a See also:uniform medium at rest spread out as See also:spheres with the source as centre . If we take one of these spheres a distance from the source very See also:great as compared with a single wave-length, and draw a radius to a point on the See also:sphere, then for some little way round that point the sphere may be regarded as a plane perpendicular to the radius or the line of propagation . Every particle in the plane will have the same displacement and the same velocity, and these will be perpendicular to the plane and parallel to the line of propagation . The waves for some little distance on each side of the plane will be practically of the same See also:size . In fact, we may neglect the divergence, and may regard them as " plane waves." We shall investigate the velocity of such plane waves by a method which is only a slight modification of a method given by W . J . M . See also:Rankine (Phil . Trans., 187o, p . 277) . Whatever the form of a wave, we could always force it to travel on with that form unchanged, and with any velocity we See also:chose, if we could apply any " See also:external " force we liked to each particle, in addition to the " See also:internal " force called into See also:play by the compressions or extensions .

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

ABC (fig . 8), and we require it to travel P B a' on in time dt to A'B'C', where AA' = Udt, the displacement of the particle originally at M must change from PM to P'M or by PP' . This change can always be effected if we can apply whatever force may be needed to produce it . We shall investigate the external force needed to make a train of plane waves travel on unchanged in form with velocity U . We shall regard the external force as applied in the form of a pressure X per square centimetre parallel to the line of propagation and varied from point to point as required in order to make the disturbance travel on unchanged in form with the specified velocity U . In addition there will be the internal force due to the change in volume, and consequent change in pressure, from point to point . Suppose that the whole of the medium is moved backwards in space along the line of propagation so that the undisturbed portions travel with the velocity U . The disturbance, or the train of waves, is then fixed in space, though fresh matter continually enters the disturbed region at one end, undergoes the disturbance, and then leaves it at the other end . Let A (fig . 9) be a point fixed in space in the disturbed region, B a fixed point where the medium is not yet disturbed, the medium A moving through A and B from right to left . Since the condition of the medium between A and B remains constant, even though the matter is continually changing, the momentum possessed by the matter between A and B is constant . Therefore the momentum entering through a square centimetre at B per second is equal to the momentum leaving through a square centimetre at A .

Now the See also:

transfer of momentum across a surface occurs in two ways, firstly by the See also:carriage of moving matter through the surface, and secondly by the force acting between the matter on one side of the surface and the matter on the other side . U cubic centimetres move in per second at B, and if the density is po the See also:mass moving in through a square centimetre is poU . But it has velocity U, and therefore momentum poU2 is carried in . In addition there is a pressure between the layers of the medium, and if this pressure in the undisturbed parts of the medium is P, momentum P per second is being transferred from right to left across each square centimetre . Hence the matter moving in is receiving on this See also:account P per second from the matter to the right of it . The total momentum moving in at B is therefore P+poU2 . Now consider the momentum leaving at A . If the velocity of a particle at A relative to the undisturbed parts is u from left to right, the velocity of the matter moving out at A is U—u, and the momentum carried out by the moving matter is p(U—u)2 . But the matter to the right of A is also receiving momentum from the matter to the left of it at the See also:rate indicated by the force across A . Let the excess of pressure due to change of volume be w, so that the total " internal " pressure is P+& . There is also the " external " applied pressure X, and the total momentum flowing out per second is X+P+-w-i-P(U—u)2 . Equating this to the momentum entering at B and subtracting P from each X I w~ P(U—u)2=poU2 .

