SOUND,1 subjectively the sense impression of the organ of
1" Sound " is an interesting example of the numerous homonymous words in the English language. In the sense in which it is treated in this article it appears in Middle English as soun, and comes through Fr. son from Lat. sonus; the d is a mere addition, as in the nautical term " bound " (outward, homeward bound) for the earlier " boun," to make ready, prepare. In the adjectival meaning, healthy, perfect, complete, chiefly used of a deep undisturbed sleep, or of a wellbased argument or doctrine, or of a 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. saints, healthy, whence the Eng. sane, insanity, sanitation, &c. Lastly, there is a group of words which etymologists are inclined to treat as being all forms of the word which in O. Eng. is sund, meaning '
' swimming." These words are for (1) the swimbladder of a fish; (2) a narrow stretch of water between an inland sea and the ocean, or between an island and the mainland, &c., cf. Souxv, THE, below; (3) to test or measure the depth of anything, particularly the depth of water in lakes or seas (see SOUNDING, below). As a substantive the term is used of a surgical instrument for the exploration of a 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 umbra, beneath the shade, is given as a parallel.
hearing, and objectively the vibratory 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 science of Acoustics, we shall consider only the 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 ear, and which there produces sound, and the modes in which various 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 body whence the sound proceeds must have been thrown, by a blow or other means, into a 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 common glassjar be struck so as to yield an audible sound, the existence of a motion of this kind may be felt by the 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 cork put in the jar will be found to dance about during the continuance of the sound; water or spirits of wine poured into the glass will, under the same circumstances, exhibit a ruffled surface. The experiment is usually performed, in a more striking manner, with a belljar and a number of small light wooden balls suspended by silk strings to a fixed frame above the jar, so as to be just in contact with the widest part of the glass. On drawing a violin 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 eye the motions of the particles of the sounding body, as, for instance, in the case of a violin 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 Time to Travel.—If we watch a man breaking stones by the roadside some distance away, we can see the hammer fall before we hear the blow. We see the steam issuing from the whistle of a distant engine long before we hear the sound. We see lightning before we hear the thunder which spreads out from the flash, and the more distant the flash the longer the interval between the two. The wellknown rule of a mile for every five seconds between flash and peal gives a fair estimate of the distance of the lightning.
Sound needs a Material Medium to Travel Through.—In order that the ear may be affected by a sounding body there must be continuous matter reaching all the way from the body to the ear. This can be shown by suspending an electric bell in the receiver of an airpump, the wires conveying the current passing through an airtight cork closing the hole at the top of the receiver. These wires form a material channel from the bell to the outside air, but if they are 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 airpump experiment, before exhaustion it travels through the glass of the receiver and the base plate. We may easily realise its transmission through a solid by putting the ear against a table and scratching the wood at some distance, and through a liquid by keeping both ears under water in a bath and tapping the side of the bath.
Sound is a Disturbance of the Wave Kind.—As sound arises in general from vibrating bodies, as it takes time to travel, and as the medium which carries it does not on the whole travel forward, but subsides into its 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 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 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 rest of the air, and flowing into this space it is extended. It makes room for the next layer of air to move back and to be extended and so on, and an 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 representation of the motion of the layers of air in a train of sound waves by means of a device due to Crova and known as " Crova's disk." A small circle, say 2 or 3 mm. radius, is drawn on a card as in fig. 2, and round this circle equidistant points, say 8 or 12, are
taken. From these points as centres, circles are drawn in 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 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 running from the centre outwards, the wave motion is still more evident. If the figure be photographed as a lantern slide which is mounted so as to turn round, the wave motion is excellently shown on the screen, the compressions and extensions being represented by the crowding in and opening out of the lines.
Another illustration is afforded by a long spiral of wire with coils, say 2 in. in diameter and z in. apart. It may be hung up by threads so as to lie horizontally. If one end is sharply pressed in, a compression can be seen running along the spring.
The Disturbance in Sound Waves is Longitudinal.—The motion of a particle of air is, as represented in these illustrations, to and fro in the direction of 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 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 die away at a very short distance from the source. But transverse disturbances may be propagated as waves in solids. For instance, if a rope is fixed at one end and held in the hand at the other end, a transverse jerk by the hand will travel as a transverse 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 compound nature being easily seen in watching the motion of a floating particle.
Displacement Diagram.—We can represent waves of longitudinal displacement by a curve, and this enables us to draw very important conclusions in a very simple way. Let a train of waves be passing from left to right in the direction 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 :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 scale. At the points ABCD there is no displacement, and the line AD through these points is called the axis. Forward displacement is represented by height above the axis, backward displacement by depth below it. In ordinary sound waves the displacement is very minute, perhaps of the order io6 cm., so that we multiply it perhaps by 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 wavelength. 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.
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 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 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 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 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 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 elasticity, defined as increase of pressure = decrease of volume per unit volume where the pressure increase is so small that this ratio is 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 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 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, 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 wavelength 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. Half the total excursion is called the 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 energy or the intensity 'of the sound of given wavelength is proportional to the square of the amplitude.
The pitch of a sound, the 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 major third, its fifth and its 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 wavelength. We can only conclude that it depends on wave form, a conclusion fully borne out by investigation. The displacement curve of the waves from a tuningfork on its resonance box, or from the human 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 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 guide in conducting and interpreting the observations.
The waves from a source surrounded by a uniform medium at rest spread out as spheres with the source as centre. If we take one of these spheres a distance from the source very great as compared with a single wavelength, and draw a radius to a point on the 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 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. 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 chose, if we could apply any " external " force we liked to each particle, in addition to the " internal " force called into play by the compressions or extensions. For instance, if we have a wave with displacement curve of form 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 transfer of momentum across a surface occurs in two ways, firstly by the 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 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 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 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+wiP(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 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 equation (6) would give us the velocity of propagation if the 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 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 Newton (Principia, bk. ii., § 8, props. 4849). He supposed that in air Boyle's 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 formula). At the standard pressure of 76 cm. of 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 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 conduction keeps the temperature constant, we must use the adiabatic relation p=kpy, whence
dp/dp =7k01 = 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 Sir G. G. Stokes (Math. and Phys. Papers, iii. 142) showed that a very small departure from the adiabatic condition would lead to a stifling of the sound quite out of accord with observation.
