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 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 long as the gas jet remains alight. The same phenomenon is exhibited by burning jets of other combustible gases, but to a less marked degree. If the flame is observed by means of a rotating mirror, similar to that used in connection with manometric flames, it will be seen that the flame is in vibration. By using the stroboscopic method of observing the flame, Töpler was able to show that in many cases at one time during each vibration the flame retires inside the jet through which the gas is supplied. It is also found that the length of the gas supply-tube bears an important part in the phenomenon. If the supply-tube is lightly plugged with cotton-wool near the jet the gas flame, although it appears just as usual, is incapable of producing vibrations, while the notes which can be obtained with any given flame depend on the length of the supply-tube and on the nature of the gas. These observations indicate that stationary waves are set up in the supply-tube. The effect of these vibrations in the supply-tube is that the emission of the gas, instead of being uniform, is intermittent, so that the size of the flame, and hence also the supply of heat to the air contained in the tube which surrounds the flame, is intermittent. Now when a column of air is in vibration and heat is supplied to the air at the moment of greatest condensation, this supply of heat will increase the force with which the gas tends to expand, i.e. to regain its normal condition of pressure. The effect of this will be similar to that 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 to increase the amplitude of the vibrations. If, on the other hand, the supply of heat takes place when the air is at its greatest rarefaction, this 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 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 r. Then, as shown in § 51, the velocity with which the element is moving is 2πn √a2 - x2, where n is the frequency of the vibrations executed by the string. Hence the kinetic energy of the element is 2nm(ax). Now the acceleration with which the element is moving when its displacement is a is hence the force acting to produce this acceleration is equal to the product of the mass into the acceleration, or 4mx. This is the force of restitution when the displacement is .r, and we see that it is proportional to the displacement x. Hence if we draw a line, OP (Fig. 260), to represent the connection 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 tor is equal to the area of the triangle OPN (§ 77). Hence, as the potential energy when the displacement FORCE OF RESTITUTION P N DISPLACEMENT is is equal to the work done in displacing the element to r 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 Hence the total energy, both potential and kinetic, of the element when the displacement is r is given by 22n2m(a2x2)+2π2n2mx2 Since this expression for the total energy does not involve the displacement, 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 TM are both zero. When d=1/2, that is, at the middle of the string, a=A, for sin 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 22nm42 sin 2dl, or Now the expression sin dl does not involve the amplitude with which the string is vibrating, neither does the expression 22nm. 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. 310. Decrease of the Amplitude of Waves with Increase of Distance from the Source.-Suppose we have a centre of disturbance A within an isotropic medium, so that the wave-fronts are spheres with A as centre. Let R1 be the radius of one of the spherical wave-fronts, and let the amplitude of the waves as they cross the surface of this sphere be . Similarly let A be the amplitude of the waves when they reach a sphere of radius R. Now if we consider a thin shell of radius R1 and thickness r, the energy due to the waves contained in this shell is proportional to the volume of the shell and to the square of the amplitude. Thus the energy is equal to where K is a constant. 4 Rx.A.K, Now the waves travel out carrying their energy with them, and when they reach the sphere of radius R, the energy contained in a shell of thickness will be 4 Rx.A. K. Now the waves which occupy this new shell are the same that some time previously occupied the shell of radius R1, and so the energy contained within the new shell must be equal to that which was contained within the old. We are here, of course, supposing that the waves can travel through the medium without any of the energy being dissipated as heat, &c., communicated to the medium. Hence, equating the energy contained within the two shells, we get That is, the amplitude decreases as the distance from the centre of disturbance. The intensity of the wave-motion being proportional to the square of the amplitude, it follows that the intensity decreases as the square of the distance from the centre of disturbance. This result is, of course, applicable to the case of sound-waves, so that the intensity of a sound varies inversely as the square of the distance from the sounding body. This only applies if the sound-waves are propagated in free air, so that the wave-fronts are spheres. If the sound is propagated along the air contained within a tube, the cross section of the wave-fronts at all distances from the source remains the same, and hence the decrease in amplitude is only due to heat produced by friction of the moving air against the sides of the tube, and such like causes. 311. Damping. When a vibrating body produces a sound, the energy of the sound-waves which travel out from the body is derived from the energy of vibration of the body; also a certain amount of the energy is converted into heat on account of viscosity of the particles of the body and friction. This loss of energy causes a gradual decrease in the amplitude of the vibrating body, unless energy is supplied to the |