(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 of the internal forces only, unchanged in form and with velocity U (E/p) . (6) The pressure X is introduced in order to show that a wave can be propagated unchanged in form . If we omitted it we should have to assume this, and See also:equation (6) would give us the velocity of propagation if the See also:assumption were justified . But a priori we are hardly justified in assuming that waves can be propagated at all, and certainly not justified in assuming that they go on unchanged by the action of the internal forces alone . If, however, we put on external forces of the required type X it is obvious that any wave can be propagated with any velocity, and our investigation shows that when U has the value in (6) then and only then X is zero everywhere, and the wave will be propagated with that velocity when once set going . It may be noted that the elasticity E is only constant for small volume changes or for small values of dy/dx . Since by See also:definition E = —v(dp/dv) = p(dp/dp) equation (6) becomes U = J (dp/dp) • (7) The value U=st(E/p) was first virtually obtained by See also:Newton (Principia, bk. ii., § 8, props . 48--49) .

He supposed that in air See also:

Boyle's See also:law holds in the extensions and compressions, or that p=kp, whence dp/dp=k=p/p . His value of the velocity in air is therefore U = J (PIP) (Newton's See also:formula) . At the See also:standard pressure of 76 cm. of See also:mercury or 1,014,000 dynes / sq. cm., the density of dry air at o° C. being taken as 0.001293, we get for the velocity in dry air at 0° C . Uo=28,00o cm.sec . (about 920 ft./sec.) approximately . Newton found 979 ft./sec . But, as we shall see, all the determinations give a value of U0 in the neighbourhood of 33,000 cm./sec., or about Io8o ft./sec . This discrepancy was not explained till 1816, when See also:Laplace (A an. de chimie, 1816, vol. iii.) pointed out that the compressions and extensions in sound waves in air alternate so rapidly that there is no time for the temperature inequalities produced by them to spread . That is to say, instead of using Boyle's law, which supposes that the pressure changes so exceedingly slowly that See also:conduction keeps the temperature constant, we must use the adiabatic relation p=kpy, whence dp/dp =7k0-1 = yp/p, and U = (yp/p) [Laplace's formula] . (8) If we take y =1.4 we obtain approximately for the velocity in dry air at 0° C, Uo=33,15o cm . /sec., which is closely in accordance with observation . Indeed See also:Sir G .

G . See also:

Stokes (Math. and Phys . Papers, iii . 142) showed that a very small departure from the adiabatic condition would See also:lead to a stifling of the sound quite out of See also:accord with observation . If we put p=kp(i-See also:Fat) in (8) we get the velocity in a See also:gas at t° C, Ui=J{yk(i+at)} . At 0° C. we have Uo=~ (-yk), and hence tit = Uo ,/ (i +at) =Uo(I+0.001841) (for small values of t) . (9) The velocity then should be See also:independent of the barometric pressure, a result confirmed by observation . For two different gases with the same value of ', but with densities at the same pressure and temperature respectively pi and p2, we should have Ui/U2=II (P2/pi), (10) another result confirmed by observation . Alteration of Form of the Waves when Pressure Changes are See also:Con- siderable.—When the value of dy/dx is not very small E is no longer constant, but is rather greater in compression and rather less in extension than yP . This can be seen by considering that the relation between p and p is given by a curve and not by a straight line . The consequence is that the compression travels rather faster, and the extension rather slower, than at the See also:speed found above . We may get some idea of the effect by supposing that for a short time the change in form is negligible .

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 the velocity is in excess, and where u is negative the velocity is in defect of the normal value . If, then, a (fig. to) represents the displacement curve of a train of waves, b will represent the pressure excess and particle velocity, and from (i I) we see that while the nodal conditions of b, with a=o and u=o, travel with velocity Al (E/p), the crests exceed that velocity by ;(y+i)u, and the hollows fall short of it by ;(y+I)u, with the result that the fronts of the pressure waves become steeper and steeper, and the train b changes into something like c .

