# 1911 Encyclopædia Britannica/Sound

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​SOUND, subjectively the sense impression of the organ of ​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 glass-jar 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 bell-jar 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 lakes 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 well-known 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 air-pump, the wires conveying the current passing through an air-tight 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 air-pump 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.
Fig. 1.
Let AB (fig. 1) 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 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 were pulled back by AB. Really, 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
Fig. 2.
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 1/2 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 ​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 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.
Fig. 3.
Thus let the particle which was at L be at l, 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 Mm. 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 10$^{−5}$ cm., so that we multiply it perhaps by 100,000 in forming the displacement curve.
Wave Length and Frequency.—If the waves are continuous and each of the same shape they form a “train,” and the displacement curve repeats itself. The shortest distance in which this repetition occurs is called the wave-length. It is usually denoted by λ. In fig. 3, AC＝λ. 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 second must occupy a length U and therefore U＝nλ.
Distribution of Compression and Extension in a Wave.—Let fig. 4 be the displacement diagram of a wave travelling from left to right.
Fig. 4.
At A the air occupies its original position, while at H it 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.
Fig. 5.
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.
Fig. 6.
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 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 1 has increased to 1+dy/dx and the increase of volume 1 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, ω the small increase of pressure, and −(dy/dx) the volume decrease, then
| E＝ω/(−dy/dx) or ω/E＝−dy/dx) | (1) |
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 (1) and (2)
| u/U＝ω/E | (3) |
which gives the particle velocity in terms of the pressure excess.
Generally, if any condition φ in the wave is carried forward unchanged with velocity U, the change of φ at a given point in time dt is equal to the change of φ as we go back along the curve a distance dx = Udt at the beginning of dt.
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 wave-length depends on the extent of the to and fro excursion of the air particles. This is obvious if we consider that the greater the vibration of the source the greater is the excursion of the air in the issuing waves, and the louder is the sound heard. 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 wave-length 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 ​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 wave-length. 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 tuning-fork 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.
Fig. 7.
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 wave-length, 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., 1870, 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
Fig. 8.
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
Fig. 9.
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 ρ$_0$ the mass moving in through a square centimetre is poll. But it has velocity U, and therefore momentum ρ$_0$U$^2$ 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+ρ$_0$U$^2$. 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 ρ(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 ῶ, so that the total “internal” pressure is P+ῶ. There is also the " external " applied pressure X, and the total momentum flowing out per second is
X+P+ῶ+ρ(U−u)$^2$.
Equating this to the momentum entering at B and subtracting P from each
| X+ῶS+ρ(U−u)$^2$＝ρ$_0$U$^2$. | (4) |
If y is the displacement at A, and if E is the elasticity, substituting for ῶ and u from (2) and (3) we get
X − Edy/dx + ρU$^2$(1+dy/dx)＝ρ$_0$U$_2$
But since the volume dx with density ρ$_0$ has become volume dx+dy with density ρ
ρ(1 + dy/dx)＝ρ$_0$.
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 − ρ$_0$(U−u)$^2$＝0, then X＝0, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity
| U＝√(E/ρ). | (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) = ρ(dp/dp) equation (6) becomes
| U＝√(dp/dρ). | (7) |
The value U = V(E/ρ) was first virtually obtained by Newton (Principia, bk. ii., § 8, props. 48–49). He supposed that in air Boyle's law holds in the extensions and compressions, or that p＝kρ, whence dp/dp＝k＝p/ρ. His value of the velocity in air is therefore
At the standard pressure of 76 cm. of mercury or 1,014,000 dynes / sq. cm., the density of dry air at 0° C. being taken as 0·001293, we get for the velocity in dry air at 0° C.
​approximately. Newton found 979 ft./sec. But, as we shall see, all the determinations give a value of U$_0$ in the neighbourhood of 33,000 cm./sec., or about 1080 ft./sec. This discrepancy was not explained till 1816, when Laplace (Ann. 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 = kργ, whence
dp/dρ＝γkργ−$^1$＝γp/ρ,
If we take γ＝1·4 we obtain approximately for the velocity in dry air at 0° C .
U$_0$＝33,150 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＝kρ(1+αt) in (8) we get the velocity in a gas at t° C,
U$_t$＝√γk(1+αt)}.
At 0° C. we have U$_0$＝√(γk), and hence U$_t$＝U$_0$√(1+αt)
＝U$_0$(1 +0·00184t) (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 γ, but with densities at the same pressure and temperature respectively ρ$_1$ and ρ$_2$, we should have
| U$_1$/U$_2$＝√(ρ$_2$/ρ$_1$), | (10) |
another result confirmed by observation.
Alteration of Form of the Waves when Pressure Changes are Considerable.—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 γP. This can be seen by considering that the relation between p and ρ 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
ω+ρ(U−u)$^2$＝ρ$_0$U$^2$.
Let us seek a more exact value for ω. If when P changes to P+ω volume V changes to V−v then (P+ω)(V−v)$^γ$ = PV$^γ$,
. t>( v , 7(7 + 1) v 2 \ Pd / 7 + 1 v \ whence M = P^+ ^J =y ^ \ l +^~ y) â–
We have U−u＝U(1−u/U)＝U(1−v/V) since u/U＝−dy/dx＝v/V. Also since ρ(V−v)＝ρ$_0$V, or ρ＝ρ$_0$/(1−v/V), then ρ(U−u)$^2$＝ Vρ$_0$U$^2$(1−v/V).
Substituting in the momentum equation, we obtain
γP/ T 1 7 + 1 . . .
If U = √(γP/ρ$_0$) is the velocity for small disturbances we may put U$_0$ for U in the small term on the right, and we have
-L t )=U(i-p/V), since Â«/U = -dy/dx = vfV. = p V, or p = p /(l-p/V), then p(U-w) 2 =
whence
U$^2$＝. . .
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 when 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＝0, 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. 10) represents the displacement curve of a train of waves, b will represent the pressure excess and particle velocity, and from (11) we see that while the nodal conditions of b, with ῶ＝0 and u＝0, travel with velocity √(E/ρ), the crests exceed that velocity by 1/4(γ + 1)u. and the hollows fall short of it by 1/4(γ + 1)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 like the breaking of a water-wave when the, crest gains on the next trough. In ordinary sound-waves the effect of the particle velocity in affecting the velocity of transmission must be very small.
Fig. 10.
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 0·1 mm. in the waves—a very extreme value. The maximum particle velocity is 2πna (where n is the frequency and a the amplitude), or 2πaU/λ. This gives maximum u＝about 8 cm./sec, which would not seriously change the form of the wave in a few wave-lengths. Meanwhile the waves are spreading out and the value of u is falling in inverse proportion to the distance from the source, so that very soon its effect must become negligible.
In loud sounds, such as a peal of thunder from a near flash, or the 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 (11), for at the very front of the disturbance u＝0 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 _of 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 1/2ρu$^2$ , where ρ is the density and u 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+ῶ, the average pressure during the change may be taken as P + 1/2ῶ, since the pressure excess for a small change is proportional to the change. Hence the work done on the air is (P + 1/2ῶ)v, and the work done per cubic centimetre is (P+Jffl)p/V. The term Pv/V added up for a complete wave vanishes, for P/V is constant and Σv＝0, since on the whole the compression equals the extension. We have then only to consider the term 1/2ῶv/V.
| But v/V | ＝u/U from equation (2) |
| and ῶ | ＝Eu/U from equation (3) |
| Then 1/2ῶv/V | ＝1/2Eu$^2$/U$^2$ = 1/2ρu$^2$ 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. cross-section is W＝$\int _{0}^{\lambda }$ρu$^2$dx. ​
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 2π/λ(x − Ut). For a discussion of this type of wave, see below.
and
$\int _{0}^{\lambda }$ ρu$^2$dx＝ρ4π$^2$U$^2$a$^2$/λ$^2$ $\int _{0}^{\lambda }$cos$^2$2π/λ(x−Ut)dx
| ＝2ρπ$^2$U$^2$a$^2$/λ | (12) |
The energy per cubic centimetre on the average is
| 2ρπ$^2$U$^2$a$^2$/λ$^2$ | (13) |
and the energy passing per second through 1 sq. cm. perpendicular to the line of propagation is
| 2ρπ$^2$U$^3$a$^2$/λ$^2$ | (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, 11, 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 inthewall 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. 11), and that they are absorbed there without reflection. Fig. 11. Let ABCD be a column of air 1 sq. cm. cross-section. The pressure on CD is equal to the momentum which it receives per second. On the whole the air within ABCD neither gains nor loses momentum, so that on the whole it receives as much through AB as it gives up to CD. Fig. 11. If P is the undisturbed pressure and P+S the pressure at AB, the momentum entering through AB per second is $\int _{0}^{1}$(P+ῶ+ρu$^2$ )dt. But $\int _{0}^{1}$Pdt = 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 $\int _{0}^{1}$(ῶ+ρu$^2$)dt. But the values of ῶ+ρu$^2$ 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 dξ is an element of the path, putting dt＝dξ/U, we have the average excess of pressure
p = $\int _{0}^{1}$(ῶ+....
Here dξ is an actual length in the disturbance. We have fi and u expressed in terms of the original length dx and the displacement dy so that we must put dξ＝dx+dy＝(1 + dy/dx)dx, and
p＝. . .
We have already found that if V changes to V−v
ῶ＝....
since r/V＝− dy/dx.
We also have ρu$^2$ = ρ$_0$u$^2$/(1 + dy/dx). Substituting these values and neglecting powers of dy/dx above the second we get
But $\int _{0}^{\text{U}}{\frac {dy}{dx}}dx=0$ since the sum of the displacements＝0. Then putting (dy/dx)$^2$ = (u/U)$^2$ , we have
| ＝1/2(γ + 1) average energy per cubic centimetre, | (15) |
a result first published by Lord Rayleigh (Phil. Mag., 1905, 10, 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, 125 m. away. He found that the time varied between 55J seconds when the wind was blowing most strongly with the sound, to 6$ 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 T and T 2 be the observed times of passage in the two directions. We have U+w = D/T$_1$ and U−w＝D/T$_2$. Adding and dividing by 2
U＝D/2(1/T$_1$+1/T$_2$).
If T$_1$ and T$_2$ are nearly equal, and if T = 1/2(T$_1$+T$_2$), this is very nearly U = D/T.
The reciprocal method was adopted in 1738 by, a commission of the French Academy (Mémoires de l'Académie des sciences, (1738). Cannons were fired at half-hour intervals, alternately at Montmartre and Montlhery, 17 or 18 m. apart. There were also two intermediate stations] at which observations were made. The times were measured by pendulum clocks. The result obtained at a temperature about 6° C. was, when converted to metres, U＝337 metres/second.
The theoretical investigation given above shows that if U is the velocity in air at tÂ° C. then the velocity U$_0$ at 0° C. in the same air is independent of the barometric pressure and that U$_0$＝U/(1+0·00184t), whence U$_0$ = 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 = 340·9 met. /sec, whence U$_0$=330·6 met. /sec. (F. J. D. Arago, Connaissance 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 ariival of an event and his perception that it has arrived, or it may be the interval between arrival and 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 extra- ordinary care and ingenuity of Regnault's work, is given in the Mémoires de l'académie des sciences, 1868, xxxvii. On page 459 of the Mémoire will be found a list of previous careful experiments on the velocity of sound.
In the open-air experiments the receiver consisted of a large ​ cone having a thin india-rubber 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 contact-piece 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 electro-magnet, actuated a style writing a record on a band of travelling smoked paper. On the same band a tuning-fork electrically maintained and a seconds clock actuating another style wrote parallel records. The circuit was continued to the gun which served as a source, and stretched across its muzzle. When the gun was fired, the circuit was broken, and the break was recorded on the paper. The circuit was at once remade. When the wave travelled to the receiver it pushed back the disk from the contact-piece, and this break, too, was recorded. The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuning-fork record. The receiving apparatus had what we may term a personal equation, for the break of contact could only take place when the membrane travelled some finite distance, exceedingly small no doubt, from the contact-piece. 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 0° C. by means of equation (10).
Regnault used two different distances, viz. 1280 metres and 2445 metres, obtaining from the first U$_0$＝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.
U$_0$＝330·71 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 0·108 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 U$_0$=330·6 met./sec, while for a diameter 0·108 it was U$_0$ = 324·25 met./sec.
Regnault also set up a shorter length of pipes of diameter 0-108 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 U$_1$ and U$_2$ in two different gases with densities ρ$_1$ and ρ$_2$ at the same temperature and pressure, and with ratios of specific heats γ$_1$, γ$_2$, theory gives
U$_1$/U$_2$＝√{γ$_1$ρ$_2$/γ$_2$ρ$_1$).
This formula was very nearly confirmed for hydrogen, carbon dioxide and nitrous oxide.
J. Violle and T. Vautier (Ann. chim. phys., 1890, vol. 19) made observations with a tube 0·7 m. in diameter, and, using Regnault's apparatus, found that the velocity could be represented by
331·3(1+C√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. §§ 347–8).