If we put p=kp(iFat) in (8) we get the velocity in a 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 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 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 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(y1i)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 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 swing of the air particles in organ pipes. Thus Mach found an amplitude 0.2 cm. when the issuing waves were 250 cm. long. The amplitude in the 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 wavelengths. 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 report of a gun, the effect may be considerable, and the rumble of the thunder and the prolonged boom of the gun may perhaps be in part due to the breakdown of the wave when the crest of maximum pressure has moved up to the front, though it is probably due in part also to 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 Regnault in his classical experiments (described below) found that the velocity of the report of a 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 cannon in different directions; he found that it rose to a maximum of 1267 ft./sec. at 70, to 90 ft. in the rear and then fell off.
A very curious observation is recorded by the Rev. G. Fisher in an appendix to Captain Parry's Journal of a Second Voyage to the Arctic Regions. In describing experiments on the velocity of sound he states'that " on one day and one day only, February 9, 1822, the officer's word of command ' fire ' was several times heard distinctly both by Captain Parry and myself about one 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 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 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 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 column 1 sq. cm.
crosssection is W =f pu2dx. 0
like the breaking of a waterwave when the, crest gains on the next trough. In ordinary soundwaves 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 sin 2 (x—Ut). For a discussion of this type of wave,
see below.
\Ve have u= dt =  Ua cos  (x—Ut),
and fopu2dx=p4n2 C 27
j1:22a2J ocos2 (x—Ut)dx
= 2pa2U2a2/? (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 (Ann. der Physik, 1903, II, p. 405). A small circular disk at one end of a torsion arm formed part of a solid wall, but was 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. crosssection. 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 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 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 Lord 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 measures by a clock or chronometer the
time elapsing between the 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 wind. For
instance, William Derham (Phil. Trans., 1708) made a series
of observations, noting the time taken by the report of a cannon
fired on Blackheath to travel across the Thames to Upminster
Church in 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 commission of the French Academy (Memoires de l'academie des sciences, (1738). Cannons were fired at halfhour 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/(1Fo•oo184t), whence Uo=332 met./sec.
In 1822 a commission of the 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. Arago, Con;iai:sancc des temps, 1825).
When the measurement of a time interval depends on an observer, his " personal equation " comes in to affect the estimation of the quantity. This is the interval between the arrival of an event and his perception that it has arrived, or it may be the interval between arrival a:_d his 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 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 extraordinary care and ingenuity of Regnault's work, is given in the Memoires de l'academie des sciences, 1868, xxxvii. On page 459 of the Memoire will be found a list of previous careful experiments on the velocity of sound.
In the openair experiments the receiver consisted of a large 25
cone having a thin indiarubber membrane stretched over its narrow end. A small metal disk was attached to the centre of the membrane and connected to earth by a fine wire. A metal contactpiece adjustable by a screw could be made to just touch a point at the centre of the disk. When contact was made it completed an electric circuit which passed to a recording station, and there, by means of an electromagnet, actuated a style writing a record on a band of travelling smoked paper. On the same band a tuningfork 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 contactpiece, 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 tuningfork record. The receiving apparatus had what we may term a personal equation, for the break of contact could only take place when the membrane travelled some finite distance, exceedingly small no doubt, from the contactpiece. But the apparatus was used in such a way that this could be neglected. In some experiments in which contact was made instead of broken, Regnault determined the personal equation of the apparatus.
To eliminate wind as far as possible reciprocal firing was adopted, the interval between the two firings being only a few seconds. The temperature of the air traversed and its humidity were observed, and the result was finally corrected to the velocity in dry air at o° C. by means of equation (ro).
Regnault used two different distances, viz. 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. Stone at the Cape of Good Hope. In this experiment the personal equations of the observers were determined and allowed for.
Velocity of Sound in Air and other Gases in Pipes.—In the memoir cited above Regnault gives an account of determinations of the velocity in air in pipes of great length and of diameters ranging from o•ro8 metres to 1.1 metres. He used various sources and the method of electric registration. He found that in all cases the velocity decreased with a diameter. The sound travelled to and fro in the pipes several times before the signals died away, and he found that the velocity decreased with the intensity, tending to a limit for very feeble sounds, the limit being the same whatever the source. 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 court at the College de France, and with this length he could use dry air, vary the pressure, and fill with other gases. He found that within wide limits the velocity was independent of the pressure, thus confirming the theory. Comparing the velocities of sound Ui and U2 in two different gases with densities p, and p2 at the same temperature and pressure,
and with ratios of specific heats '2, theory gives
Ui/U2 = 11 {Yip2/Y2pi
This formula was very nearly confirmed for hydrogen, carbon dioxide and nitrous oxide.
J. Violle and T. Vautier (Ann. chim. phys., 189o, vol. 19) made observations with a tube o•7 M. in diameter, and, using Regnault's apparatus, found that the velocity could be represented by
331.3(1 P),
where P is the mean excess of pressure above the normal. According to von Helmholtz and Kirchhoff the velocity in a tube should be less than that in free air by a quantity depending on the diameter of the tube, the frequency of the note used, and the viscosity of the gas (Rayleigh, Sound, vol. ii. §§ 3478).
Correcting the velocity obtained in the 0•7 M. tube by Kirchhoff's formula, Violle and Vautier found for the velocity in open air at o° C.
Uo=331•Io met./sec.
with a probable error estimated at o•ro metre.
It is obvious from the various experiments that the velocity of sound in dry air at o° C. is not yet known with very great accuracy. At present we cannot assign a more exact value than
IIo = 331 metres per second.
Violle and Vautier made some later experiments on the propagation of musical sounds in a tunnel 3 metres in diameter (Ann. chim. phys., I9o5, vol. 5). They found that the velocity of propagation of different musical sounds was the same. Some curious effects were observed in the formation of harmonics in the rear of the primary tone used. These have yet to. find an explanation.
Velocity of Sound in Water.—The velocity in water was measured by J. D. Colladon and J. K. F. Sturm (Ann. chim. phys., 1827 (2), 36, p. 236) in the water of Lake Geneva. A bell under water was struck, and at the same instant some gunpowder was flashed in air above the bell. At a station more than 13 kilometres away a sort of big eartrumpet, closed by a membrane, was placed with the membrane under water, the tube rising above the surface. An observer with his ear to the tube noted the interval between the arrival of flash and sound. The velocity deduced at 8.1° C. was U=1435 met./sec., agreeing very closely with the value calculated from the formula U2 = E/p.