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 of the air particles in organ pipes . Thus See also:Mach found an amplitude 0.2 cm. when the issuing waves were 250 cm. long . The amplitude in the See also:pipe was certainly much greater than in the issuing waves . Let us take the latter as o. f mm. in the waves—a very extreme value . The maximum particle velocity is 2lrna (where n is the frequency and a the amplitude), or 2aaU/A . This gives maximum u =about 8 cm./sec., which would not seriously change the form of the wave in a few wave-lengths . Meanwhile the waves are spreading out and the value of u is falling in inverse proportion to the distance from the source, so that very soon its effect must become negligible . In loud sounds, such as a peal of thunder from a near flash, or the See also:report of a See also:gun, the effect may be considerable, and the rumble of the thunder and the prolonged See also:boom of the gun may perhaps be in part due to the breakdown of the wave when the See also:crest of maximum pressure has moved up to the front, though it is probably due in part also to See also:echo from the surfaces of heterogeneous masses of air . But there is no doubt that with very loud explosive sounds the normal velocity is quite considerably exceeded . Thus See also:Regnault in his classical experiments (described below) found that the velocity of the report of a See also:pistol carried through a pipe diminished with the intensity, and his results have been confirmed by J . Violle and T .

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

cannon in different directions; he found that it See also:rose to a maximum of 1267 ft./sec. at 70, to 90 ft. in the See also:rear and then See also:fell off . A very curious observation is recorded by the Rev . G . See also:Fisher in an appendix to See also:Captain See also:Parry's See also:Journal of a Second Voyage to the See also:Arctic Regions . In describing experiments on the velocity of sound he states'that " on one See also:day and one day only, See also:February 9, 1822, the officer's word of command ' See also:fire ' was several times heard distinctly both by Captain Parry and myself about one See also:beat of the chronometer [nearly half a second] after the report of the gun." This is hardly to be explained by equation (I1), for at the very front of the disturbance u=o and the velocity should be normal . The Energy in a Wave Train.—The energy in a train of waves carried forward with the waves is partly See also:strain or potential energy due to change of volume of the air, partly kinetic energy due to the motion Qf the air as the waves pass . We shall show that if we sum these up for a whole wave the potential energy is equal to the kinetic energy .

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 pressure during the change may be taken as P+Za, since the pressure excess for a small change is proportional to the change . Hence the See also:work done on the air is (P-+aa)v, and the work done per cubic centimetre is (P+Ja)v/V . The term Pv/V added up for a complete wave vanishes, for P/V is constant and Ev=o, since on the whole the compression equals the extension . We have then only to consider the term Zav/V . But v/V =u/U from equation (2) and a = Eu/U from equation (3) Then Zav/V =iEu2/U2=Zpu2 from equation (6) Then in the whole wave the potential energy equals the kinetic energy and the total energy in a complete wave in a See also:column 1 sq. cm . See also:cross-See also:section is W =f pu2dx . 0 like the breaking of a water-wave when the, crest gains on the next trough . In ordinary sound-waves the effect of the particle velocity in affecting the velocity of transmission must be very small . (if) We may find here the value of this when we have a train of waves in which the displacement is represented by a sine curve of amplitude a, viz. y=a See also:sin 2 (x—Ut) . For a discussion of this type of wave, see below . \Ve have u= dt = -- Ua See also:cos - (x—Ut), and fopu2dx=p4n-2 C 27 See also:j1:22a2J ocos2 (x—Ut)dx = 2pa-2U2a2/ ?

(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. der Physik, 1903, II, p . 405) . A small circular disk at one end of a torsion See also:arm formed part of a solid See also:wall, but was See also:free to move through a hole in the wall slightly larger than the disk . When intense sound waves impinged on the wall, the disk moved back through the hole, and by an amount showing a pressure of the order given by the following investigation Suppose that a train of waves is incident normally on the surface S (fig . II), and that they are absorbed there without reflection . Let ABCD be a column of air I sq. cm. cross-section . The pressure on CD is equal to the C momentum which it receives per second . On the whole the air S within ABCD neither gains nor loses momentum, so that on the whole it receives as much through AB as it gives up to CD . If P is the undisturbed pressure and P+a the pressure at AB, the momentum entering through AB per second is f 1(P+&+pu2)dt . But f iPdt= P is the normal pressure, and as we only wish to find the excess we may leave this out of account .