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 0° C.
U$_0$＝3310 met./sec.
with a probable error estimated at ± 0·10 metre.
It is obvious from the various experiments that the velocity of sound in dry air at 0° C. is not yet known with very great accuracy. At present we cannot assign a more exact value than
U$_0$＝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., 1905, 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 ear-trumpet, 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 U$^2$＝E/ρ.
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 ear-trumpet 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).
Fig. 12. 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 9 with the tangent at A. Let ON( = OP cos θ) 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 θ with the tangent will, with its reflections, touch a larger circle. Hence all rays between ±θ will be confined in the space between the outer dome and a circle of radius OP cos θ, 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 wave-length. 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. (1) 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.
Fig. 13. 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 15½°. 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 15½°, 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 (Proc. Roy. Soc., 1874, 22, p. 531) first pointed out that refraction would result from a variation in the temperature of the air at different heights. The velocity of sound in air is independent of the pressure, but varies with the temperature, its value at t° C. being as we have seen
U＝U$_0$(1 + 1/2at),
where U is the velocity at 0° 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
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.
Refraction by Wind.
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 from the surface upwards (Brit. Assoc. Rep., 1857, p. 22). It is, of course, a matter of common observation that the wind increases in velocity from the surface upwards, An excellent illustration of this increase was pointed out by F. Osier in the shape of old clouds; their upper portions always appear dragged forward and they lean over, as it were, in the ​ direction in which the wind is going. The same kind of thing happens with sound-wave 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 well-known 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 equally-spaced 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.
Fig. 15.
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 5 is not quite in the plane of the rails. The wave from D has travelled to a circle of radius nearly equal to DF, that from C to a circle of radius nearly CF, and so on. As these “secondary waves” return to S their distance apart is nearly equal to twice the distance between the rails, and the observer then hears a note of wave-length nearly 2EF. But if an observer is stationed at S' the waves will be about half as far apart and will reach him with nearly twice the frequency, so that he hears a note about an octave higher. As he travels further round the frequency increases still more. The railings in fact do for sound what a diffraction grating does for light.
Frequency and Pitch.
Sounds may be divided into noises and musical notes. A mere 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 equally-spaced 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 projecting teeth 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 funnel-shaped 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 A$_3$, 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 tuning-fork usually employed for regulating concert-pitch. 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 60 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 80, the number of taps imparted to the card every second will amount to 44×80 or 3520. This, therefore, is the number of vibrations corresponding to the note A$_3$ . If we divide this by 2$^3$ 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, C$_1$ are 5/4, 3/2 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 to revolve with moderate rapidity. This disk is perforated with small round holes arranged in circles about the centre of the disk. In the first series of circles, reckoning Seebeck’s Siren. 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, 10, 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, E$_1$, G$_1$, C$_2$, &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 10, 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 (5/4). Similar results are obtainable by means of the remaining perforations.
A still simpler form of siren may be constituted with a good spinning-top, 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 intended for supplying the cavity of the cylinder with compressed air or other gas, or even liquid. The top 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 the air repeated, and hence the rapidity of rotation increased. This will go on
continually as long as air is supplied to the cylinder, and the velocity of rotation of the upper plate will be accelerated up to a certain maximum, at which it may be maintained by keeping the force of the current constant.
Now, it is evident that each coincidence of the perforations in the two plates is followed by a non-coincidence, during which the air-current is shut off, and that consequently, during each revolution of the upper plate, there occur n alternate passages and interceptions of the current. Hence arises the same number, of successive impulses of the external air immediately in contact with the movable plate, which is thus thrown into a state of vibration at the rate of n for every revolution of the plate. The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value. If, then, we can determine the number m of revolutions performed by the plate in every second, we shall at once have the number of vibrations per second corresponding to the audible note by multiplying m by n.
For this purpose the axis is furnished at its upper part with a screw working into a toothed wheel, and 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 100 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 90n, 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 readily ascertained. In it the fixed and movable plates are each furnished with four concentric series of perforations, dividing the circumferences into different aliquot parts, as, for example, 8, 10, 12, 16. Beneath the lower Dove’s Siren. 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 air-current through any one or more of the series of perforations in the fixed plate. 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 (CEGC$_1$). The inventor has given to this instrument the name of the many-voiced siren.
Helmholtz (Sensations of Tone, ch. viii.) further adapted the siren for more extensive use, by the addition to Dove's instrument of another chest containing its own fixed 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).
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 tuning-fork, and if the fork, while in vibration, is moved rapidly over a glass plate coated with lamp-black, 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 fork. The same result will be obtained 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 vibration-value of the note yielded by the fork. Or, if the same plate be moved in contact with two tuning-forks, 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 tuning-forks being properly fixed to a frame above.
Greater accuracy may be attained with a revolving-drum chronograph first devised by Thomas Young (Lect. on Nat. Phil., 1807, i. 190), consisting of a cylinder which may be coated with lamp-black, or, better still, a metallic cylinder round which a blackened sheet of paper is wrapped. 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 speaking-trumpet, 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 feather-fibre, 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 tuning-fork 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., 1880, ix. 394). The motion of the fork is maintained by the clock acting through an escapement, and the dial registers both the number of vibrations of the fork and the seconds, minutes and 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 1°, the frequency being exactly 256 at 26·2° C. Hence the frequency may be put as 256 {1−0·000113 (t−26·2)}.
(From Lord Rayleigh's Theory of Sound, by permission of Macmillan & Co., Ltd.)
Fig. 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., 1880, pt. i. p. 1), a cylinder is ruled with equidistant white lines parallel to the axis on a black ground. It is set so that it can be turned at any desired and determined speed about a horizontal axis, M‘Leod and Clarke’s Stroboscopic Method. and when going fast enough it appears grey. Imagine 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 right-hand 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 Â± 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—·00011 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 several soft-iron armatures fixed at equal distances round its circumference. The wheel rotates between the poles of an electro-magnet, which is fed by an intermittent current such as that which is working an electrically maintained tuning-fork (see infra). If the wheel be driven at such rate that the armatures move one place on in about the period of the current, then on putting on the current the electro-magnet 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 electro-magnet. 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 1/8 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 1/8 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 + 1)th passage of the hole. Then the wheel makes 8 N + 1 revolutions in N clock beats, and the fork makes 32 N + 4 vibrations in the same time. If the clock is going exactly right, this gives a frequency for the fork of 32 + 4/N. If the fork has rather less frequency than 32 then the flashes appear to move forward and the frequency will be 32−4/N. In Rayleigh's experiment the 32 fork was made to drive electrically one of frequency about 128, and somewhat as with the phonic wheel, the frequency was controlled so as to be exactly four times that of the 32 fork. A standard 128 fork could then be compared either optically or by beats with the electrically driven fork.
Scheibler’s Tonometer.—When two tones are sounded together with frequencies not very different, " beats " or swellings-out of the sound are heard of frequency equal to the difference of frequencies of the two tones (see below). Johann Heinrich Scheibler (1777–1838) tuned two forks to an exact octave, and then prepared a number of others dividing the octave into such small steps that the beats between each and the next could be counted easily. Let the forks be numbered 0, 1, 2, . . . N. If the frequency of 0 is n, that of N is 2n. Suppose that No. 1 makes m$_1$ beats with No. 0, that No. 2 makes m$_2$ beats with No. 1, and so on, then the frequencies are
n, n+m$_1$, n+m$_1$+m$_2$ , . . ., n+m$_1$+m$_2$+ . . . + m$_N$.
Since n+m$_1$+m$_2$ + . . . + m$_N$=2n, n=m$_1$+m$_2$+ . . . +m$_N$, 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. 10, it might have frequency either 3 above or below that of No. 10. But if it lies above No. 10 it will beat less often with No. 11 than with No. 9; if below No. 10 less often with No. 9 than with No. 11. Suppose it lies between No. 10 and No. 11 its frequency is that of No. 10+3.
Manometric Flames.—This is a device due to Koenig (Phil. Mag., 1873, 45) an d represented diagrammatically in fig. 19. f is a flame
Fig. 19.
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.
​ 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 key-board 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 key-note.
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 t the intervals from the key-note, the frequency ratios with the key-note, the successive frequency ratios and the successive intervals are as follows:—
Note
Interval with C
C
D
second
E major third
F fourth
G fifth
A major sixth
B
seventh
C
octave
Frequency
i
i
i
t
I
V
2
Successive fre-
1
10 9
tt
1
10 9
i
II
quency ratios. Successive in- tervals
major tone
minor tone
major semi- tone
major tone
minor tone
major tone
major semi- tone
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 j[, which is the product of the ratios for a fourth f, and a third f . That is, if we add intervals we must multiply frequency ratios 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 i, 2, 3 or 5 as numerator and denomina- tor, 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 9/8 = 3/2×3/2×1/2, and we may regard it as an ascent through two fifths in succession and then a descent through an octave. The third 5/4=5×1/2×1/2 or ascent through an interval 5/1, which has no special name, and a descent through two octaves, and so on.
Now suppose we take G as the key-note 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
Frequency ratios with C = I . Frequency ratios of diatonic scale
with G = 1
Frequency ratios with C = i, G = f
G
H
We see that all but two notes coincide with notes on the scale of C. But instead of A = 5/8 we have 27/16, and instead of /= f we have ff . The interval between f and H = 1 & + f = I h is termed a " comma," and is so small that the same note on an instrument may serve for both. But the interval between f and f-ji = 44 â– *" t = iff i s quite perceptible, and on the piano, for instance, a separate string must be provided above /. This note is / sharp, and the interval Ml is termed a sharp.
Taking the successive key-notes 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 key-note, besides a small difference at d, we have as the fourth from it f X i = l i, making with B = V an interval \&%, and requiring a new note, B flat. This does not coincide with A sharp which is the octave below the seventh from Bor VxVXj= Hi- It makes with
it an interval = V -=- flHr = M4I, 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 key-note, another note, E flat, is required. E flat as key-note 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 key-notes, key-board instruments would have to be provided with a great number of separate strings or pipes, and the corresponding keys would be required. The construc- tion 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' a" 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 CDEFGABc. The next octave below is Ci Di Ei Fi Gi Ai Bi C, and each preceding octave has another accent as suffix.
The standard frequency for laboratory work is c = i28, 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:—
| C | D | E | F | G | A | B | c |
| Ut$_1$ | Re$_1$ | M$_1$ | Fa$_1$ | Sol$_1$ | La$_1$ | Si$_1$ | Ut$_2$. |
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 Ut$_3$ 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. Jt 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 16,000 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., 1880, x. 1). 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 wave- length 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 λ where U is the velocity of sound, λ the wave-length, and n the frequency, it follows that the forward frequency is greater than the backward frequency.
The more general case of motion of source, medium and receiver ​ 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.
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 pitch-pipe in one end of a few feet of rubber tubing and blowing through the other end while the tubing is whirled round the head. An observer in the plane of the motion can easily hear a change in the pitch as the pitch-pipe moves to and from him.
Musical Quality or Timbre.—Though a musical note has definite pitch or frequency, notes of the same pitch emitted by different instruments have quite different quality or timbre. The three characteristics of a longitudinal periodic disturbance are its amplitude, the length after which it repeats itself, and its form, which may be represented by the shape of the displacement curve. Now the amplitude evidently corresponds to the loudness, and the length of period corresponds to the pitch or frequency. Hence we must put down the quality or timbre as depending on the form.
The simplest form of wave, so far as our sensation goes—that is, the one giving rise to a pure tone—is, we have every 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
y = a sin -r-(x—e), we see that j> = o, for x=e, e-\-\\, e + %\,
e + 'X, and so on, that y is + from x—e to x = e-\-\\, —from e + i\ to e + l\, 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.
Fig. 21.
In this curve ABCD are nodes. OA = e is termed the epoch, being the distance from O of the first ascending node. AC is the shortest distance after which the curve begins to repeat itself; this length X is termed the wave-length. The maximum height of the curve HM=Â« is the amplitude. If we transfer O to A,
e = o, and the curve may be represented by y=a sin -r^x.
A
If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch Â« = OA at any time / will be VI, so that the value of y may be represented as
y=asm^(x-XJt). (16)
The velocity perpendicular to the axis of any point on the curve at a fixed distance x from O is
dy 2ir\Ja 2tt. TT
37= X" cos T (*-UQ. (17)
The acceleration perpendicular to the axis is d?y_ 4 ^U^ . 2x. dP X 5—sm-jj-U-UQ
47T 2 U 2
which is an equation characteristic of simple harmonic motion.