Experiments on the velocity of sound in iron have been made on lengths of iron piping by J. B. Biot, and on telegraph wires by Wertheim and Brequet. The experiments were not satisfactory, and it is sufficient to say that the results accorded roughly with the value given by theory.
Reflection of Sound.
When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached. Echo is a familiar example of this. The laws of reflection of sound are identical with those of the reflection of light, viz. (1) the planes of incidence and reflection are coincident, and (2) the angles of incidence and reflection are equal. Experiments may be made with plane and curved mirrors to verify these laws, but it is necessary to use short waves, in order to diminish diffraction effects. For instance, a ticking watch may be put at the focus of a large concave metallic mirror, which sends a parallel " beam " of sound to a second concave mirror facing the first. If an eartrumpet is placed at the focus of the second mirror the ticking may be heard easily, though it is quite inaudible by direct waves. Or it may be revealed by placing a sensitive flame of the kind described below with its nozzle at the focus. The flame jumps down at every tick.
Examples of reflection of sound in buildings are only too frequent. In large halls the words of a speaker are echoed or reflected from flat walls or roof or floor; and these reflected sounds follow the direct sounds at such an interval that syllables and words overlap, to the confusion of the speech and the annoyance of the audience.
Some curious examples of echo are given in Herschel's article on " Sound " in the Encyclopaedia Metropolitana, but it appears that he is in error in one case. He states that in the whispering gallery in St Paul's, London, " the faintest sound is faithfully conveyed from one side to the other of the dome but is not heard at any intermediate point." In some domes, for instance in a dome at the university of Birmingham, a sound from one end of a diameter is heard very much more loudly quite close to the other end of the diameter than elsewhere, but in St Paul's Lord Rayleigh found that " the abnormal loudness with which a whisper is heard is not confined to the position diametrically opposite to that occupied by the whisperer, and therefore, it would appear, does not depend materially upon the symmetry
of the dome. The whisper seems to creep round the gallery horizontally, not necessarily along the shorter arc, but rather along that arc towards which the whisperer faces. This is a consequence of the very unequal audibility of a whisper in front and behind the speaker, a phenomenon which may easily be observed in the open air" (Sound, ii. § 287).
Let fig. 12 represent a horizontal section of the dome through the source P. Let OPA be the radius through P. Let PQ represent a ray of sound making the angle B with the tangent at A. Let ON(=OP cos 0) be the perpendicular on PQ. Then the reflected ray QR and the ray reflected at R, and so on, will all touch the circle drawn with ON as radius. A ray making an angle less than 0 with the tangent will, with its reflections, touch a larger circle. Hence all rays between =0 will be confined in the space between the outer dome and a circle of radius OP cos 0, and the weakening of intensity will be chiefly due to vertical spreading.
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 earthquake disturbances is probably affected by the curvature of the surface of the globe, which may act like a whispering gallery.
In some cases of echo, when the original sound is a compound musical note, the octave of the fundamental tone is' reflected much more strongly than that tone itself. This is explained by Rayleigh (Sound, ii. § 296) as a consequence of the irregularities of the reflecting surface. The irregularities send back a scattered reflection of the different incident trains, and this scattered reflection becomes more copious the shorter the wavelength. Hence the octave, though comparatively feeble in the incident train, may predominate in the scattered reflection constituting the echo.
Refraction of Sound.
When a wave of sound travelling through one medium meets a second medium of a different kind, the vibrations of its own particles are communicated to the particles of the new medium, so that a wave is excited in the latter, and is propagated through it with a velocity dependent on the density and elasticity of the second medium, and therefore differing in general from the previous velocity. The direction, too, in which the new wave travels is different from the previous one. This change of direction is termed refraction, and takes place, no doubt, according to the same laws as does the refraction of light, viz. (I) The new direction or refracted ray lies always in the plane of incidence, or plane which contains the incident ray (i.e. the direction of the wave in the first medium), and the normal to the surface separating the two media, at the point in which the incident ray meets it; (2) The sine of the angle between the normal and the incident ray bears to the sine of the angle between the normal and the refracted ray a ratio which is constant for the same pair of media. 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 bent in towards the normal, or the reverse, according as the velocity of propagation in the former exceeds or falls short of that in the latter. Thus, for instance, sound is refracted towards the perpendicular when passing into air from water, or into carbonic acid gas from air; the converse is the case when the passage takes place the opposite way.
It further follows, as in the analogous case
of light, that there is a certain angle termed
the critical angle, whose sine is found by dividing the less by
the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflection back into the first body, if the velocity in that body is less than that in the other body, and if the angle of incidence exceeds the limiting angle.
The velocities in air and water being respectively 1090 and 4700 ft. the limiting angle for these media may be easily shown to be slightly above 15Y. Hence, rays of sound proceeding from a distant source, and therefore nearly parallel to each other, and to PO (fig..13), the angle POM being greater than 15a°, will not pass into the water at all, but suffer total reflection. Under such circumstances, the report of a gun, however powerful, should be inaudible by an ear placed in the water.
Acoustic Lenses.—As light is concentrated into a focus by a convex glass lens (for which the velocity of light is less than for the air), so sound ought to be made to converge by passing through a convex lens formed of carbonic acid gas. On the other hand, to produce convergence with water or hydrogen gas, in both which the velocity of sound exceeds its rate in air, the lens ought to be concave. These results have been confirmed experimentally by K. F. J. Sondhauss (Pogg. Ann., 1852, 85. p. 378), who used a collodion lens filled with carbonic acid. He found its focal length and hence the refractive index of the gas, C. Hajech (Ann. chim. phys., 1858, (iii). vol. 54) also measured the refractive indices of various gases, using a prism containing the gas to be experimented on, and he found that the deviation by the prism agreed very closely with the theoretical values of sound in the gas and in air.
