are so strong as to completely drown the fundamental, which is very weak.

The clarionet consists essentially of an open reed-pipe, the length of which can be altered by opening and shutting various valves. The note given is very rich in overtones, and it is owing to the presence of these overtones that this instrument owes its characteristic sound.

In the cornet and horn there is no reed attached to the instrument, but the lips of the performer vibrate and perform the functions of a reed. In the cornet, the length of the tube can be altered at will, and thus the different notes can be obtained. In the horn and bugle the length of the tube remains constant, but the performer alters the manner in which his lips vibrate so as to make the pipe give its different overtones, so that in these instruments all the possible notes that can be obtained are the overtones of the fundamental note which the pipe will give.

CHAPTER VI

SUPPLY OF ENERGY TO A SOUNDING BODY

RESONANCE

C

308. Vibrations Maintained by Heat. In the case of organpipes, which are the only sources of sound which we have considered which are capable of giving a maintained note, the energy necessary to maintain the vibrations, and make up for the energy which is radiated as sound-waves, is supplied by the blast of air used to make the pipe "speak." We have now to consider other methods by which the energy which is communicated to the air as sound-waves by a sounding body can be supplied. In the case of a tuning-fork, say, the energy necessary to supply the sound-waves is derived from the loss of energy of motion of the prongs, so that the sound gradually dies out. In some cases, however, the necessary energy is supplied in the form of heat. The most familiar case of sound being produced by heat is Trevelyan's rocker. This instrument consists of a piece of copper or iron, the cross section of which is shown at CD (Fig. 268), which is heated and then rested on a block of lead, AB. Under these circumstances the rocker gives out a musical note. The cause of the vibrations is the expansion of the lead owing to the heat conducted from the rocker. Suppose the rocker to be resting on the edge C more heavily than on D, the result will be that heat will flow more rapidly into the lead at C than at D. This heat will cause the lead to expand immediately under the edge C, and this expansion will tilt the rocker over on to the other edge. The conduction of heat will now be greater at the edge D, so that the lead will now expand under it and cause the rocker to tilt back on to the first edge, when the whole process will be repeated. Thus the rocker is set in vibration and gives out a note, the energy being supplied by the heat of the rocker; in fact the arrangement forms a heat engine in which the rocker is the source and the lead block the sink, and some of the heat of the source is converted into energy of motion, while at the same time a portion of the heat passes from a higher to a lower temperature, that is, passes to the sink.

A

D

B

FIG. 268.

If a jet of hydrogen gas be placed within a vertical tube open at both ends, then in general a loud note will be produced, which will continue as

are so stro

weak.

The c
which ca
given is

overtor
In

but t'

In t'

diff

tul

lit

tl

(

414

as the gas jet remains alight

inbration.

is intermittent, so

Sound

The same

· phenomenon is exhibited by

that the size of the flame, and hence also the supply of

If

bets of other combustible gases, but to a less marked degree. in connection with manometric flames, it will be seen that the flame is the dame is observed by means of a rotating mirror, similar to that used By using the stroboscopic method of observing the flame, vibration the flame retires inside the jet through which the gas is Tipler was able to show that in many cases at one time during each suppled. It is also found that the length of the gas supply-tube bears an with cotton-wool near the jet the gas flame, although it appears just as important part in the phenomenon. If the supply-tube is lightly plugged obtained with any given flame depend on the length of the supply-tube usual, is incapable of producing vibrations, while the notes which can be the supply-tube is that the emission of the gas, instead of being uniform, waves are set up in the supply-tube. The effect of these vibrations in and on the nature of the gas. These observations indicate that stationary mittent. Now when a column of air is in vibration and heat is supplied heat to the air contained in the tube which surrounds the flame, is internormal condition of pressure. The effect of this will be similar to that increase the force with which the gas tends to expand, i.e. to regain its to the air at the moment of greatest condensation, this supply of heat will produced when a pendulum is struck a blow at the end of its swing tending to drive it back towards its position of rest, namely, it will tend supply of heat takes place when the air is at its greatest rarefaction, this to increase the amplitude of the vibrations. will tend to resist the return of the air to its condition of rest, and will therefore tend to check the vibrations. Just as in the case of the pendulum, if it is struck a blow tending to check its motion as it is passing through its position of rest, the amplitude will decrease. Hence, if the periodic increase in the size of the flame always occurs at the instant when the air, in that portion of the tube near the flame, is, owing to the natural vibrations of the column of air in the tube, at its maximum condensation, the amplitude of the vibrations will be increased or at any rate maintained. If, however, the increase in size of the flame occurs sometimes at the instant of maximum condensation and sometimes at that of maximum rarefaction, that is, if the natural periods of the column of gas in the supply-tube and of the column of air in the tube are not commensurate, the heat will sometimes assist the vibrations and sometimes oppose. Hence, under these circumstances, the vibrations of the air in the tube will not on the whole be maintained by the heat, and so will die out. It will thus be evident why it is necessary that the length of the supply-tube and the position of the flame should bear definite relations to the length of the tube in order that a sound may be produced. When a plug of cotton-wool is placed in the supply-tube vibrations can no longer take place in the gas contained in the tube, and so the variations in the

