Page:1902 Encyclopædia Britannica - Volume 25 - A-AUS.pdf/73

 ACOUSTICS harmonics are most readily heard if we fortify the ear by an air cavity with a natural period equal to that of Resonators. the harmonic to be sought. The form used by Helmholtz is a globe of thin brass 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 card-board tube closed at one end, with the open end near the ear, will often suffice, and it may be tuned by more or less covering up the open end. If the harmonic corresponding to the resonator is present its tone swells out loudly. This resonance is a particular example of the general principle that a vibrating system will be set in vibration by any periodic force applied to it, and ultimately in the period of the force, its own natural vibrations gradually dying down. 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. 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 ser es ex en ^Fourier ^ t ds over the whole range of audible analyser, pitch. We need not here enter into the question of the structure constituting these resonators. Each of them is supposed to have its own 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 (Rayleigh, Sound, ii. § 388) thought that if the intensity of x’esponse 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 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. Determinations of the pressure changes, or extent of excursion of the air, in sounding organ pipes have been made by Kundt (Fogg. Ann. cxxxiv. 1868, p. ofvibra^ or Tbpler and Boltzmann {Fogg. Ann. cxli., tion. Rayleigh, Sound, § 422a), and Mach {OptischAhustiscken Versuche). 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

53

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 stroboscopic method being used to get the position of the vapour line at a definite part of the vibration. Mach found an excursion of OH cm. at the end of an open pipe 125 cm. long. The amplitude found by the other observers was of the same order. Lord Rayleigh has made experiments by two different methods to determine the amplitude of vibration in waves only just audible {Sound, ii. § 384). It will Minimum be sufficient to indicate the second method, amplitude A fork of frequency 256 was used as the of audible source. The energy of this fork with a given soundsamplitude 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 42T eTea/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 x 10~7 cm. Other forks gave results not very different. M. Wien (Wied. Ann. xxxvi. 1889, p. 834) by another method has obtained a smaller value still, though one of the same order. He used a globular resonator in which the pressure variations could be determined by a contrivance something like a delicate aneroid barometer. From the pressure changes the amplitude was known, and from Helmholtz’s theory of the resonator the amplitude in the external exciting waves was calculated. All the results show that “ the streams of energy required to influence the eye and ear are of the same order of‘magnitude.” The simple theory of the vibration of air in pipes, due to Bernouilli, practically assumes that a stationary wave is formed with a node at a closed end and a loop at an open end. But (as pointed out v'brnt‘ons in O. A. § 85) the condition for a loop at an pipes.0 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 not 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. § 307), that for a cylindrical tube of radius R, provided with a flat extended flange, the loop may be regarded as about