The maximum velocity of a particle in the wave-train is the amplitude of dy/dt. It is, therefore,
Â« m = 2jrUa/X =27rÂ«a. (19)
xxv. 15
The maximum pressure excess is the amplitude of co = EÂ«/U = (E/U)dy/dt. It is therefore
<3 m = (E / U) 2ifVa/\—2vnpVa. (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. _
Suppose that any periodic sound disturbance, consisting of plane waves, is being propagated in the T " eorem 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
FlG. 22.
shortest distance after which the repetition occurs, so that CLDME is merely AHBKC moved on a distance AC. Then AC=X is the wave-length 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 wave-lengths X, JX, JX, fX. . . 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 -r-{x−e)+b sin ^(x−f)+c sin -j-(x−g)+&c. (21)
where the terms may be infinite in number, but always have wave-lengths submultiples of the original or fundamental wave-length 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 wave-length X/Â», but evidently also after a distance equal to the fundamental wave-length 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 wave-lengths equal to the original wave-length 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 wave-lengths; 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
Fig. 23. us imagine it to form half a wave-length of the extended train ZGAHBKC, on an indefinitely extended stretched string, the values of y at equal distances from A (or from B) being equal and opposite. Then, as we shall prove later, the vibrations of the string may be 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 bet-veen A ​ and B. Now we may resolve these trains by Fourier’s theorem into harmonics of wave-lengths λ, 1/2λ, 1/3λ, &c., where λ=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 geometrical analysis. Using the term “note” for the sound produced by a periodic disturbance, there is no doubt that a well-trained ear can resolve a note into pure tones of frequencies equal to those of the fundamental and its harmonics. Resonators. 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.
Fig. 24.—Helmholtz Resonator.
This resonance is a particular example of the general principle that a vibrating system will be set in vibration by any periodic force applied to it, and ultimately in the period of the bration and force, its own natural vibrations gradually dying down. Resonance 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 no dissipation of energy, the vibration would increase indefinitely when the periods coincided. But there is always leakage of energy either through friction or through wave-emission, 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 Mẍ = μx, where μx is the restoring force to the centre of vibration. Putting μ/M＝n$^2$ the equation becomes ẍ+n$^2$x＝0, whence x＝A sin nt, and the period is 2π/n.
Now suppose that in addition to the internal force represented by −μx, an external harmonic force of period 2π/p is applied. Representing it by—P sin pt, the equation of motion is now
| ẍ+n$^2$x +P/M sin pt＝0. | (22) |
Let us assume that the body makes vibrations in the new period 2-irp, and let us put x = B sin pt; substituting in (22) we have -p*B+n>B + P/M=o, whence
B＝P/M · 1/p$^2$ −n$^2$
and the “forced” oscillation due to −P sin pt is
| x＝ P/M · 1/p$^2$ −n$^2$ | (23) |
If p>n the motion agrees in phase with that which the applied force alone would produce, obtained by putting n＝0. If p<n the phases are opposite. If p＝n the amplitude becomes infinite. This is the case of " resonance." The amplitude does not, of course, become infinite in practice. There is always loss of energy by dissipation in the vibrating machinery and by radiation into the medium, and the amplitude only increases until this loss is balanced by the gain from the work done by the applied force.
According to Helmholtz, the ear probably contains within it a series of resonators, with small intervals between the periods of the successive members, while the series extends over the whole range of audible pitch. We need not here enter into the question of the structure constituting these resonators. Each of them is supposed to have its own The Ear as a Fourier Analyser. natural frequency, and to be set into vibration when the ear receives a train of waves of that frequency. The vibration in some way arouses the sensation of the corresponding tone. But the same resonator will be appreciably though less affected by waves of frequency differing slightly from its own. Thus Helmholtz from certain observations (Sound, ii. § 388) thought that if the intensity of response by a given resonator in the ear to its own tone is taken as 1, then its response to an equally loud tone a semitone different may be taken as about j^. According to this theory, then, when a pure tone is received the auditory apparatus corresponding to that tone is most excited, but the apparatus on each side of it is also excited, though by a rapidly diminishing amount, as the interval increases. If the sensations corresponding to these neighbouring elements are thus aroused, we have no such perception as a pure tone, and what we regard as a pure tone is the mean of a group of sensations. The sensitiveness of the ear in judging of a given tone must then correspond to the accuracy with which it can judge of the mean.
Measurements of Intensity of Sound or Loudness.—Various devices have been successfully employed for making sounds of determinate loudness in order to test the hearing of partially deaf people. But the converse, the measurement of the loudness of a sound not produced at our will, is by no means so easy. If we compare the problem with that of measuring the illumination due to a source of light, we see at once how different it is. In sound sensation we have nothing corresponding to white light. A noise such as the roar due to traffic in a town may correspond physically in that it could probably be resolved into a nearly continuous series of wave-lengths, but psychically it is of no interest. We do not use such noise, but rather seek to avoid it. We certainly do not wish to measure its loudness, and even if we did it might be difficult to fix on any unit of noisiness. Probably we should be driven to a purely physical unit, the stream of energy proceeding in any direction, and if the noise were great enough we might measure it possibly by the pressure against a surface.
The intensity of the stream of energy passing per second through a square centimetre when a given pure tone is sounded is more definite and can be measured. There are two practical methods. In the one, the energy of vibration of the source is measured, and the rate at which that energy decreases is observed. The amount radiated out in the form of sound waves is deduced, and hence the energy of the stream at any distance is known. In the other, the waves produce a measurable effect on a vibrating system of the same frequency, and the amplitude in the waves can be deduced.
The first may be illustrated by Lord Rayleigh’s experiments to determine the amplitude of vibration in waves only just audible (Sound, ii. § 384). He used two kinds of experiment, but it will be sufficient here to indicate the second. A fork of frequency 256 was used as the source. The energy of this fork with a given amplitude of vibration could be calculated from its dimensions and elasticity, and the amplitude was observed by measuring with a microscope the line into which the image of a starch grain on the prong was drawn by the vibration. The rate of loss of energy was calculated from the rate of dying down of the vibration. This rate of loss for each amplitude was determined (1) when the fork was vibrating alone, and (2) when a resonator was placed with its mouth under the free ends of the fork. The difference in loss in the two cases measured the energy given up to and sent out by the resonator as sound. The amplitude of the fork was observed when the sound just ceased to be audible at 27·4 metres away, and the rate of energy emission from the resonator was calculated to be 42·1 ergs / second. Assuming this energy to be propagated in hemispherical waves, it is easy to find the quantity per second going through 1 sq. cm. at the distance of the listener, and thence from the energy in a wave, found above, to determine the amplitude. The result was an amplitude of 1·27×10$^{−7}$ cm. Other forks gave results not very different.
In a later series of experiments Lord Rayleigh (Phil. Mag., 1907, 14, p. 596) found that the least energy stream required to excite sensation did not vary greatly between frequencies of 512 and 256, ​ but that the stream required increased rapidly as the frequency was reduced below 256.
The second method may be illustrated by the experiments of M. Wien (Wied. Ann., 1889, xxxvi. 834). He used a spherical Helmholtz resonator resounding to the tone to be measured. The orifice which is usually placed to the ear was enlarged and closed by a corrugated plate like that of an aneroid barometer, and the motion of this plate was indicated by means of a mirror which had one edge fixed, while the other was attached to a style fixed to the centre of the plate. When the plate vibrated the mirror was vibrated about the fixed edge, and the image of a reflected slit was broadened out into a band, the broadening giving the amplitude of vibration of the plate. From subsidiary experiments (for which the original memoir must be consulted) the pressure variations within the resonator could be calculated from the movements of the plate. The open orifice of the resonator was then exposed to the waves from a source of its own frequency. Helmholtz’s theory of the resonator (Rayleigh, Sound, ii. § 311) gives the pressure variations in the incident waves in terms of those in the resonator, and so the pressure variation and the amplitude of vibration in the waves to be measured were determined.
For minimum audible sounds Wien found a somewhat smaller value of the amplitude than Rayleigh. It is remarkable that, as Lord Rayleigh says, “the streams of energy required to influence the eye and the ear are of the same order of magnitude.” Wien also used the apparatus to find the decrease of intensity with increase of distance, and found that it was somewhat more rapid than the inverse square law would give.
In a later series of experiments (Science Abst. vi. 301) Wien used a telephone plate, of which the amplitude could be determined from the value of the exciting current, and he found that the smallest amplitude audible was 6·3 × 10$^{−10}$ cm.
W. Zernov (Ann. d. Physik, 1906, 21, p. 131) compared the indications of Wien’s resonator manometer with those of V. Altberg’s sound pressure apparatus and found very satisfactory agreement.
Stationary Waves.—As a preliminary to the investigation of the modes of vibration of certain sources of sound we shall consider the formation of “stationary waves.” These are not really waves in the ordinary sense, but the disturbance arising from the passage through the medium in opposite directions of two equal trains. The medium is divided up into sections between fixed points, and these sections vibrate. We can form stationary waves with ease by fixing one end of a rope—say 20 ft. long—and holding the other end in the hand. When the hand is moved to and fro transversely waves are sent along the rope and reflected at the fixed end. The direct and reflected systems are practically equal, and by suitably timing the vibrations of the hand for each case the rope may be made to vibrate as a whole, as two halves, as three-thirds and so on. When it vibrates in several sections, each section moves in the opposite way to its neighbours.
Let us suppose that two trains of sine waves of length λ and amplitude a are travelling in opposite directions with velocity U. We may represent the displacement due to one of the trains by
| y$_1$ ＝a sin 2π/λ(x − Ut). | (24) |
where x is measured as in equation (16) from an ascending node as A in fig. 21. If we measure / from an instant at which the two trains exactly coincide, then as U for the other train has the opposite sign, its displacement is represented by
| y$_2$＝a sin 2π/λ(x+Ut). | (25) |
The sum of the disturbance is obtained by adding (24) and (25)
. 2ir r ,, . 2x
y = yi+y2 = 2a cos ylf sin -r-x, (26)
At any given instant t this is a sine curve of amplitude 2a cos (2?r/X) Vt, and of wave-length X, and with nodes at x = o, JX, X, . . . , that is, there is no displacement at these nodes whatever the value of /, and between them the displacement is always a sine curve, but of amplitude varying between +2a and −2a. The ordinate of the curve changes sign as we pass through a node, so that successive sections are moving always in opposite directions and have opposite displacements. Each section then vibrates, and its amplitude goes through all its values in time given by 2?rtjT/X = 27r, or T=X/U, and the frequency is U/X. We may represent such a train of " stationary waves " by fig. 25, where the curves give the
Fig. 25. two extreme amplitudes. The points A, B, C, D are termed “nodes,” and the points half-way between them " loops."
The general character of these results may be obtained by a graphic construction. Let fig. 26 (1) represent a wave-length of each train when they are coincident. It is sufficient to take a single wave-length. The dotted curve represents the superposition, which simply doubles each ordinate. Divide the wave-length into, say, eight equal parts as marked. Then move one train marked (I) 1/8λ to the right, and the other train (II) 1/8λ to the left, introducing new parts of each train at one end, and sending out old parts at the other. Then we get fig. 26 (2), the dotted curve representing
Fig. 26. the resultant with amplitude 1/V2 that of (1). Another movement of jX in each direction gives (3) with resultant a straight line, and so on for (4) and (5). In (5) the displacement is evidently equal and opposite to that in (1). Further displacement will give the figures (4). (3)1 (2), (1) again, but with (I) and (II) interchanged. When we get back to (1) each train has been displaced through X and the period is X/U. Further, the original nodes are always at rest, and the intervening sections vibrate to and fro.
The vibrations of certain sources of sound may be represented, at least as a first approximation, as consisting of stationary waves, and from a consideration of the rate of propagation of waves along these sources we can deduce their frequency when we know their length.
Sources of Sound.
Elementary Theory of Pipes.—The longitudinal vibration of air in cylindrical pipes is made use of in various wind instruments. We shall deduce the modes of vibration of the air column in a cylindrical pipe from the consideration that the air in motion within the pipe forms some part of a system of stationary waves, one train being formed by the exciter of the disturbance, and the other being formed by the reflection of the train at the end of the pipe.
In order to justify the use of stationary waves we must show that two such trains can move in opposite directions over the same ground without modifying each other so long as the displacement in either is small. For this it is necessary that the total force on an element due to the sum of the displacements should be equal to the sum of the forces due to the two displacements considered separately. The medium then acts for the second train just as if it were undisturbed by the first. It is sufficient then to show that the excess of pressure at any point is the sum of the excesses due to either train separately.
If ω is the total pressure excess, and if y is the total displacement at x, then Â« = Exchange of volume ÷ original volume = − Kdyjdx. If yi and yt are the two separate displacements and if y = y\-\-yi, then w = − E (dyi/dx + dy^/dx) = wi + a?. This proves the proposition. It is a case of the principle of superposition of small disturbances.
Let us suppose that a system of stationary waves is formed in the air in a pipe of indefinite length, and let fig. 27 represent a part of the system. At the nodes A, B, C, D, E there is no displacement, but there are maximum volume and pressure changes. Consider, for instance, the point B. When the displacement is represented by AHBKC the particles on each side of B are displaced towards it ​ giving a compression, and since the slope is steepest there, or −dy/dx a maximum, the compression is also a maximum there. When the displacement is represented by AH′BK′'C the particles on each side of B are displaced from it, giving an extension, and since the slope is again the steepest, the extension is a maximum.