If, on the other hand, the

size of the flame, which are necessary if the vibrations in the air column are to be kept up, are not produced.

309*. The Energy of a Vibrating String.-When a string is vibrating transversely, it possesses energy due to its condition. When it is at its maximum elongation on either side of its position of rest, it is momentarily at rest, and so its energy is entirely potential, that is, is stored up owing to the deformation of the string. When the string is passing through its position of rest, its energy is entirely kinetic.

Let the mass of unit length of the string be m, then as the string vibrates each unit of length will vibrate backwards and forwards in a simple harmonic motion. Let the amplitude of the vibrations executed by an element of the string of unit length, and therefore of mass m, be a, and let its displacement from its position of rest at a given instant be x. Then, as shown in § 51, the velocity with which the element is moving is

where n is the frequency of the vibrations executed by the string. Hence the kinetic energy of the element is

2π2n2m(a2 – x2).

Now the acceleration with which the element is moving when its displacement is ris

hence the force acting to produce this acceleration is equal to the product of the mass into the acceleration, or

42n2mx.

This is the force of restitution when the displacement is r, and we see that it is proportional to the displacement a Hence if we draw a line, OP (Fig. 260), to represent the connec

tion between the displacement and the force of restitution, it will be a straight line passing through the origin, for when the displacement is zero, so is the force of restitution.

If NP represents the force of restitution when the displacement is x, the work which has been done against the force of restitution to displace the element to r is equal to the area of the triangle OPN (§ 77). Hence, as the potential energy when the displacement

x

FORCE OF RESTITUTION

P

N

DISPLACEMENT
FIG. 269.

is r is equal to the work done in displacing the element to from its position of rest, the potential energy is

But NP is the force of restitution when the displacement is r, so that the potential energy is

22n2mx2.

Hence the total energy, both potential and kinetic, of the element when the displacement is x is given by

2π2n2m(a2 − x2)+2π2n2mx2
=2=2n2ma2.

Since this expression for the total energy does not involve the displacement r, we see that the total energy remains constant throughout the vibration, as of course it must, and we simply have changes from the potential to the kinetic form, and vice versa, during the motion.

To find the total energy of the whole string we have to add together the energy due to all the elements, so that the total energy is

where the amplitude a varies from element to element.

To proceed any further we must make some assumption as to the relation between the amplitudes of the different parts of the string. If / is the length of the string and A is the amplitude at the centre, then, if the string is vibrating in its fundamental form, we may represent the amplitude of a point at a distance d from one end by the expression

a= A sin dl.

When do or d=l, that is, at the ends, a is zero, for sin o and sin are both zero. When d=7/2, that is, at the middle of the string, a=4, for sin π/2 = 1. Hence the expression does give us the correct values of the amplitude at the ends and the centre. Substituting this expression for a, we get the total energy equal to

Now the expression sin dl does not involve the amplitude with which the string is vibrating, neither does the expression 2′′22, Hence the total energy of a vibrating string is proportional to the square of the amplitude A with which the centre is vibrating.

Now the only scientific method of measuring the intensity of the vibrations of a body is to consider the energy which the body possesses on account of these vibrations. Hence we see that the intensity of the vibrations of a string are proportional to the square of the amplitude of the vibrations.

By a similar line of argument it can be shown that in the case of all vibrations the energy is proportional to the square of the amplitude. Hence the intensity of all vibrations is proportional to the square of the amplitude.