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 of the path, putting dt=d/U, we have the average excess of pressure p=J 0(w+pu2)dt=U f o (co+pu2)d . Here dA is an actual length in the disturbance . We have iv and u expressed in terms of the original length dx and the displacement dy so that we must put d=dx+dy = (I+dy/dx)dx, and p=U f o (~+pu2) (1+dz) dx . We have already found that if V changes to V—v =yP (V+y 1V2) =poU2 —dz+y21 (d ) 2 since v/V= —dy/dx . We also have pu2=p0u2/(t+dy/dx) . Substituting these values and neglecting See also:powers of dy/dx above the second we get p — U) o poU2 € —ax+ry 2 I (dx) 2 dx . But f u dxdx=o since the sum of the displacements=o . Then putting (dy/dx)2 = (u/U)2, we have =7 r• U f o Pou2dx = 1(7+1) average energy per cubic centimetre, (15) a result first published by See also:Lord See also:Rayleigh (Phil . Mag., 1905, to, P .

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 by a See also:clock or chronometer the time elapsing between the See also:receipt of the flash, which passes practically instantaneously, and the receipt of the report . The distance divided by the time gives the velocity of the sound . The velocity thus obtained will be affected by the See also:wind . For instance, See also:William See also:Derham (Phil . Trans., 1708) made a See also:series of observations, noting the time taken by the report of a cannon fired on See also:Blackheath to travel across the See also:Thames to Upminster See also:Church in See also:Essex, 121 M. away . He found that the time varied between 551 seconds when the wind was blowing most strongly with the sound, to 63 seconds when it was most strongly against the sound . The value for still air he estimated at 1142 ft. per second .

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 of the See also:French See also:Academy (Memoires de l'academie See also:des sciences, (1738) . Cannons were fired at half-See also:hour intervals, alternately at Montmartre and Montlhery, 17 or 18 M. apart . There were also two intermediate stations! at which observations were made . The times were measured by pendulum clocks . The result obtained at a temperature about 6° C. was, when converted to metres, U=337 metres/second .

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 des Longitudes made a series of experiments between Montlhery and Villejuif, 11 m. apart . Cannons were fired at the two stations at intervals of five minutes . Chronometers were used for timing, and the result at 15.9° C. was U=31.0.9 met./sec., whence Uo=33o•6 met./sec . (F . J . D . See also:Arago, Con;iai:sancc des temps, 1825) . When the measurement of a time interval depends on an observer, his " See also:personal equation " comes in to affect the estimation of the quantity . This is the interval between the arrival of an event and his See also:perception that it has arrived, or it may be the interval between arrival a:_d his See also:record of the arrival . This personal equation is different for different observers . It may differ even by a considerable fraction of a second .

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, different, that for the flash being most likely less than that for the sound . In a long series of experiments carried out by V . Regnault in the years 1862 to 1866 on the velocity of sound in open air, in air in pipes and in various other gases in pipes, he sought to eliminate personal equation by dispensing with the human element in the observations, using electric receivers as observers . A short account of these experiments is given in Phil . Mag., 1868, 35, p . 161, and the full account, which serves as an excellent example of the extra-ordinary care and ingenuity of Regnault's work, is given in the Memoires de l'academie des sciences, 1868, See also:xxxvii . On See also:page 459 of the Memoire will be found a See also:list of previous careful experiments on the velocity of sound . In the open-air experiments the receiver consisted of a large 25 See also:cone having a thin See also:india-See also:rubber membrane stretched over its narrow end . A small See also:metal disk was attached to the centre of the membrane and connected to See also:earth by a fine wire . A metal contact-piece adjustable by a See also:screw could be made to just See also:touch a point at the centre of the disk . When contact was made it completed an electric See also:circuit which passed to a recording station, and there, by means of an electro-magnet, actuated a See also:style See also:writing a record on a See also:band of travelling smoked See also:paper .

On the same band a tuning-fork electrically maintained and a seconds clock actuating a