Fig. 27.
At the loops, for instance at H, the displacement is a maximum. The tangent to the displacement curve is always parallel to the axis, that is for a small distance the successive particles are always equally displaced, and therefore always occupy the same volume. This means that at the loops while the motion is greatest there are no pressure changes. .
We have now to select such portion of this system as will suit the conditions imposed by any actual pipe. There are three distinct types which we will consider in succession.
1. Pipe Closed at One End, Open at the Other.—At the closed end there is no motion, for the pressure always constrains the air to remain in contact with the end. The closed end is therefore a node. At the open end, as a first approximation to be corrected later, there are no pressure changes, for any tendency to excess can be relieved by immediate expansion into the outer air, and any tendency to defect can be filled up by an inrush from the outer air. The open end is therefore a loop. It is to be noted that the exciter of the vibrations is in general at the open end, and that the two trains forming the stationary system consist of the direct waves from the exciter travelling into the tube, and the waves reflected back from the
In fig 27 we may have the length AH occupying the tube. In <his case AH = |X 1 = i, the length of the tube, and the frequency j(l = U/Xi = U/4/. But we may also have a shorter wave-length \ 2 such that the length AK occupies the tube. In this case AK = |X 2 = Z, and the frequency w 2 = U/X 2 = 311/4/. With a still shorter wave-length X 3 we may have the length AL occupying the tube and AL = |X 3 =Z, and the frequency n% = U/X 3 = 5U/4/, and so on, as we take succeeding loops for the open end.
In fig. 28 are represented the stationary wave systems of the first four modes, and any of the succeeding ones are easily drawn.
condition for a loop at the open end, that of no pressure variation, cannot be exactly fulfilled. This would require that the air outside should have no mass in order that it should at once move out and relieve the air at the end of the pipe from any excess of pressure, or at once move in and fill up any defect. There are variations, therefore, at the open end, and these are such that the loop may be regarded as situated a short distance outside the end of the pipe. It may be noted that in practice there is another reason for pressure variation at the end of the pipe. The stationary wave method regards the vibration in the pipe as due to a series of waves travelling to the end and being there reflected back down the pipe. But the reflection is not complete, for some of the energy comes out as waves; hence the direct and reflected trains are quite equal, and cannot neutralize each other at the loop.
The position of the loop has not yet been calculated for an ordinary open pipe, but Lord Rayleigh has shown (Sound, ii. Â§ 307) that for a cylindrical tube of radius R, provided with a flat extended flange, the loop may be regarded as about 0-82 R, in advance of the end. That is, the length of the pipe must be increased by 0-82 R before applying Bernoulli's theory. This is termed the " end correction."
Fig. 28. The reader will be able to make out the simultaneous motions and pressures at various points. It is obvious that the nodes are alter- nately in compression and extension, or vice versa, and that for 4 X on each side of a node the motion is either to it on both sides or from it on both sides. ,,
The first mode of vibration gives the " fundamental tone, and the succeeding modes are termed " overtones." The whole series forms the series of odd harmonics. A " stopped pipe " in an organ is a pipe of this type, and both the fundamental and the overtones may occur simultaneously when it is blown.
We may illustrate the successive modes of vibration by using as pipe a tall cylindrical jar, and as exciter a vibrating tuning-fork held over the mouth. The length of the pipe may be varied by pouring in water, and this is done until we get maximum resonance of the pipe to the fork. Thus if a fork 11(3 = 256 is used, the length of pipe for the fundamental at 0Â° C. is about 33,000/4X256 = 33 cms. If a fork Sol 4 = 768 is used the pipe resounds to it according to the mode of the first overtone. If the temperature is <Â° the length for given frequency must be increased by the factor 1 +0-001844.
Correction to Length at the Open End.—The approximate theory of pipes due to Bernoulli assumes a loop at the open end, but the
Fig. 29.
Using this result Rayleigh found the correction for an unflanged open end by sounding two pipes nearly in unison, each provided with a flange, and counting the beats. Then the flange was removed from one and the beats were again counted. The change in virtual length by removal of the flange was thus found, and the open end correction for the unflanged pipe was o-6 R. This correction has also been found by David James Blaikley by direct experiment (Phil. Mag., 1879, 7, p. 339). He used a tube of variable length and determined the length resounding to a given fork, (1) when the closed end was the first node, (2) when it was the second node. If these lengths are h and h, then l$_2$− /a = JX and i(h−h)−li is the correction for the open end. The mean value found was 0·576 R.
2. Pipe Open at Both Ends.—Each end is a loop. We must there- fore select a length of fig. 27 between two loops. The fundamental mode is that in. which H and K represent the ends of the pipe. In this case HK = JXi=/, and the frequency is Â»i = U/Xi = U/2Z. There is a node in the middle. In the next mode H and L represent the ends and HL = X 2 = Z and re 2 = U/X 2 = 2U/2/. In the third mode HM = ^X 3 = / and Â» 3 = U/X 3 = 3U/2/, and so on.
In fig. 29 are represented the stationary wave systems of the first four modes. The whole series of fundamental and overtones gives the complete set of harmonics of frequencies proportional to 1, 2, 3, 4, . . . , and wave-lengths proportional to 1, Â§, \, \. . . .
A metal or brass tube will serve as such a pipe, and may be excited by a suitable tuning-fork held at one end. To, obtain the virtual length we must add the correction for each open end, probably about 1-2 radius. If the frequency is 256 the corrected length for the fundamental is about (33,000/2X256) (1 +-001840 at P. The pipe will also resound to forks of frequencies 512, 768, 1024 and so on.
An open " flue " organ pipe is of this type. The wind rushing through the slit S (fig. 30) maintains the vibration in a way to be discussed later, and the opening O makes the lower end a loop.
The modes of vibration in an open organ pipe may be exhibited by means of Koenig's manometric flames (Phil. Mag., 1873, vol. 45). The pipe is provided with manometric flames at its middle point, and at one-quarter and three-quarters of its length. When the pipe is blown softly the fundamental is very predominant, and I there is a node at the middle point. The flame there is much ​ affected by the nodal pressure changes, while the other two vibrate only slightly. If, however, the pipe is blown strongly, the funda- mental dies away, and the first overtone is predominant. Then the middle point is a loop, and the middle flame is only slightly affected, while the other two, now being at nodes, vibrate strongly.
3. Pipe Closed at Both Ends. â€” The two ends in such a pipe are nodes. It is evident that the overtones will Follow the same rule as for a pipe opened at both ends. This case is not exactly realized in practice, but it is closely approximated to in Kundl's dust-tube. A glass tube, the " dust-tube," 3 ft. or more in length, and perhaps 1 in. in diameter, has a little lycopodium powder introduced, and the powder is allowed to run all along the tube, which is then fixed horizontally. A closely-fitting adjustable piston is provided at one end. A glass or metal rod, the " sounder," is clamped at its middle point, and fixed along the prolongation of the axis of the dust-tube as in fig. 31, a loosely- fitting cork or card piston being fixed on one end of the sounder, which is inserted within the dust-tube. The other end of the sounder is stroked outwards with a damp cloth so as to make it sound its funda- mental. Stationary waves are formed in the air in the dust-tube if the length is rightly adjusted by the closely-fitting piston, and the lycopodium dust collects at the nodes in little heaps, the first being at the fixed end and the last just in front of the piston on the sounder. The stationary wave system adjusts itself so that its motion agrees with that of the sounder, which is therefore not exactly at a node. If UÂ« is the velocity of longitudinal waves along the sounder, and I the length of the sounder, the frequency of vibration is VJ2I. If L is the distance between successive dust-heaps, i.e. half a wave- length, the frequency in the air is U/2L, where U is the velocity of sound in the pipe. Then, since the frequencies are the same, U/2L = U./2/orL// = U/U,.
Fig. 30.
Fig. 31.
The velocities in different gases may be compared by this appara- tus by filling the dust-tube with the gases in place of air. If L; is the internodal distance and Ui the velocity in a gas, L and U being the corresponding values for air, we have Ui/U = L,/L.
Kundt's dust-tube may also be employed for the determination of the ratio of the specific heats of a gas or vapour. If U is the Specific velocity of sound in a gas at pressure P with density p, Heats' and if waves of length X and frequency N are propagated (/alio. through it, then the distance between the dust-heaps is
d ~2~2N~2N \ p '
where 7 is the ratio of the two specific heats. If d is measured for two gases in succession for the same frequency N, we have
72_ PzPi iÂ£_
7i PiPz di" where the suffixes denote the gases to which the quantities relate. If ti is known this gives 72. Kundt and Warburg applied the method to find 7 for mercury vapour (Pogg. Ann., 1876, 157, p. 356), using a double form of the apparatus in which there are two dust- tubes worked by the same sounding rod. This rod is supported at \ and J of its length where it enters the two dust-tubes, as represented diagrammatically in fig. 32. It is stroked in the middle so as to
1
amplitude are both double those in either train, so that the same relation holds.
Determinations of the pressure changes, or extent of excursion of the air, in sounding organ pipes have been made by A- Kundt (Pogg. Ann., 1868, 134, p. 163), A. J. I. Topler and . ,,, . . L. loltzmann (Pogt /nn. : vol. 141, or Rayleigh, Â«Â£fÂ°' Sound, ii. Â§ 422a), and E. Mach (Optisch-akustischen _ Versuche, 1873). Mach's method is perhaps the most direct. The pipe was fixed in a horizontal position, and along the top wall ran a platinum wire wetted with sulphuric acid. When the wire was heated by an electric current a fine line of vapour descended from each drop. The pipe was closed at the centre by a membrane which prevented a through draught, yet permitted the vibrations, as it was at a node. The vapour line, therefore, merely vibrated to and fro when the pipe was sounded. The extent of vibration at different parts of the pipe was studied through a glass side wall, a strohoscopic method being used to get the position of the vapour line at a definite part of the vibration. Mach found an excursion of 0-4 cm. at the end of an open pipe 123 cm. long. The amplitude found by the other observers was of the same order. For the vibration of air in other cavities than long cylindrical pipes we refer to Rayleigh's Sound, vol. ii. chs. 12 and 16.
Propagation of Waves in Pipes of Circular Section. â€” Helmholtz investigated the velocity of propagation of sound in pipes, taking into account the viscosity of the air (Rayleigh, Sound, ii. Â§ 347), and Kirchhoff investigated it, taking into account both the viscosity and the heat communication between the air and the walls of the pipe (loc. cit. ii. Â§ 350). Both obtained the value for the velocity
U V~RV2 J rNp) ' where U is the velocity in free air, R is the radius of the pipe, N the frequency, and p the air density. C is a constant, equal to the coefficient of viscosity in Helmholtz's theory, but less simple in Kirchhoff's theory. Experiments on the velocity in pipes were carried out by H. Schneebeli (Pogg. Ann., 1869, 156, p. 296) and by T. J. Seebeck (Pogg. Ann., 1870, 139, p. 104) which accorded with this result as far as R is concerned, but the diminution of velocity was found to be more nearly proportional to N - *. Kundt also obtained results in general agreement with the formula (Rayleigh, Sound, ii. Â§ 260). He used his dust-tube method.
Elementary Theory of the Transverse Vibration of Musical Strings.
We shall first investigate the velocity with which a disturbance travels along a string of mass m per unit length when it is stretched with a constant tension T, the same at all points. We shall then show that on certain limitations two trains of disturbance may be superposed so that stationary waves may be formed, and thence we shall deduce the modes of vibration as with pipes.
Fig. 32.
excite its second mode of vibration. The method ensures that the two frequencies shall be exactly the same. In the mercury experi- ment the sounding rod was sealed into the dust-tube, which was exhausted of air, and contained only come mercury and some quartz dust to give the heaps. It was placed in a high temperature oven, where the mercury was evaporated. The second tube containing air was outside. When a known temperature was attained the sounder was excited, and d\ and d\ could be measured. From the temperature, Pa/pj was known, and 72/71 could then be found. Taking 71 = 1-41, 72 was determined to be 1-66. Lord Rayleigh and Sir William Ramsay (Phil. Trans. A. 1895, pt. i. p. 187) also used a single dust-tube with a sounder to find 7 for argon, and again the value was i-66.
Determinations of Pressure Changes and Amplitude of Vibrations in Pipes. â€” If the maximum pressure change is determined, the amplitude is given by equation (20), viz. w m = 2imapU, for in the stationary wave system the pressure change and the
Fig. 33.