Osborne Reynolds (Prot. Roy. Soc., 1874, 22, p. 53,) first pointed out that refraction would result from a variation in the temperature of the air at different heights. The velocity rai of sound in air is independent of the pressure, ijTempefracion Lion. e but varies with the temperature, its value at t° C.
being as we have seen
FIG; 14.
the upper part, moving faster, gains on the lower, and the front tends to swing round as shown by the successive positions in a 2, 3 and 4; that is, the sound tends to come down to the surface. This is well illustrated by the remarkable horizontal carriage of sound on a still clear frosty morning, when the surface layers of air are decidedly colder than those above. At sunset, too, after a warm day, if the air is still, the cooling of the earth by radiation cools the lower layers, and sound carries excellently over a level surface. But usually the lower layers are warmer than the upper layers, and the velocity below is greater than the velocity above. Consequently a wave front such as b 1 tends to turn upwards, as shown in the successive positions b 2, 3 and 4. Sound is then not so well heard along the level, but may still reach an elevated observer. On a hot summer's day the temperature of the surface layers may be much higher than that of the higher layers, and the effect on the horizontal carriage of sound may be very marked.
It is well known that sound travels far better with the wind than against it. Stokes showed that this effect is one of refraction, due to variation of velocity of the air Refraction from the surface upwards (Brit. Assoc. Rep., 1857, p. by wind. 22). It is, of course, a matter ofcommon observation
that the wind increases in velocity from the surface upwards. An excellent illustration of this increase was pointed out by F. Osier in the shape of old clouds; their upper portions always appear dragged forward and they lean over, as it were, in the
A
U=Uo(i+aat),
where U° is the velocity at o° C., and a is the coefficient of
expansion •00365. Now if the temperature is higher overhead
than at the surface, the velocity overhead is greater. If a wave
front is in a given position, as a 1 (fig. 14), at a given instant
_ • 3 4
Ill
direction in which the wind is going. The same kind of thing happens with soundwave fronts when travelling with the wind.
The velocity of any part of a wave front relative to the ground will be the normal velocity of sound + the velocity of the wind at that point. Since the velocity increases as we go upwards the front tends to swing round and travel downwards, as shown in the successive positions a 1, 2, 3 and 4, in fig. 14, where we must suppose the wind to be blowing from left to right. But if the wind is against the sound the velocity of a point of the wave front is the normal velocity—the wind velocity at the point, and so decreases as we rise. Then the front tends to swing round and travel upwards as shown in the successive positions b 1, 2, 3, and 4, in fig. 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 head and not be heard at all.
Diffraction of Sound Waves.
Many of the wellknown phenomena of optical diffraction may be imitated with sound waves, especially if the waves be short. Lord Rayleigh (Scientific Papers, iii. 24) has given various examples, and we refer the reader to his account. We shall only consider one interesting case of sound diffraction which may be easily observed. When we are walking past a fence formed by equallyspaced vertical rails or overlapping boards, we may often note that each footstep is followed by a musical ring. A sharp clap of the hands may also produce the effect. A short impulsive wave travels towards the fence, and each rail as it is reached by the wave becomes the centre of a new secondary wave sent out all round, or at any rate on the front side of the fence.
S'
Let S (fig. 15) be the source very nearly in the line of the rails ABCDEF. 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 wavelength 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 noise is an irregular disturbance. If we study the source producing it we find that there is no regularity of vibration. A musical note always arises from a source which has some regularity of vibration, and which sends equallyspaced waves into the air. A given note has always the same frequency, that is to say, the hearer receives the same number of waves per second whatever the source by which the note is produced. Various instruments have been devised which produce any desired note, and which are provided with methods of counting the frequency of vibration. The results obtained fully confirm the general law that " pitch," or the position of the note in the musical scale, depends solely on its frequency. We shall now describe some of the methods of determining frequency.
Savart's toothed wheel apparatus, named after Felix Savart (1791–1841), a French physicist and surgeon, consists of a brass wheel, whose edge is divided into a number of equal projectingteeth distributed uniformly over the circumference, and which is capable of rapid rotation about an axis perpendicular to its plane and passing through its centre, by means of a series of multiplying wheels, the last of which is turned round by the hand. The toothed wheel being set in motion, the edge of a card or of a funnelshaped piece of common notepaper is held against the teeth, when a note will be heard arising from the rapidly succeeding displacements of the air in its vicinity. The pitch of this note will rise as the rate of rotation increases,_ and becomes steady when that rotation is maintained uniform. It may thus be brought into unison with any sound of which it may be required to determine the corresponding number of vibrations per second, as for instance the note A3, three octaves higher than the A which is indicated musically by a small circle placed between the second and third lines of the G clef, which A is the note of the tuningfork usually employed for regulating concertpitch. A3 may be given by a piano. Now, suppose that the note produced with Savart's apparatus is in unison with A3, when the experimenter turns round the first wheel at the rate of 6o turns per minute or one per second, and that the circumferences of the various multiplying wheels are such that the rate of revolution of the toothed wheel is thereby increased 44 times, then the latter wheel will perform 44 revolutions in a second, and hence, if the number of its teeth be 8o, the number of taps imparted to the card every second will amount to 44X80 or 3520. This, therefore, is the number of vibrations corresponding to the note A3. If we divide this by 2' or 8, we obtain 440 as the number of vibrations answering to the note A. If, for the single toothed wheel, be substituted a set of four with a common axis, in which the teeth are in the ratios 4: 5: 6: 8, and if the card be rapidly passed along their edges, we shall hear distinctly produced the fundamental chord C, E, G, Ci and shall thus satisfy ourselves that the intervals C, E; C, G and C, Ci are 1, z and 2 respectively.
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 siren of L. F. W. A. Seebeck (1805–1849) is the simplest form of apparatus thus designated, and consists of a large circular disk mounted on a central axis, about which it may be made Seebeck's to revolve with moderate rapidity. This disk is per Shia
forated with small round holes arranged in circles
about the centre of the disk. In the first series of circles, reckoning from the centre the openings are so made as to divide the respective circumferences, on which they are found, in aliquot parts bearing to each other the ratios of the numbers 2, 4, 5, 6, 8, to, 12, 16, 20, 24, 32, 40, 48, 64. The second series consists of circles each of which is formed of two sets of perforations, in the first circle arranged as 4:5, in the next as 3:4, then as 2:3, 3:5, 4:7. In the outer series is a circle divided by perforations into four sets, the numbers of aliquot parts being as 3 : 4 : 5 : 6, followed by others which we need not further refer to.