Let AB (fig. 33) represent the string with the ends AB fixed. Let a disturbance once set going travel along unchanged in form from A to B with velocity U. Then move AB from right to left with this velocity, and the disturbance remains fixed in space. Take a point P in the disturbed part, and a point Q which the disturbance has not yet reached. Since the conditions in the region PQ remain always the same, the momentum perpendicular to AB entering the region at Q is equal to the momentum perpendicular to AB leaving the region at P. But, since the motion at Q is along AB, there is no momentum there perpendicular to AB. So also there is on the whole none in that direction leaving at P. Let the tangent at P make angle with AB. The velocity of the string at P parallel to PM is U sin <Â£, and the mass of string passing P is jmU per second, so that wU ! sin is carried outper second. But the tension at P is T, parallel to the tangent, and T sin <j> parallel to PM, and through this â€” T sin <Â£ is the momentum passing out at P per second. Since the resultant is zero, Â«U J sin <j>~ T sin <j> =0, or U 2 =T/m.
Now keep AB fixed, and the disturbance travels with velocity U. We might make this investigation more general by introducing a force X as in the investigation for air, but it hardly appears necessary.
To form stationary waves two equal trains must be able to travel in opposite directions with equal velocities, and to be superposed. We must show then that the force called out by the sum of the disturbances is equal to the sum of the forces called out by each train separately.
In order that the velocity shall remain unchanged the tension T must remain the same. This implies that the disturbance is so small that the length is not appreciably altered. The component of T ​
acting parallel to the axis or straight string is Tdx/ds, and when the disturbance is sufficiently small the curve of displacement is so nearly parallel to the axis that dx/ds=l, and this component is T. The component of T perpendicular to the axis is Tdy]ds = Tdy/dx. Now if yi and yi are the displacements due to the two trains separately, and y = yi+y2, the two separate forces are Tdyi/dx and Tdy>/dx, while that due to y is Tdy/dx. But since y = yi+y-2, Tdy/dx = Tdyi/dx+Tdy 1 /dx, or the condition for superposition holds when the displacement is so small that we may put dx/ds=i. Evidently this comes to neglecting <t> a . Let two trains of _ equal waves moving in opposite directions along such a string of indefi- nite length form the stationary system of fig. 27. Since the nodes are always at rest we may represent the vibration of a
given string by the length between any two nodes. The fundamental mode is that in which A and B represent the ends of the string. In this case AB = |Xi=/ the length, and the frequency m = U/Xi = V/2I = (i/2/)V(T/m)- The middle of the string is a loop. In the next mode A and C represent the ends and AC = X 2 = / and Â«j = U/X 2 = 2U/2/ = (2/2/)V(T/m). In the third mode A and D represent the ends and AD = |X 3 = ; and re 3 = U/X 3 = 3U/2/ = (3/2ZH (T/m) and so on. In fig. 34 the stationary wave systems of the first four modes are represented.
The complete series of harmonics are possible modes. The experimental demonstration of these results is easily made by the sonometer or monochord (%â€¢ 35)- A string is fixed at C on the top of a hollow box, and
Fig. 34.
by proper adjustment of its length and tension vibrates in unison and divides itself into one or more loops or ventral segments easily discernible by a spectator. If the length of the thread be kept invariable, a certain tension will give but one ventral segment ; the fundamental note of the thread is then of the same pitch as the note of the body to which it is attached. By reducing the tension to one quarter of its previous amount, the number of ventral segments will be seen to be increased to two, indicating that the first harmonic of the thread is now in unison with the solid, and consequently that its fundamental is an octave lower than it was with the former tension ; thus confirming the law that n varies as VT. In like manner, on further lowering the tension to one ninth, three ventral segments will be formed, and so on.
The law that, caeteris paribus, n varies inversely as the thickness may be tested by forming a string of four lengths of the single thread used before, and consequently of double the thickness of the latter, when, for the same length and tension, the compound thread will exhibit double the number of ventral segments presented by the single thread.
The other laws admit cf similar illustration.
Longitudinal Vibrations of Wires and Rods.
Subject to a limitation which we shall examine later, the velocity of a longitudinal disturbance along a wire or rod depends only on the material of the rod, and not upon the cross-section. Since the forces called into play by an extension or compression of the material are proportional to the cross-section, it follows that if we consider any case and then another case in which, with the same longitudinal disturbance, the cross-section is doubled, the force in the second case is doubled as well as the mass to be moved. The acceleration therefore remains the same, and the velocity is unaltered. We shall find the velocity of propagation, just as in previous cases, from the consideration of transfer of momentum.
Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right. Let us move the rod from right to left, so that the undisturbed parts move with velocity U. Then the disturbance remains fixed in space. Let A be a point in
Fig. 35. passes over two edges AB, which serve as the fixed ends, and then over a pulley P, being stretched by a weight W. Between A and B a " bridge " D, i.e. another edge slightly higher than A or B, can be inserted in any position, which is determined by a graduated scale. The effective length of the string is then AD. Keeping the same tension, it may be shown that nl is constant by finding n for various lengths. Keeping AD constant and varying W it may be shown that Â« Â»VW. Lastly, by using different strings, it may be shown that, with the same T and I, n x V(i/m). ..
The various modes of vibration may also be exhibited. If D is removed and the string is bowed in the middle, the fundamental is brought out. If it is touched in the middle with a feather, the edge of a card, or the finger nail, and bowed a quarter of the way along the octave, the first overtone comes out. Each of the first few harmonics may be easily obtained by touching the string at the first node of the harmonic required, and bowing at the first loop, and the presence of the nodes and loops may be verified by putting light paper riders of shape A on the string at the nodes and loops. When the harmonic is sounded the riders at the loops are thrown off, while those at the nodes remain seated.
Not only may the fundamental and its harmonics be obtained seoarately, but they are also to be heard simultaneously, particularly the earlier ones, which are usually more prominent than those higher in the series. A practised ear easily discerns the coexistence of these various tones when a pianoforte or violin string is thrown into vibration. It is evident that, in such case, the string, while vibrating as a whole between its fixed extremities, is at the same time executing subsid ! ary oscillations about its middle point, its points
y
Fig. 36. of trisection, &c, as shown in fig. 36, for the fundamental and the first harmonic. When a string is struck or bowed at a point, any harmonic with a node at that point is absent. Since the quality of the note sounded depends on the mixture of harmonics, the quality therefore is to some extent dependent on the point of excitation.
A highly ingenious and instructive method for illustrating the laws of musical strings was contrived by F. E. Melde. It consists i.i attaching to the loop or ventral segment of a vibrating body, e.g. a tuning-fork or a bell-glass, a silk or cotton thread, the other extremity being either fixed or passing over a pulley and supporting weights by which the thread may be stretched to any degree required. The vibrations of the larger mass are communicated to the thread, which
â€” A dx
u
Fig. 37.
the disturbance, and B a point in the undisturbed part. The material between A and B, though continually changing, is always in the same condition, and therefore the momentum within AB is constant. Hence the amount carried out at A is equal to that carried in at B.
Now momentum is transferred in two ways, viz. by the force acting between contiguous portions of a body and by the transfer of moving matter. At B there is only the latter kind, and since the transfer of matter is powoU, where po is the undisturbed density and So is the undisturbed cross-section, since its velocity is U the passage of momentum per second is poSnlh 1 . At A, if the velocity of the disturbance relative to undisturbed parts of the rod is u from left to right, the velocity relative to A is U-Â«. If p is the density at A, and Â£ the cross-section, then the momentum carried past A is pco(U â€” w) 2 . But if y is the displacement at A, dy/dx is the extension at A, and the force acting is a pull across A equal to Ydody/dx, where Y is Young's modulus of elasticity. Then we have
Ywody/<fc-fpu(U-w) 2 = poÂ£oU 2 . (27)
Now u/V = -dy/dx, (28)
for the particle at A moves over dy backwards, while the disturbance moves over U. Also since dx has been stretched to dx-j-dy
pa> (dx 4- dy ) = po&odx or pH(i+dyjdx) =p Â«o. (29)
Substituting from (28) in (27)
Yfiog + pfiU 2 (1 + g) 2 = po&U*, (30)
and substituting from (29) in (30)
Y3og + PoffloU 2 (1 + g) = poSoU 2 , (31)
whence â€¢ Yfio = po<SoU 2 ,
or U 2 =Y/p, (32)
where now p is the normal density of the rod. The velocity with which the rod must travel in order that the disturbance may be fixed in space is therefore U = V(Y/p), or, if the rod is kept fixed, this is the velocity with which the disturbance travels.
This investigation is subject to the limitation that the diameter of the cross-section must be small compared with the wave-length. When the rod extends or contracts longitudinally it contracts or ​ extends radially and in the ratio a, known as Poisson's ratio, which in metals is not far from \. Let us suppose that the rod is circular, of radius r, and that the radial displacement of the surface is i\. The longitudinal extension is-dy/dx, and therefore the radial contraction
, , , , Tr , . 2tt , TTl . 2-jrtrra 2ir , _.,.
is i)jr = adyldx. If then y =a sin â€” (xâ€” vt), i)= â€” r â€” cos-r-(;c â€” <Jt).
If r is of the order of A, rj is of the order of y; and the kinetic energy of the radial motion is of the same order as that of the longitudinal motion. But our investigation entirely leaves this out of account, and is therefore faulty. In fact, the forces are then no longer parallel to the axis. There are shears of the order dri/dx and the simple Young's modulus system can no longer be taken to represent the actual condition (see Rayleigh, Sound, i. Â§ 157). But keeping r/X small we may as before form stationary waves, and it is evident that the series of fundamental and overtones will be just as with the air in pipes, and we shall have the same three types â€” fixed at one end, free at both ends, fixed at both ends â€” with fundamental frequencies respectively
W^hV^ and W7-
It
The overtones will be obvious.
For an iron wire Y/p is about io 12 /4, so that for a frequency of 500 in a wire fixed at both ends a length about 5 metres is required. If the wire is stretched across a room and stroked in the middle with a damp cloth the fundamental is easily obtained, and the first harmonic can be brought out by stroking it at a quarter the length from one end. A glass or brass rod free at both ends may be held by the hand in the middle and excited by stroking one end outwards with a damp cloth. If it is clamped at one-quarter and three- quarters of the length from the ends, and is stroked in the middle, the first harmonic sounds.
Young's modulus may be obtained for the material of a rod by clamping it in the middle and obtaining the frequency of the funda- mental when Y = 4l 2 n 2 p.
The value thus obtained is generally appreciably greater than that obtained by a statical method in which the rod is pulled out by an applied tension.
Rods of different materials may be used as sounders in a Kundt's dust tube, and their Young's moduli may be compared, since: â€”
... , . . . . ., length of rod
velocity in rod = velocity in air X j*-7 râ€” ? -3 â€” rr
' distance between dust-heaps.
'Torsional Vibrations of Rods and Wires.—The velocity of propagation of a torsional disturbance along a wire of circular section may be found by the transfer of momentum method, remembering that we must now replace linear momentum by angular momentum. Let the disturbance be supposed to travel unchanged in form from left to right with velocity U. Now suppose that the wire or rod is moved from right to left with velocity U. The disturbance is then fixed in space. Let A be a point in the disturbance and B a point in the undisturbed portion. The condition of the matter between A and B remains constant, though fresh matter keeps coming in at B and an equal quantity leaves at A. Hence the angular momentum of the part between A and B remains constant, or as much enters at B as leaves at A. But at B there is no torsion, and no torsion couple of one part of the wire on the next. So that no angular momentum enters at B, and therefore on the whole none leaves at A. The transfer of angular momentum through A is of two kinds—first, that due to the passage of rotating matter, and, secondly, that due to the couple with which matter to the right of A acts upon matter to the left of A. The mass of matter moving through A per second is piro 2 U, where a is the radius of the wire and $\rho$ is its density. If $\theta$ is the angle of twist, the angular velocity is $d\theta /dt$. The radius of gyration of the section is ${\frac {1}{2}}a^{2}$. Hence the angular momentum conveyed per second outwards is ${\frac {1}{2}}\rho \pi a^{4}\mathrm {U} d\theta /dt$. The couple due to the twist of a wire of length $l$ through $\sigma$ is $\mathrm {G} ={\frac {1}{2}}n\pi a^{4}\sigma /l$, and we may put $\sigma /l=d\theta /dx$. Since no angular momentum goes out on the whole
lmra 4 d8/dx + -2pTra 4 Ud8/dt=o. (33)
But the condition of unchanged form requires that the matter shall twist through (ddjdx)dx while it is travelling over dx, i.e. in time dx/V. ~, dd dx dB , dd .,dd
Thcn . Tt u = -s:/ ,cor 3i = - u s-
Substituting in (33) we get
U 2 = n/p. ( 34 )
If we now keep the wire at rest the disturbance travels along it with velocity U = V(n/p), and it depends on the rigidity and density of the wire and not upon its radius.
It is easy to deduce the modes of vibration from stationary waves as in the previous cases. If a rod is clamped at one end and free at the other, the fundamental frequency is (i//)V (nip). For iron n!p is of the order io 11 , so that the frequency for a rod I metre long is about 3000. When a cart wheel is ungreased it produces a very . high note, probably due to torsional vibrations of the axle.