The disk being started, then by means of a tube held at one end between the lips, and applied near to the disk at the other, or more easily with a common bellows, a blast of air is made to fall on the part of the disk which contains any one of the above circles. The current being alternately transmitted and shut off, as a hole passes on and off the aperture of the tube or bellows, causes a vibratory motion of the air, whose frequency depends on the number of times per second that a perforation passes the mouth of the tube. 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 spinningtop, a perforated card disk, and a tube for blowing with.
The siren of C. Cagniard de la Tour is founded on the same principle as the preceding. It consists of a cylindrical chest of brass, the base of which is pierced at its centre with an opening in which is fixed a brass tube projecting outwards, and Siren of intended for supplying the cavity of the cylinder with Cagniard de compressed air or other gas, or even liquid. The top is Tour. of the cylinder is formed of a plate perforated near its edge by holes distributed uniformly in a circle concentric with the plate, and which are cut obliquely through the thickness of the plate. Immediately above this fixed plate, and almost in contact with it, is another of the same dimensions, and furnished with the same number, n, of openings similarly placed, but passing obliquely through in an opposite direction from those in the fixed plate, the one set being inclined to the left, the other to the right.
This second plate is capable of rotation about an axis perpendicular to its plane and passing through its centre. Now, let the movable plate be at any, time in a position such that its holes are immediately above those in the fixed plate, and let the bellows by which air is forced into the cylinder (air, for simplicity, being
supposed to be the fluid employed) be put in action; then the air in its passage will strike the side of each opening in the movable
plate in an oblique direction (as shown in
fig. 16), and will therefore urge the latter
to rotation round its centre. 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 noncoincidence, during which the aircurrent 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 driving it round, during each revolution of the plate, through a space equal to the interval between two teeth. An index resembling the hand of a watch partakes of this motion, and points successively to the divisions of a graduated dial. On the completion of each revolution of this toothed wheel (which, if the number of its teeth be 100, will comprise Too revolutions of the movable plate), a projecting pin fixed to it catches a tooth of another toothed wheel and turns it round, and with it a corresponding index which thus records the number of turns of the first toothed wheel. As an example of the application of this siren, suppose that the number of revolutions of the plate, as shown by the indices, amounts to 5400 in a minute, that is, to 90 per second, then the number of vibrations per second of the note heard amounts to Son, or (if number of holes in each plate=8) to 720.
H. W. Dove (1803–1879) produced a modification of the siren by which the relations of different musical notes may be more
Doves readily ascertained. In it the fixed and movable plates
siren. are each furnished with four concentric series of per
forations, dividing the circumferences into different aliquot parts, as, for example, 8, lo, 12, 16. Beneath the lower or fixed plate are four metallic rings furnished with holes corresponding to those in the plates, and which may be pushed round by projecting pins, so as to admit the aircurrent through any one or more of the series of perforations in the fixed elate. Thus may be obtained, either separately or in various combinations, the four notes whose vibrations are in the ratios of the above numbers, and which therefore form the fundamental chord (CEGC1). The inventor has given to this instrument the name of the manyvoiced siren.
Helmholtz (Sensations of Tone, ch. viii.) further adapted the
siren for more extensive use, by the addition to Dove's instrument
Helmholtz's of another chest con
Double taining its own fixed
Siren, and movable perforated
plates and perforated rings, both the movable plates being driven by the same current and revolving about a common axis. Annexed is a figure of this instrument (fig. 17).
Graphic Methods.—The relation between the pitch of a note and the frequency of the corresponding vibrations has also been studied by graphic methods. Thus, if an elastic metal slip or a pig's bristle be attached to one prong of a tuningfork, and if the fork, while in vibration, is moved rapidly over a glass plate coated with lampblack, the attached style touching the plate lightly, a wavy line will be traced on the plate answering to the vibrations to and fro of the
tained with a stationary fork and a movable glass plate; and, if the time occupied by the plate in moving through a given distance can be ascertained and the number of complete undulations exhibited on the plate for that distance, which is evidently the number of vibrations of the fork in that time, is reckoned, we shall have determined the numerical vibrationvalue of the note yielded by the fork. Or, if the same plate be moved in contact with two tuningforks, we shall, by comparing the number of sinuosities in the one trace with that in the other, be enabled to assign the ratio of the corresponding numbers of vibrations per second. Thus, if the one note be an octave higher than the other, it will give double the number of waves in the same distance. The motion of the plate may be simply produced by dropping it between two vertical grooves, the tuningforks being properly fixed to a frame above.
Greater accuracy may be attained with a revolvingdrum chronograph first devised by Thomas Young (Led. on Nat. Phil., 1807, i. 190), consisting of a cylinder which may be coated The Revolwith lampblack, or, better still, a metallic cylinder T e Revol round which a blackened sheet of paper is wrapped. v'ng . The cylinder is mounted on an axis and turned round, while the style attached to the vibrating body is in light contact with it, and traces therefore a wavy circle, which, on taking off the paper and flattening it, becomes a wavy straight line. The superiority of this arrangement arises from the comparative facility with which the number of revolutions of the cylinder in a given time may be ascertained. In R. Koenig's arrangement (Quelques experiences d'acoustique, p. 1) the axis of the cylinder is fashioned as a screw, which works in fixed nuts at the ends, causing a sliding as well as a rotatory motion of the cylinder. The lines traced out by the vibrating pointer are thus prevented from overlapping when more than one turn is given to the cylinder. In the phonautograph of E. L. Scott (Comptes rendus, 1861, 53, p. 108) any sound whatever may be made to record its trace on the paper by means of a large parabolic cavity resembling a speakingtrumpet, which is freely open at the wider extremity, but is closed at the other end by a thin stretched membrane. To the centre of this membrane is attached a small featherfibre, which, when the reflector is suitably placed, touches lightly the surface of the revolving cylinder. Any sound (such as that of the human voice) transmitting its rays into the reflector, and communicating vibratory motion to the membrane, will cause the feather to trace a sinuous line on the paper. If, at the same time, a tuningfork of known number of vibrations per second be made to trace its own line close to the other, a comparison of the two lines gives the number corresponding to the sound under consideration. The phonograph (q.v.) may be regarded as an instrument of this class, in that it records vibrations on a revolving drum or disk.
Lissajous Figures.—A mode of exhibiting the ratio of the frequencies of two forks was devised by Jules Antoine Lissajous (1822–1880). On one prong of each fork is fixed a small plane mirror. The two forks are fixed so that one vibrates in a vertical, and the other in a horizontal, plane, and they are so placed that a converging beam of light received on one mirror is reflected to the other and then brought to a point on a screen. If the first fork alone vibrates, the point on the screen appears lengthened out into a vertical line through the changes in inclination of the first mirror, while if the second fork alone vibrates, the point appears lengthened out into a horizontal line. If both vibrate, the point describes a curve which appears continuous through the persistence of the retinal impression. Lissajous also obtained the figures by aid of the vibration microscope, an instrument which he invented. Instead of a mirror, the objective of a microscope is attached to one prong of the first fork and the eyepiece of the microscope is fixed behind the fork. Instead of a mirror the second fork carries a bright point on one prong, and the microscope is focused on this. If both forks vibrate, an observer looking through the microscope sees the bright point describing Lissajous figures. If the two forks have the same frequency, it is easily seen that the figure will be an ellipse (including as limiting cases, depending on relative amplitude and phase, a circle and a straight line). If the forks are not of exactly the same frequency the ellipse will slowly revolve, and from its rate of revolution the ratio of the frequencies may be determined (Rayleigh, Sound, i. § 33). If one is the octave of the other a figure of 8 may be described, and so on. Fig. 18 shows curves given by intervals of the octave, the twelfth and the fifth.
The kaleidophone devised by Charles Wheatstone in 1827 gives these figures in a simple way. It consists of a straight rod clamped in a vice and carrying a bead at its upper free end. The bead is illuminated and shows a bright point of light. 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. hours. By comparison with a clock of known rate
the total number of vibrations of the fork in any time may be accurately determined. One prong of the fork carries a microscope objective, part of a vibration microscope, of which the eyepiece is fixed at the back of the clock and the Lissajous figure
made by the clock fork and any other fork may be observed. With this apparatus Koenig studied the effect of temperature on a standard fork of 256 frequency, and found that the frequency decreased by 0.0286 of a vibration for a rise of I°, the frequency being exactly 256 at 26.2° C. Hence the frequency may be put as 256 11—0.000113 (l—26.2)1.
(From Lord Rayleigh's Theory of Sound, by permission of Macmillan & Co., Ltd.) Flo. 18.
Koenig also used the apparatus to investigate the effect on the frequency of a fork of a resonating cavity placed near it. He found that when the pitch of the cavity was below that of the fork the pitch of the fork was raised, and vice versa. But when the pitch of the cavity was exactly that of the fork when vibrating alone, though it resounded most strongly, it did not affect the frequency of the fork. These effects have been explained by Lord Rayleigh (Sound, i. § 117).
In the stroboscopic method of H. M'Leod and G. S. Clarke, the
full details of which will be found in the original memoir (Phil.
Trans., 188o, pt. i. p. I), a cylinder is ruled with equi
M'Leodand distant white lines parallel to the axis on a black
Clarke's ground. It is set so that it can be turned at any de
Strobo sired and determined speed about a horizontal axis,
scopk: and when going fast enough it appears grey. Imagine
Method. now that a fork with black prongs is held near the cylinder with its prongs vertical and the plane of vibration parallel to the axis, and suppose that we watch the outer outline of the righthand prong. Let the cylinder be rotated so that each white line moves exactly into the place of the next while the prong moves once in and out. Hence when a white line is in a particular position on the cylinder, the prong will always be the same distance along it and cut off the same length from view. The most will be cut off in the position of the lines corresponding to the furthest swing out, then less and less till the furthest swing in, then more and more till the furthest swing out, when the appearance will be exactly as at first. The boundary between ,the grey cylinder and the black fork will therefore appear wavy with fixed undulations, the distance from crest to crest being the distance between the lines on the cylinder. If the fork has slightly greater frequency, then a white line will not quite reach the next place while the fork is making its swing in and out, and the waves will travel against the motion of the cylinder. If the fork has slightly less frequency the waves will travel in the opposite direction, and it is easily seen that the frequency of the fork is the number of
white lines passing a point in a second the number of waves passing the point per second. This apparatus was used to find the temperature coefficient of the frequency of forks, the value obtained—•000it being the same as that found by Koenig. Another important result of the investigation was that the phase of vibration of the fork was not altered by bowing it, the amplitude alone changing. The method is easily adapted for the converse determination of speed of revolution when the frequency of a fork is known.
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 softiron armatures fixed at equal distances
round its circumference. The wheel rotates between
Phonic
Wheel. the poles of an electromagnet, which is fed by an
intermittent current such as that which is working an electrically maintained tuningfork (see infra). If the wheel be driven at such rate that the armatures move one place on in about the period of the current, then on putting on the current the electromagnet controls the rate of the wheel so that the agreement of period is exact, and the wheel settles down to move so that the electric driving forces just supply the work taken out of the wheel.
If the wheel has very little work to do it may not be necessary to apply driving power, and uniform rotation may be maintained by the electromagnet. In an experiment described by Rayleigh such a wheel provided with four armatures was used to determine the exact frequency of a driving fork known to have a frequency near 32. Thus the wheel made about 8 revolutions per second. There was one opening in its disk, and through this was viewed the pendulum of a clock beating seconds. On the pendulum was fixed an illuminated silver bead which appeared as a bright point of light when seen for an instant. Suppose now an observer to be looking from a fixed point at the bead through the hole in the phonic wheel, he will see the bead as 8 bright points flashing out in each beat, and in succession at intervals of a second. Let us suppose that he notes the positions of two of these next to each other in the beat of the pendulum one way. If the fork makes exactly 32 vibrations and the wheel 8 revolutions in one pendulum beat, then the positions will be fixed, and every two seconds, the time of a complete pendulum vibration, he will see the two positions looked at flash out in succession at an interval of s second. 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 swellingsout 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+mlhm2+ . . . + mN2f, 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 character its reflection is viewed in a revolving mirror. For this purpose four vertical mirrors are arranged round the vertical sides of a cube which is rapidly revolved about a vertical axis. The flame then appears toothed as shown. If several notes are present the flame is jagged by each. Interesting results are obtained by singing the different vowels into a funnel substituted for the resonator in the figure.
448
If two such flames are placed one under the other they may be excited by different sources, and the ratio of the frequencies may be approximately determined by counting the number of teeth in each in the same space.
The Diatonic Scale.