The torsional vibrations of a wire are excited when it is bowed. If small paper rings are put on a monochord wire they rotate through these vibrations when the wire is bowed.
Transverse Vibrations of Bars or Rods. â€” When a bar or rod is of considerable cross-section, a transversal disturbance calls into play forces due to the strain of the material much more important than the forces due to any tension which is ordinarily applied. The velocity of a disturbance along such a bar, and its modes of vibration, depend therefore on the elastic properties of the material and the dimensions of the bar. We cannot investigate the vibrations in an elementary manner. A full discussion will be found in Rayleigh's Sound, vol. i. ch. 8. We shall only give a few results.
The cases interesting in sound are those in which (1) the bar is free at both ends, and (2) it is clamped at one end and free at the other.
For a bar free at both ends the fundamental mode of vibration has two nodes, each 0-224 of the length from the end. The next mode has a node in the middle and two others each 0-132 from the end. The third mode has four nodes 0-094 an d 0-357 from each end, and so on. The frequencies are nearly in the ratios 3 2 : 5 2 : 7 2 . . . . Such bars are used in the harmonicon.
When one end is clamped and the other is free the clamped end is always a node. The fundamental mode has that node only. The next mode has a second node 0-226 from the free end; the next, nodes at o- 132 and 0-5 from the free end, and so on. The frequencies are nearly in the ratios 1:6-25:17-5. Such bars are used in musical boxes and as free reeds in organ pipes.
The most important example of this type is the tuning-fork, which may be regarded as consisting of two parallel bars clamped together at the base. The first overtone has frequency 6-25 that of the fundamental, and is not in the harmonic series. If the fork be mounted on a resonance box or held in front of a cavity resounding to the fundamental and not to the first overtone, the fundamental is brought out in great purity.
Vibrations of Plates. â€” These are for the most part interesting rather from the point of view of elasticity than of sound. We shall not attempt to deal with the theory here but shall describe only the beautiful mode of exhibiting the regions of vibration and of rest devised by E. F. F. Chladni (1756-1827). As usually arranged, a thin metal plate is screwed on to the top of a firm upright post at the centre of the plate, which is horizontal. White sand is lightly scattered by a pepper-box over the plate. The plate is then bowed at the edge and is thrown into vibration between nodal lines or curves and the sand is thrown from the moving parts or ventral segments into these lines, forming " Chladni's figures." The development of these figures by a skilful bower is very fascinating. As in the case of a musical string, so here we find that the pitch of the note is higher for a given plate the greater the number of ventral segments into which it is divided ; but the converse of this does not hold good, two different notes being obtainable with the same number of such segments, the position of the nodal lines being, however, different.
The upper line of annexed figures shows how the sand arranges itself in three cases, when the plate is square. The lower line gives the same in a sort of idealized form. Fig. 38, i, corresponds to the
\ 1
3
Fig. 38.
lowest possible note of the particular plate used; fig. 38, 2, to the fifth higher; fig. 38, 3, to the tenth or octave of the third, the numbers of vibration in the same time being as 2 to 3 to 5.
If the plate be small, it is sufficient, in order to bring out the simpler sand-figures, to hold the plate firmly between two fingers of the same hand placed at any point where at least two nodal lines meet, for instance the centre in (1) and (2), and to draw a violin bow downwards across the edge near the middle of a ventral segment. But with larger plates, which alone will furnish the more complicated figures, a clamp-screw must be used for fixing the plate, and, at the same time, one or more other nodal points ought to be touched with the fingers while the bow is being applied. In this way, any of the possible configurations may be easily produced.
By similar methods, a circular plate may be made to exhibit nodal lines dividing the surface by diametral lines into four or a greater, but always even, number of sectors, an odd number being incompatible with the general law of stationary waves that the parts of a body adjoining a nodal line on either side must always vibrate oppositely to each, other.
Another class of figures consists of circular nodal lines along with diametral lines (fig. 39). ​ Circular nodal lines unaccompanied by intersecting lines cannot be produced in the manner described; but may be got either. by drill- ing a small hole through the centre, and drawing a horse-hair along its edge to bring out the note, or by attaching a long thin elastic rod to the centre of the plate, at right angles to it, holding the rod by the middle and rubbing it lengthwise with a bit of cloth powdered with
Fig. 39. resin, till the rod gives a distinct note; the vibrations are com-
municated to the plate, which consequently vibrates transversely, and causes the sand to heap itself into one or more concentric rings.
Paper, parchment, or any other thin membrane stretched over a square, circular, &c, frame, when in the vicinity of a sufficiently powerful vibrating body, will, through the medium of the air, be itself made to vibrate in unison, and, by using sand, as in previous instances, the nodal lines will be depicted to the eye, and seen to vary in form, number and position with the tension of the plate and the pitch of the originating sound. The membrana tympani or drum of the ear, has, in like manner and on the same principles, the property of repeating the vibrations of the external air which it communicates to the internal parts of the ear.
Bells may be regarded as somewhat like circular plates vibrating with radial nodes, and with the edges turned down. Lord Rayleigh has shown that there is a tangential motion as well as a motion in and out. Ordinarily when a bell is struck the impulse primarily excites the radial motion, and the tangential motion follows as a matter of course.. When a finger-glass (an inverted bell), is excited by passing the finger round the circumference, the tangential motion is primarily excited and the radial follows it. Some discussion of the vibrations of bells will be found in Rayleigh, Sound, vol. i. ch. 10 (see also Bells).
Singing Flames. â€” A " jet tube," i.e. a tube a few inches long with a fine nozzle at the top, is mounted as in fig. 40, so as to rise out of a vessel to which coal-gas, or, better, hydro- gen, is supplied. The supply is regulated so that when the gas is lighted the flame is half or three-quarters of an inch high. A " sound- ing tube," say an inch in diameter, and some- what more than twice the length of the jet tube, is then lowered over the flame, as in the figure. When the flame is at a certain distance within the tube the air is set in vibration, and the sounding tube gives out its fundamental note continuously. The flame aDpears to lengthen, but if the reflection is viewed in a vertical mirror revolving about a vertical axis or in Koenig s cube of mirrors, it is seen that the flame is really intermittent, jumping up and down once with each vibration, sometimes apparently going within the jet tube at its lowest point. For a given jet tube there is a position of maximum efficiency easily ob- tained by trial. The jet tube, for a reason which will be given when we consider the maintenance of vibrations, must be less than half the length of the sounding tube.
A series of pipes of lengths to give any desired series of notes may be arranged. If two tubes in unison are employed, a pretty example of resonance may be obtained. One is adjusted so as just not to sing. The other is then made to sing and frequently the first will be set singing also.
Sensitive Flames and Jets. â€” When a flame is just not flaring, any one of a certain range of notes sounded near it may make it flare while the note is sounding. This was first noticed by John Le Conte (Phil. Mag., 1858, 15, p. 235), and later by W. F. Barrett {PhU. Mag., 1867, 33, p. 216). Barrett found that the best form of burner for ordinary gas pressure might be made of glass tubing about I in. in diameter contracted to an orifice / 8 inch in diameter, the orifice being nicked by a pair of scissors into a V-shape. The flame rises up from the burner in a long thin column, but when an appropriate note is sounded it suddenly drops down and thickens. Barrett further showed by using smoke jets that the flame is not essential. John Tyndall (Sound, lecture vi. Â§ 7 seq.) describes a number of beautiful experiments with jets at higher pressure than ordinary, say 10 in. of water, issuing from a pinhole steatite burner. The flame may be 16 in. high, and on receiving a suitably high sound it suddenly drops down and roars. The sensi- tive point is at the orifice. Lord Rayleigh (Sound, ii. Â§ 370), using as a source a " bird-call," a whistle of high frequency, formed a series of stationary waves by reflection at a flat surface. Placing the sensitive flame at different parts of this train, he found that it was excited, not at the nodes where the pressure varied, but at the loops where the motion was the greatest and where there was little pressure change. In his Sound (ii. ch. 21) he has given a theory of the sensitiveness. When the velocity of the jet is gradually increased there is a certain range of velocity for which the jet is unstable,
-Supply
Fig. 40. â€” Singing Flame.
so that any deviation from the straight rush-out tends to increase as the jet moves up. If then the jet is just on the point of insta- bility, and is subjected as its base to alternations of motion, the sinuosities impressed on the jet become larger and larger as it flows out, and the flame is as it were folded on itself. Another form of sensitive jet is very easily made by putting a piece of fine wire gauze 2 or 3 in. above a pinhole burner and igniting the gas above the gauze. On adjusting the gas so that it burns in a thin column, just not roaring, it is extraordinarily sensitive to some particular range of notes, going down and roaring when a note is sounded. _ If a tube be placed over such a flame it makes an excellent singing tube. The flame of an incandescent gas mantle if turned low is frequently sensitive to a certain range of notes. Such a flame may jump down, for instance, to each tick of a neighbouring clock.
Savart's Liquid Jets. â€” If a jet of water issues at an angle to the horizontal from a round pinhole orifice under a few inches pressure, it travels out as an apparently smooth cylinder for a short distance, and then breaks up into drops which travel at different rates, collide, and scatter. But if a tuning-fork of appropriate frequency be set vibrating with its stalk in contact with the holder of the pipe from which the jet issues, the jet appears to go over in one continuous thread. Intermittent illumination, however, with frequency equal to that of the fork shows at once that the jet is really broken up into drops, one for each vibration, and that these move over in a steady procession. The cylindrical form of jet is unstable if its length is more than ir times its diameter, and usually the irregular disturbances it receives at the orifice go on growing, and ultimately break it up irregularly into drops which go out at different rates. But, if quite regular disturbances are impressed on the jet at intervals of time which depend on the diameter and speed of outflow (they must be somewhat more than v times its diameter apart), these disturbances go on growing and break the stream up into equal drops, which all move with the same velocity one after the other. An excellent account of these and other jets is given in C. V. Boys' Soap Bubbles, lecture iii.
Maintenance of Vibrations. â€” When a system is set vibrating and left to itself, the vibration gradually dies away as the energy leaks out either in the waves formed or through friction. In order that the vibration may be maintained, a periodic force must be applied either to aid the internal restoring force on the return journey, or weaken it on the outgoing journey, or both. Thus if a pendulum always receives a slight impulse in the direction of motion just about the lowest point, this is equivalent to an increase of the restoring force if received before passage through the lowest point, and to a decrease if received after that passage, and in either case it tends to maintain the swing. If the bob of the pendulum is iron, and if a coil is placed just below the centre of swing, then, if a current passes through the coil, while and only while the bob is moving towards it, the vibration is maintained. If the current is on while the bob is receding the vibration is checked. If it is always on it only acts as if the value of gravity were increased, and does not help to maintain or check the vibration, but merely to shorten the period. In a common form of electrically maintained fork, the Electrically fork is set horizontal with its prongs in a vertical Maintained plane, and a small electro-magnet is fixed between pork. them. The circuit of the electro-magnet is made and broken by the vibration of the fork in different ways â€” say, by a wire bridge attached to the lower prong which dips into and lifts out of two mercury cups. The mercury level is so adjusted that the circuit is just not made when the fork is at rest. When it is set vibrating contact lasts during some part of the outward and some part of the inward swing. But partly owing to the delay in making contact through the carriage down of air on the contact piece, and partly owing to the delay in establishing full current through self, induction, the attracting force does not rise at once to its full valu& in the outgoing journey, whereas in the return journey the mercury tends to follow up the contact piece, and the full current continues up to the instant of break. Hence the attracting force does more work in the return journey than is- done against it in the outgoing, and the balance is available to increase the vibration.