It is not necessary here to deal generally with the various musical scales. We shall treat only of the diatonic scale, which is the basis of European music, and is approximated to as closely as is consistent with convenience of construction in keyboard instruments, such as the piano, where the eight white notes beginning with C and ending with C an octave higher may be taken as representing the scale with C as the keynote.
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 keynote, the frequency ratios with the keynote, 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 fourth from C to F and then through a third from F to A, the frequency ratio of A to C is i, which is the product of the ratios for a fourth , and a third . That is, if we add intervals we must multiply frequency ratio§ to obtain the frequency ratio for the interval which is the sum of the two.
The frequency ratios in the diatonic scale are all expressible either as fractions, with 1, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.
Thus the second = 2 X X a, and we may regard it as an ascent through two fifths in succession and then a descent through an octave. The third t= 5X 1 X z or ascent through an interval , which has no special name, and a descent through two octaves, and so on.
Now suppose we take G as the keynote and form its diatonic scale. If we write down the eight notes from G to g in the key of C, their frequency ratios to C, the frequency ratios required by the diatonic scale for G, we get the frequency ratios required in the last line:
Notes on scale of C G A B c d e f g
Frequency ratios with C = I . . . $ s is" 2 4 2 3
Frequency ratios of diatonic scale I a s ~5 2
with G= 1 . ...
Frequency ratios with C=I, G=y . i[ ras 2 2 1s 3
We see that all but two notes coincide with notes on the scale of C. But instead of A = i we have Imo, and instead of f = we have I. The interval between and ft = = ft is termed a " comma," and is so small that the same note on an instrument may serve for both. But the interval between 1 and
H = 141 is quite perceptible, and on the piano, for instance, a separate string must be provided above f. This note is f sharp, and the interval I is termed a sharp.
Taking the successive keynotes D, A, E, B, it is found that besides small and negligible differences, each introduces a new sharp, and so we get the five sharps, C, D, F, G, A, represented nearly by the black keys.
If we start with F as keynote, besides a small difference at d, we have as the fourth from it * X = , making with B = R an interval +i;, and requiring a new note, B flat. This does not coincide with A sharp which is the octave below the seventh from B or Y X X = M. It makes with[DIATONIC SCALE
it an interval = V =  = ffii, rather less than a comma; so that the same string in the piano may serve for both. If we take the new note B flat as keynote, another note, E flat, is required. E flat as keynote introduces another flat, and so on, each flat not quite coinciding with a sharp but at a very small interval from it.
It is evident that for exact diatonic scales for even a limited number of keynotes, keyboard instruments would have to be provided with a great number of separate strings or pipes, and the corresponding keys would be required. 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 compromise has to be made, and the note has to be tuned so as to make the compromise as little unsatisfactory as possible. At present twelve notes are used in the octave, and these are arranged at equal intervals 2. This is termed the equal temperament scale, and it is obviously only an approach to the diatonic scale.
Helmholtz's Notation.—In works on sound it is usual to adopt Helmholtz's notation, in which the octave from bass to middle C is written c d e f g a b c'. The octave above is c' d' e' f' g' a' b' c". The next octave above has two accents, and each succeeding octave another accent. The octave below bass C is written C D E F G A B c. The next octave below is CI D1 El F1 GI Al B1 C, and each preceding octave has another accent as suffix.
The standard frequency for laboratory work is c=128, so that middle C'=256 and treble c" = 512.
The standard for musical instruments has varied (see PITCH, MUSICAL). Here it is sufficient to say that the French standard is a' =435 with c" practically 522, and that in England the pitch is somewhat higher.
The French notation is as under:
CD E F G A B c
Ut1 Rel. MI Fai Soli Lai Si, Ut2.
The next higher octave has the suffix 2, the next higher the suffix 3, and so on. French forks are marked with double the true frequency, so that Uts is marked 512.
Limiting Frequencies for Musical Sounds.—Until the vibrations of a source have a frequency in the neighbourhood of 30 per second the ear can hear the separate impulses, if strong enough, but does not hear a note. ,It is not easy to determine the exact point at which the impulses fuse into a continuous tone, for higher tones are usually present with the deepest of which the frequency is being counted, and these may be mistaken for it. Helmholtz (Sensations of Tone, ch. ix.) used a string loaded at the middle point so that the higher tones were several octaves above the fundamental, and so not likely to be mistaken for it; he found that with 37 vibrations per second a very weak sensation of tone was heard, but with 34 there was scarcely anything audible left. A determinate musical pitch is not perceived, he says, till about 40 vibrations per second. At the other end of the scale with increasing frequency there is another limiting frequency somewhere about 20,000 per second, beyond which no sound is heard. But this limit varies greatly with different individuals and with age for the same individual. Persons who when young could hear the squeaks of bats may be quite deaf to them when older. Koenig constructed a series of bars forming a harmonicon, the frequency of each bar being calculable, and he found the limit to be between 16,00o and 24,000.
The Number of Vibrations needed to give the Perception of Pitch.—Experiments have been made on this subject by various workers, the most extensive by W. Kohlrausch (Wied. Ann., 188o, x. I). He allowed a limited number of teeth on the arc of a circle to strike against. a card. With sixteen teeth the pitch was well defined; with nine teeth it was fairly determinate; and even with two teeth it could be assigned with no great error. His remarkable result that two waves give some sense of pitch, in fact a tone with wavelength equal to the interval between the waves, has been confirmed by other observers.
Alteration of Pitch with Motion of Source or Hearer: Doppler's Principle.—A very noticeable illustration of the alteration of pitch by motion occurs when a whistling locomotive moves rapidly past an observer. As it passes, the pitch of the whistle falls quite appreciably. The explanation is simple. The engine follows up any wave that it has sent forward, and so crowds up the succeeding waves into a less distance than if it remained at rest. 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 wavelength, 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 sixth
Frequency 199 $ pies 2
Successive {re is 1 6 S 18 $ 1
quency ratios.
Successive in major minor major major minor major major
tervals . . tone tone semi tone tone tone semi
tone tone
may be treated very easily if the motions are all in the line joining source and receiver. Let S (fig. 20) be the source at a given instant, and let its frequency of vibration, or the number of waves it sends out per second, be n. Let S' be its position one second later, its velocity being u. Let R be the receiver at a given instant, R' its position a second later, its velocity being v. Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.