In the organ pipe â€” as in the common whistle â€” a thin sheet of air is forced through a narrow slit at the bottom of the embouchure and impinges against the top edge, which is made very g-_ a _ p/ De sharp. The disturbance made at the commencement of the blowing will no doubt set the air in the pipe vibrating in its own natural period, just as any irregular air disturbance will set a suspended body swinging in its natural period, but we are to consider how the vibration is maintained when once set going. When the motion due to the vibration is up along the pipe from the em- bouchure, the air moves into the pipe from the outside, and carries the sheet-like stream in with it to the inside of the sharp edge. This stream does work on the air, aiding the motion. When the motion is reversed and the air moves out of the pipe at the embouchure, the sheet is deflected on to the outer side of the sharp edge, and no work is done against it by the air in the pipe. Hence, the stream of air does work during half the vibration and this is not abstracted during the other half, and so it goes on increasing the motion until the supply of energy in blowing is equal to the loss by friction and sound. ​
The maintenance of the vibration of the air in the singing tube has been explained by Lord Rayleigh (Sound, vol. ii. Â§ 322 h) as due â€ž. , to the way in which the heat is communicated to the
a & n X vibrating air. When the air in a pipe open at both
lube. ends is vibrating in its simplest mode, the air is
alternately moving into and out from the centre. During the quarter swing ending with greatest nodal pressure, the kinetic energy is changed to potential energy manifested in the increase of pressure. This becomes again kinetic in the second quarter swing, then in the third quarter it is changed to potential energy again, but now manifested in the decrease of pressure. In the last quarter it is again turned to the kinetic form. Now suppose that at the end of the first quarter swing, at the instant of greatest pressure, heat is suddenly given to the air. The pressure is further increased and the potential energy is also increased. There will be more kinetic energy formed in the return journey and the vibration tends to grow. But if the heat is given at the instant of greatest rare- faction, the increase of pressure lessens the difference from the un- disturbed pressure, and lessens the potential energy, so that during the return less kinetic energy is formed and the vibration tends to die away. And what is true for the extreme points is true for the half periods of which they are the middle points; that is, heat given during the compression half aids the vibration, and during the extension half damps it. Now let us apply this to the singing tube. Let the gas jet tube be of somewhat less than half the length of the singing tube, and let the lower end of the jet tube be in a wider tube or cavity so that it may be regarded as an " open end." When the air in the singing tube is singing, it forces the gas in the jet tube to vibrate in the same period and in such phase that at the nozzle the pressure in both tubes shall be the same. The lower end of the jet tube, being open, is a loop, and the node may be regarded as in an imaginary prolongation of the jet tube above the nozzle. It is evident that the pressure condition will be fulfilled only if the motions in the two tubes are in the same direction at the same time, closing into and opening out from the nodes together. When the motion is upwards gas is emitted ; when the motion is downwards it is checked. The gas enters in the half period from least to greatest pressure. But there is a slight delay in ignition, partly due to expulsion of incombustible gas drawn into the jet tube in the previous half period, so that the most copious supply of gas and heat is thrown into the quarter period just preceding greatest pressure, and the vibration is maintained. If the jet tube is somewhat longer than half the sounding tube there will be a node in it, and now the condi- tion of equality of pressure requires opposite motions in the two at the nozzle, for their nodes are situated on opposite sides of that point. The heat communication is then chiefly in the quarter vibration just preceding greatest rarefaction, and the vibration is not maintained.
Interference of Sound.
When two trains of sound waves travel through the same medium, each particle of the air, being simultaneously affected by the disturbances due to the different waves, moves in a different manner than it would if only acted on by each wave singly. The waves are said mutually to interfere. We shall exemplify this subject by considering the case of two waves travelling in the same direction through the air. We shall then obviously be led to the following results: â€”
If the two waves are of equal length X, and are in the same phase (that is, each producing at any given moment the same ,.---.^ .-â– '"â– -. state of motion in the
air particles), their com- bined effect is equivalent to that of a wave of the same length X, but by which the excursions of the particles are increased, being the sum of those due to the two component waves respectively, as In fig. 41, 1.
If the two interfering waves, being still of same length X, be in opposite phases, or so that one is in advance of the other by Â£X, and consequently one produces in the air the opposite state of motion to the other, then the resultant wave is one of the same length X, but the excursions of the particles are decreased, being the difference between those due to the component waves as in fig. 41, 2. If the amplitudes of vibration which thus mutually interfere are moreover equal, the effect is the total mutual destruction of the vibratory motion.
Thus we learn that two musical notes, of the same pitch, conveyed to the ear through the air, will produce the effect of a single note of the same pitch, but of increased loudness, if they are in the same phase, but may affect the ear very slightly, if at all, when in opposite phases. If the difference of phase
be varied gradually from zero to-X. the resulting sound will
2
gradually decrease from a maximum to a minimum.
Among the many experimental confirmations which may be adduced of these proportions we will mention the following: â€”
Take a circular plate, such as is available for the production of Chladni's figures, and cut out of a sheet of pasteboard a piece of the shape ABOCD (fig. 42), consisting of two circular quadrants of the same diameter as the plate. Let, now, the plate be made in the usual manner to vibrate so as to exhibit two nodal lines coinciding with two rectangular diameters. If the ear be placed right above the centre of the plate, the sound will be scarcely audible. But, if the pasteboard be interposed so as to intercept the vibrating segments AOB, DOC, the note becomes much more dis- tinct. The reason of this is, that the segments of the plate AOD, BOC always vibrate in the same direction, but oppo- sitely to the segments AOB, DOC. Hence, when the pasteboard is in its place, there are two waves of same phase starting from the two former segments, and reaching the ear after equal distances of transmission through the air, are again in the same phase, and produce on the ear a conjunct impression. But when the paste- board is removed, then there is at the ear opposition of phase between the first and the second pair of waves, and consequently a minimum of sound.
A tubular piece of wood shaped as in fig. 43. Â» nd having a piece of thin membrane stretched over the openingat the top C, some dry sand being strewn over the membrane, is so placed over a circular or rectangular vibrating plate that the ends A, B lie over the segments of the plate, such as AOD, COB in the previous figure, which are in the same state of motion. The sand at C will be set in violent movement. But if the same ends A, B be placed over oppositely vibrating segments (such as AOD, COD), the sand will be scarcely, if at all, affected.
If a tuning-fork in vibration be turned round before the ear, four positions will be found in which it will be inaudible, owing to the mutual interference of the oppositely vibrating prongs of the fork. On interposing the hand between the ear and either prong of the fork when in one of those positions, the sound becomes audible, because then one of the two interfering waves is cut off from the ear. This experiment may be varied by holding the fork over a glass jar into which water is poured to such a depth that the air-column within reinforces the note of the fork when suitably placed, and then turning the fork round.
Helmholtz's double siren is well calculated for the investigation of the laws of interference of sound. For this purpose a simple mechanism is found in the instrument, by means of which the fixed upper plate can be turned round and placed in any position relatively to the lower one. If, now, the apparatus be so set that the notes from the upper and lower chest are in unison, the upper fixed plate may be placed in four positions, such as to cause the air-current to be cut off in the one chest at the exact instant when it is freely passing through the other, and vice versa. The two waves, therefore, being in opposite phases, neutralize one another, and the result is a faint sound. On turning round the upper chest into any inter- mediate position, the intensity of the sound will increase up to a maximum, which occurs when the air in both chests is being admitted and cut off contemporaneously.
If two organ pipes in unison are mounted side by side on a wind- chest with their ends close together, and are blown for a very short time, they sound. But if the blowing is continued, usually in less than a second the sound dies away to a small fraction of that due to either alone. Yet the air within the pipes is vibrating more vigor- ously than ever, but in opposite phases in the two pipes. This may be shown by furnishing the pipes with manometric flames placed in the same vertical line. When the flames are viewed in a revolving mirror and the pipes are blown, each image of one flame lies between two images of the other. The essential fact, as pointed out by Lord Rayleigh (Scientific Papers, i. 409), is not the common wind chest, but the nearness of the open ends, so that the outrush from one pipe can supply the inrush to the other, and the converse. If, the two pipes are slightly out of tune when sounded separately together they sound a common note which may be higher than that due to either alone. Lord Rayleigh (loc. cit.) points out that this ​
is due to reduction of the end correction. When the air rushes out from one pipe, it has not to force its way into the open air, but finds a cavity being prepared for it close at hand in the other pipe, and so the extensions and compressions at the ends are more easily reduced. Even the longer pipe may be effectively shorter than the corrected shorter pipe when sounding alone.
Beats.
When two notes are not quite in unison the resulting sound is found to alternate between a maximum and minimum of loudness recurring periodically. To these periodical alternations has been given the name of Beats. Their origin is easily explic- able. Suppose the two notes to correspond to 200 and 203 vibrations per second; at some instant of time, the air particles, through which the waves are passing, will be similarly displaced by both, and consequently the joint effect will be a sound of some intensity. But, after this, the first or less rapidly vibrating note will fall behind the other, and cause a diminution in the joint displacements of the particles, till, after the lapse of one- sixth of a second, it will have fallen behind the other by half a vibration. At this moment, therefore, opposite displacements will be produced of the air particles by the two notes, and the sound due to them will be at a minimum. This will be followed by an increase of intensity until the lapse of another sixth of a second, when the less rapidly vibrating note will have lost another half-vibration relatively to the other, or one vibration reckoning from the original period of time, and the two component vibrations will again conspire and reproduce a maximum effect. Thus, an interval of one-third of a second elapses between two successive maxima or beats, and there are produced three beats per second. By similar reasoning it may be shown that the number of beats per second is always equal to the difference between the numbers of vibrations in the same time corresponding to the two interfering notes. The more, therefore, these are out of tune the more rapidly will the beats follow each other.
The formation of beats may be illustrated by considering the disturbance at any point due to two trains of waves of equal amplitude a and of nearly equal frequencies n$_1$ n$_2$. If we measure the time from an instant at which the two are in the same phase the resultant disturbance is
y＝a sin 2πn$_1$t+a sin 2πn$_2$t ＝ 2a cos π(n$_1$−n$_2$)t sin 7r(n$_1$+n$_2$)t,
which may be regarded as a harmonic disturbance of frequency (n$_1$+n$_2$)/2 but with amplitude 2a cos ir(Â«i â€” n 2 )2 slowly varying with the time. Taking the squares of the amplitude to represent the intensity or loudness of the sound which would be heard by an ear at the point, this is
4a 2 cos 2 ir(Â»iâ€” ni)t
＝ 2a 2 ji-(-cos 2t(Â»iâ€” rh)t),
a value which ranges between o and 4a 2 with frequency tti â€” Â«2. The sound swells out and dies down niâ€”nz times per second, or there are Â«i â€” n? beats per second. If, instead of considering one point in a succession of instants, we consider a succession of points along the line of propagation at the same instant, we evidently have waves of amplitude varying from 2a down to o, and then up to 20 again in distance U/(wiâ€” Â«j).
The phenomena of beats may be easily observed with two organ- pipes put slightly out of tune by placing the hand near the open end of one of them, with two musical strings on a resonant chest, or with two tuning-forks of the same pitch mounted on their resonance boxes, or held over a resonant cavity (such as a glass jar), one of the forks being put out of tune by loading one prong with a small lump of beeswax. In the last instance, if the forks are fixed on one solid piece of wood which can be grasped with the hand, the beat will be actually felt by the hand. If one prong of each fork be furnished with a small plain mirror, and a beam of light from a luminous point be reflected successively by the two mirrors, so as to form an image on a distinct screen, when one fork alone is put in vibration, the image will move on the screen and be seen as a line of a certain length. If both forks are in vibration, and are prefectly in tune, this line may either be increased or diminished permanently in length according to the difference of phase between the two sets of vibrations. But if the forks be not quite in tune then the length of the image will be found to fluctuate between a maximum and a minimum, thus making the beats sensible to the eye. The vibro- graph is also well suited for the same purpose, and so in an especial manner is Helmholtz's double siren, in which, by continually turning round the upper box, a note is produced by it more or less out of tune with the note formed by the lower chest, according as the handle is moved more or less rapidly, and most audible beats ensue. We
have already explained how beats are used on Scheibler's tonometer to give a series of forks of known frequencies. Beats also afford an excellent practical guide in the tuning of instruments, but more so for the higher notes of the register, inasmuch as the same number of beats are given by a smaller deviation from unison by two notes of high pitch than by two notes of low pitch. Thus, two low notes of 32 and 30 vibrations respectively, whose interval is therefore || or j4,i.e. a semitone, give two beats per second, while the same number of beats are given by notes of 32X16 (four octaves higher than the first of the preceding) or 512, and 514 vibrations, which are only slightly out of tune.
Beats and Dissonance. â€” As the interval between two tones, and consequently the number of beats, increases the effect on the ear becomes more and more unpleasant. The sound is jarring and harsh, and we term it a " dissonance " or " discord." In the middle notes of the musical register the maximum harshness occurs when the beats are about 30. Thus the interval b'c" with frequencies 495 and 528, giving 33 beats in a second, is very dissonant. But the interval b'\>c" gives nearly twice as many beats and is not nearly so dissonant. The minor third a'c" with 88 beats per second shows scarcely any roughness, and when the beats rise to 132 per second the result is no longer unpleasant.
We are then led to conclude that beats are the physical founda- tion for dissonance. The frequency of beats giving maximum dissonance rises as we rise higher in the musical scale, and falls as we descend. Thus b"c'" and b'\>c" have each 66 beats per second, yet the former is more dissonant than the latter. Again b'c" and CG have each 33 beats per second, yet the latter interval is practi- cally smooth and consonant. This beat theory of dissonance was first put forward by Joseph Sauveur (1653-1716) in 1700. Robert Smith (Harmonics, 2nd ed., 1759, p. 95) states that Sauveur " in- ferred that octaves and other simple concords, whose vibrations coincide very often, are agreeable and pleasant because their beats are too quick to be distinguished, be the pitch of the sounds ever so low; and on the contrary, that the more complex consonances whose vibrations coincide seldom are disagreeable because we can distinguish their slow beats; which displease the ear, says he, by reason of the inequality of the sound. And in pursuing this thought he found that those consonances which beat faster than six times in a second are the very same that musicians treat as concords; and that others which beat slower a re the discords ; and he adds that when a consonance is a discord at a low pitch and a concord at a high one, it beats sensibly at the former pitch but not at the latter." But Sauveur fixed the limiting number of beats for the discord far too low, and again he gave no account of dissonances such as the seventh, where the frequency of the beats between the funda- mentals is far beyond the number which is unpleasant. Smith, though recognizing the unpleasantness of beats, could not accept Sauveur's theory, and, indeed, it received no acceptance till it was rediscovered by Helmholtz, to whose investigations, recorded in his Sensations of Tone, we owe its satisfactory establishment.