+ + , +
I FIG. 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 pitchpipe 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 pitchpipe 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 reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y=a sin m(x—e). If we put this in the form
2'r
y=a sin (x—e), we see that y=o, for x=e, e+iX, e+IX,
e+IIX, and so on, that y is + from x=e to x=e+4X, —from e+zX to e+ X, and so on, and that it alternates between the values+a and—a.
The form of the curve is evidently as represented in fig. 21, and it may easily be drawn to exact scale from a table of sines.
K FIG. 2I.
In this curve ABCD are nodes. OA=e is termed the epoch, being the distance from 0 of the first ascending node. AC is the shortest distance after which the curve begins to repeat itself ; this length X is termed the wavelength. 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 = 3cos—(x—Ut).
dt — X X
The acceleration perpendicular to the axis is
z 2a
dtz = 4rr sin X (x—Ut)
47r2U2   X2 Y
which is an equation characteristic of simple harmonic motion.
The maximum velocity of a particle in the wavetrain is the amplitude of dy/dt. It is, therefore,
= 2 irU a/X = 22rna. (19) XXV. 15
The maximum pressure excess is the amplitude of =Eu/U _ (E/U)dy/dt. It is therefore
=(E1U)2zrUa/X=2irnpUa. (20) We have already found the energy density in the train and the energy stream in equations (13) and (14).
The chief experimental basis for supposing that a train of longitudinal waves with displacement curve of this kind arouses the sensation of a pure tone is that the more nearly a source is made to vibrate with a single simple harmonic motion, and therefore, presumably, the more nearly it sends out such a harmonic train, the more nearly does the note heard approximate to a single pure tone.
Any periodic curve may be resolved into sine or harmonic curves by Fourier's theorem. Fourier's
Suppose that any periodic sound disturbance, consist Theorea. ing of plane waves, is being propagated in the
direction ABCD (fig. 22). Let it be represented by a displacement curve AHBKC. Its periodicity implies that after a certain distance the displacement curve exactly repeats itself. Let AC be the
H L
shortest distance after which the repetition occurs, so that CLDME is merely AHBKC moved on a distance AC. Then AC=X is the wavelength or period of the curve. Let ABCD be drawn at such level that the areas above and below it are equal; then ABCD is the axis of the curve. Since the curve represents a longitudinal disturbance in air it is always continuous, at a finite distance from the axis, and with only one ordinate for each abscissa.
Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wavelengths X, IX, IX, III... if the amplitudes a, b, c,..and the epochs e, f, g.... are suitably adjusted, and the proof of the theorem gives rules for finding these quantities when the original curve is known. We may therefore put
y=a sin T (x—e)+b sin 41 (x—f)+c sin 6 (x—g)+&c. (21)
where the terms may be infinite in number, but always have wavelengths submultiples of the original or fundamental wavelength X. Only one such resolution of a given periodic curve is possible, and each of the constituents repeats itself not only after a distance equal to its own wavelength X/n, but evidently also after a distance equal to the fundamental wavelength X. The successive terms of (21) are called the harmonics of the first term.
It follows from this that any periodic disturbance in air can be resolved into a definite series of simple harmonic disturbances of wavelengths equal to the original wavelength and its successive submultiples, and each of these would separately give the sensation of a pure tone. If the series were complete we should have terms which separately would correspond to the fundamental, its octave, its twelfth, its double octave, and so on. Now we can see that two notes of the same pitch, but of different quality, or different form of displacement curve, will, when thus analysed, break up into a series having the same harmonic wavelengths; but they may differ as regards the members of the series present and their amplitudes and epochs. We may regard quality, then, as determined by the members of the harmonic series present and their amplitudes and epochs. It may, however, be stated here that certain experiments of Helmholtz appear to show that the epoch of the harmonics has not much effect on the quality.
Fourier's theorem can also be usefully applied to the disturbance of a source of sound under certain conditions. The nature of these conditions will be best realized by considering the case of a stretched string. It is shown below how the vibrations of a string may be deduced from stationary waves. Let us here suppose that the string AB is displaced into the form AHB (fig. 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 wavelength 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 wavelengths 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 equivalent to saying that all disturbances will travel along the system with the same velocity. 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 correspondence with this Resonators. geometrical analysis. Using the term " note " for the
sound produced by a periodic disturbance, there is no doubt that a welltrained ear can resolve a note into pure tones of frequencies equal to those of the fundamental and its harmonics. If, for instance, a note is struck and held down on a piano, a little practice enables us to hear both the octave and the twelfth with the fundamental, especially if we have previously directed our attention to these tones by sounding them. But the harmonics are most readily heard if we fortify the ear by an air cavity with a natural period equal to that of the harmonic to be sought. The form used by Helmholtz is a glove of thin brass (fig. 24) with a large hole at one end of a diameter, at the other end of which the brass is drawn out into a short, narrow tube that can be put close to the ear. But a cardboard tube closed at one end, with the open end near the ear, will often suffice, and it may be tuned by more or less covering up the open end. If the harmonic corresponding to the resonator is present its tone swells out loudly.
This resonance is a particular example of the general principle that a vibrating system will be set in vibration by any periodic Forced VI force applied to it, and ultimately in the period of the :mafiosi and force, its own natural vibrations gradually dying down. Resonan,:e. 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 friction or through waveemission, so that the vibration only increases up to the point at which the leakage of energy balances the energy put in by the applied force. Further, the greater the dissipation of energy the less is the prominence of the amplitude of vibration for exact coincidence over the amplitude when the periods are not quite the same, though it is still the greatest for coincidence.
The principle of forced vibration may be illustrated by a simple case. Suppose that a mass M is controlled by some sort of spring, so that moving freely it executes harmonic vibrations given by MX = —µx, where ux is the restoring force to the centre of vibration. Putting µ/M =n2 the equation becomes k}n2x=o, whence x=A sin nt, and the period is 24n.
Now suppose that in addition to the internal force represented by —µx, an external harmonic force of period tar/p is applied. Representing it by —P sin pt, the equation of motion is now
in2x+m sin pt=o.
Let us assume that the body makes vibrations in the new period tarp, and let us put x=B sin pt; substituting in (22) we have —p2B+n2B+P/M =o, whence
B = M
P
p21 n2J
and the " forced " oscillation due toP sin pt is
x= P sin bt
M p2n2
(23)
If p> n the motion agrees in phase with that which the applied force alone would produce, obtained by putting n =o. If p 

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