Suppose that we start with two simple tones in unison; there is perfect consonance. If one is gradually raised in pitch beating begins, at first easily countable. But as the pitch of the one rises the beats become a jar too frequent to count, and only perhaps to a trained ear recognizable as beats. The two tones are now dissonant, and, as we have seen, about the middle of the scale the maximum dissonance is when there are between 30 and 40 beats per second. If the pitch is raised still further the dissonance lessens, and when there are about 130 beats per second the, interval is con- sonant. If all tones were pure, dissonance at this part of the scale would not occur if the interval were more than a third. But we have to remember that with strings, pipes and instruments gener- ally the fundamental tone is accompanied by overtones, called also " upper partials," and beating within the dissonance range may occur between these overtones.
Thus, suppose a fundamental 256 has present with it overtone harmonics 512, 768, 1024, 1280, &c, and that we sound with it the major seventh with fundamental 480, and having harmonics 960, 1440, &c. The two sets may be arranged thus c 256 512 768 1024 1280
b 480 960 1440,
and we see that the fundamental of the second will beat 32 times per second with the first overtone of the first, giving dissonance. The first overtone of the second will beat 64 times per second with the third of the first, and at such height in the scale this frequency will be unpleasant. The very marked dissonance of the major seventh is thus explained. We can see, too, at once how the octave is such a smooth consonance. Let the two tones with their harmonic overtones be
256
512 768 1024 1280 1536 512 1024 1536.
The fundamental and overtones of the second all coincide with overtones of the first.
Take as a further example the fifth with harmonic overtones as under
256 512 768 1024 1280 1536 384 768 1 1 52 1536. ​ The fundamental and overtones of the second either coincide with or fall midway between overtones in the first, and there is no approach to a dissonant frequency of beats, and the concord is perfect.
But obviously in either the octave or the fifth, if the tuning is imperfect, beats occur all along the line wherever the tones should coincide with perfect tuning. Thus it is easy to detect a want of tuning in these intervals.
The harshness of deep notes on instruments rich in overtones may be explained as arising from beats between successive overtones. Thus, if a note of frequency 64 is sounded, and if all the successive overtones are present, the difference of frequency will be 64, and this is an unpleasant interval when we get to the middle of the scale, say to overtones 256 and 320 or to 512 and 576. Thus Helmholtz explains the jarring and braying which are sometimes heard in bass voices. These cases must serve to illustrate the theory. For a full discussion see his Sensations of Tone, ch. 10.
Dissonance between Pure Tones.—When two sources emit only pure tones we might expect that we should have no dissonance when, as in the major seventh, the beat frequency is greater than the range of harshness. But the interval is still dissonant, and this is to be explained by the fact that the two tones unite to give a third tone of the frequency of the beats, easily heard when the two primary tones are loud. This tone may be within dissonance range of one of the primaries. Thus, take the major seventh with frequencies 256 and 480. There will be a tune frequency 480−256 = 224, and this will be very dissonant with 256.
The tone of the frequency of the beats was discovered by Georg Andreas Sorge in 1740, and independently a few years later by Giuseppe Tartini, after whom it is named. It may easily be heard when a double whistle with notes of different pitch is blown strongly, or when two gongs are loudly sounded close to the hearer. It is heard, too, when two notes on the harmonium are loudly sounded. Formerly it was generally supposed that the Tartini tone was due to the beats themselves, that the mere variation in the amplitude was equivalent, as far as the ear is concerned, to a superposition on the two original tones of a smooth sine displacement of the same periodicity as that variation. This view has still some supporters, and among its recent advocates are Koenig and Hermann. But it is very difficult to suppose that the same sensation would be aroused by a truly periodic displacement represented by a smooth curve, and a displacement in which the period is only in the amplitude of the to-and-fro motion, and which is represented by a jagged curve. No explanation is given by the supposition; it is merely a statement which can hardly be accepted unless all other explanations fail.
Combination Tones.—Helmholtz has given a theory which certainly accounts for the production of a tone of the frequency of the beats and for other tones all grouped under the name of “combination tones”; and in his Sensations of Tone (ch. 11) he examines the beats due to these combination tones and their effects in producing dissonance. The example we have given above of the major seventh must serve here. The reader is referred to the full discussion by Helmholtz. We shall conclude by a brief account of the ways in which combination tones may be produced. There appears to be no doubt that they are produced, and the only question is whether the theory accounts sufficiently for the intensity of the tones actually heard.
Combination tones may be produced in three ways: (1) In the neighbourhood of the source; (2) in the receiving mechanism of the ear; (3) in the medium conveying the waves.
1. We may illustrate the first method by taking a case dis- cussed by Helmholtz (Sensations of Tone, app. xvi.) where the two sources are reeds or pipes blown from the same wind-chest. Let us suppose that with constant excess of pressure, p, in the wind-chest, the amplitude produced is proportional to the pressure, so that the two tones issuing may be represented by pa sin 2-u-nit and pb sin 2-*nit. Now as each source lets out the wind periodi- cally it affects the pressure in the chest so that we cannot re- gard this as constant, but may take it as better represented by p + \a sin (2irnit-\-e)+nb sin (2irn 2 t+f). Then the issuing dis- turbance will be
[p+λ sin (2wn i t-\-e)+yb sin (2irnit-\-f)\\a sin 2irnit+b sin 2irÂ« 2 /j = pa sin 2wnit + pb sin 2jrn 2 /
-t cos e cos (4irn l /+e) + -~ cos/ j- cos (47rtt 2 Â«+/) + cos {2ir(Â»i Th)t+e) cos {2w(ni + n 2 )t+e]
H cos J2ir(Ki ni)t f\ cos (27t(m 1 +M 2 )<+/)
(35)
Thus, accompanying the two original pure tones there are (1) the octave of each; (2) a tone of frequency (n$_1$−n$_2$ ); (3) a tone of frequency (ni+n 2 ). The second is termed by Helmholtz the difference tone, and the third the summation tone. The amplitudes of these tones are proportional to the products of a and b multiplied by X or m- These combination tones will in turn react on the pressure and produce new combination tones with the original tones, or with each other, and such tones may be termed of the second, third, &c, order. It is evident that we may have tones of frequency
hn$_1$kn$_2$hn$_1$−kn$_2$hn$_1$+kn$_2$,
where h and k are any integers. But inasmuch as the successive orders are proportional to X X 2 X 3 , or m m 2 m 3 , and X and m are small, they are of rapidly decreasing importance, and it is not certain that any beyond those in equation (35) correspond to our actual sensations. The combination tones thus produced in the source should have a physical existence in the air, and the amplitudes of those represented in (35) should be of the same order. The conditions assumed in this investigation are probably nearly realized in a harmonium and in a double siren of the form used by Helmholtz, and in these cases there can be no doubt that actual objective tones are produced, for they may be detected by the aid of resonators of the frequency of the tone sought for. If the tones had no existence outside_ the ear then resonators would not increase their loudness. There is not much difficulty in detecting the difference tone by a resonator if it is held, say, close to the reeds of a harmonium, and Helmholtz succeeded in detecting the summation tone by the aid of a resonator. Further, Rücker and Edser, using a siren as source, have succeeded in making a fork of the appropriate pitch respond to both difference and summation tones (Phil. Mag., 1895. 39. p. 341. But there is no doubt that it is very difficult to detect the summation tone by the ear, and many workers have doubted the possibility, notwithstanding the evidence of such an observer as Helmholtz. Probably the fact noted by Mayer (Phil. Mag., 1878, 2, p. 500, or Rayleigh, Sound, § 386) that sounds of considerable intensity when heard by themselves are liable to be completely obliterated by graver sounds of sufficient force goes far to explain this, for the summation tones are of course always accompanied by such graver sounds.
2. The second mode of production of combination tones, by the mechanism of the receiver, is discussed by Helmholtz (Sensations of Tone, App. xii.) and Rayleigh (Sound, i. § 68). It depends on the restoring force due to the displacement of the receiver not being accurately proportional to the displacement. This want of proportionality will have a periodicity, that of the impinging waves, and so will produce vibrations just as does the variation of pressure in the case last investigated. We may see how this occurs by- supposing that the restoring force of the receiving mechanism is represented by λx+μx$^2$, where x is the displacement and μx$^2$ is very small. Let an external force F act on the system, and for simplicity suppose its period is so great compared with that of the mechanism that we may take it as practically in equilibrium with the restoring force. Then F = λx+μx$^2$. Now μx$^2$ is very small compared with λx, so that x is nearly equal to F/λ, and as an approximation, F=λx+μF$^2$/λ$^2$ , or * = F/X-mF 2 /X 3 . Suppose now that F = a sin 2πn$_1$t+ b sin 2πn$_2$t, the second term will evidently produce a series of combination tones of periodicities 2n$_1$, 2n$_2$, n$_1$−n$_2$, and n$_1$+n$_2$, as in the first method. There can be no doubt that the ear is an unsymmetrical vibratorrand that it makes combination tones, in some such way as is here indicated, out of two pure tones. Probably in most cases the combination tones which we hear are thus made, and possibly, too, the tones detected by Koenig, and by him named “beat-tones.” He found that if two tones of frequencies p and q are sounded, and if q lies between Np and (N + 1)p, then a tone of frequency either (N + 1)p−q, or of frequency q−Np, is heard. The difficulty in Helmholtz ? s theory is to account for the audibility of such beat tones when they are of a higher order than the first. Riicker and Edser quite failed to detect their external existence, so that apparently they are not produced in the source. If we are to assume that the tones received by the ear are pure and free from partials, the loudness of the beat- tones would appear to show that Helmholtz's theory is not a complete account.
3. The third mode of production of combination tones, the production in the medium itself, follows from the varying velocity of different parts of the wave, as investigated at the beginning of this article. It is easily shown that after a time we shall have to superpose on the original displacement a displacement proportional to the square of the particle velocity, and this will introduce just the same set of combination tones. But probably in practice there is not a sufficient interval between source and hearer for these tones to grow into any importance, and they can at most be only a small addition to those formed in the source or the ear.
Bibliography.—For the history of experimental and theoretical acoustics see F. Rosenberger, Geschichte der Physik (1882-1890); J. C. Poggendorff, Geschichte der Physik (1879) ; and E. Gerland and F. Traumüller, Geschichte der physikalischen Experimentierkunst (1899). The standard treatise on the mathematical theory is Lord Rayleigh's Theory of Sound (2nd ed., 1894) ; this work also contains an account of experimental verifications. The same author's Scientific Papers contains many experimental and mathematical contributions to the science. H. von Helmholtz treats the theoretical aspects of sound in his Vorlesungen uber die mathematischen ​Principien der Akustik (1898), and the physiological and psychical aspects in his Die Lehre von den Tonempfindungen (1st ed., 1863; 5th ed., 1896), English translation by A. J. Ellis, On the Sensations of Tone (1885). Sedley Taylor, Sound and Music (1882), contains a simple and excellent account of Helmholtz’s theory of consonance and dissonance. R. Koenig, Quelques expériences d’acoustique (1882) describes apparatus and experiments, intended to show, in opposition to Helmholtz, that beats coalesce into tones, and also that the quality of a note is affected by alteration of phase of one of its component overtones relative to the phase of the fundamental. Lamb, The Dynamical Theory of Sound (1910), is intended as a stepping-stone to the study of the writings of Helmholtz and Rayleigh. Barton, A Text-Book on Sound (1908), aims to provide students with a text-book on sound, embracing both its experimental and theoretical aspects. J. H. Poynting and J. J. Thomson, Sound (5th ed., 1909), contains a descriptive account of the chief phenomena, and an elementary mathematical treatment. John Tyndall, Sound (5th ed., 1893), originally delivered as lectures, treats the subject descriptively, and is illustrated by a large number of excellent experiments. Good general accounts are given in J. L. G. Violle, Cours de physique, tome ii., “Acoustique”; A. Winkelmann, Handbuch der Physik, Band ii., “Akustik”; Müller-Pouillet, Lehrbuch der Physik, (1907), ii. 1; L. A. Zellner, Vorträge über Akustik (1892), pt. 1, physical; pt. 2, physiological; R. Klimpert, Lehrbuch der Akustik (1904–1907); A. Wüllner, Lehrbuch der Experimentalphysik (1907), 6th ed., vol. i.; and C. L. Barnes, Practical Acoustics (1898), treats the subject experimentally.  (J. H. P.)