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F = — Bmp2 cos pt— Bkp sin pt + Bh (cos pt + cos pt);
Let F denote the first harmonic term of F (that is, the sum of the terms in cos pt and sin pt), while F, denotes the term in cos 3pt. Then, since
When m, k, h, a, and B are given, the minimum value of the right-hand side corresponds to
which diminishes continually as p2 increases. Hence in general terms we may say that to produce a given amplitude of velocity requires the least amplitude of impressed periodic force when the periodic time of the force is somewhat shorter than would be the natural period of the system for infinitesimal vibrations if the frictional term were abolished. And returning to the case of our strings, we may infer that a similar result will hold good. Hence a force of given finite amplitude will excite in a given string the greatest amplitude of velocity when the period of the force is somewhat shorter than the "natural" period of the string, and hence when a periodic disturbance of finite amplitude reaches the ear, the string whose resonance is excited most strongly will have a "natural" period longer than the period of the disturbance.
7. For example, let the note c be sounded with very small intensity, and let that part of the basilar membrane which responds most strongly be called the c-string. If then the same note c is made to sound close to the ear with considerable intensity, the strongest resonance will be excited no longer in the c-string, but in some string of longer natural period, such as the Bb-string. Now according to Helmholtz's theory
of sound-perception our estimate of pitch depends entirely on localization of the most strongly agitated portion of the basilar membrane, so that in the case just considered, when the note e was made to change from a very small to a very much greater intensity, we should expect to hear not only an increase of loudness, but also a lowering of pitch.
8. At the same time it is not to be supposed that the agitation of the "c-string" is lessened when the intensity of the note c is augmented. The effect must necessarily be in the other direction, so that such a note powerfully sounded close to the ear must perceptibly excite a tract of the basilar membrane corresponding to a considerable range of pitch (the place of maximum disturbance being probably not far from the middle of this range).
9. We must even suppose that an increase in the intensity of a note without change of frequency causes the "range of stimulation" (as we may call it) to extend to patches a little higher than before, as well as to those a good deal lower.
10. In connexion with § 8 it may be remarked that, when the ear is kept close to the resonant cavity of a stronglyvibrating c-fork, the impression of pitch obtained is far from being very definite. One pitch or another within an appreciable range may be heard by a mere effort of attention. On the other hand, a more definite impression is obtained on alternating between smaller and greater intensities of the same note. Thus, on raising or lowering the head, as mentioned in § 3, the interval between the soft note and the loud note may appear at first to be about a minor third, when the fork is sounding strongly. As the amplitude of vibration dies down, the interval diminishes, and it is possible to say pretty definitely when it is just a whole tone, and when it is only a semitone; until, as the note in either position of the head becomes nearly inaudible, the apparent difference of pitch is obliterated.
11. If the orifice of the ear remote from the fork is left open, the sound reaching that ear will be less intense than is heard by the other ear, and the corresponding pitch will be higher. Though of course beats are entirely absent*, it might be thought that two distinct pitches would be heard simultaneously, but this requires a distinct effort of attention. The general impression is of a pitch intermediate between those which would be heard by the two ears separately, so that on
*In the discussion Prof. S. P. Thompson suggested that, if the observations were valid, beats should occur between the notes heard by the two ears; but a consideration of the physical conditions will show that nothing of this kind is to be expected.
closing the further ear a distinct lowering of pitch takes place. For this reason the effects described in § 3 are most marked when the less stimulated ear is kept closed.
12. I have not yet mentioned an effect noticed by an observer in whom hearing with one ear was not normal. With the less sensitive ear the usual effect was reversed; that is, on bringing the ear close to the resonance-cavity of the c-fork, the pitch, instead of falling, appeared to rise by about a semitone. Even taking § 9 into account, this result seems rather anomalous.
13. Though variations of pitch accompanying variations of loudness must frequently have been observed, the physiological influences at work do not appear to have been suspected *. And yet these subjective influences must be by no means negligible in the case of wind-instrument players; and even from players of stringed instruments I think I have heard that it is easier to judge of another player's intonation than to be quite certain about one's own. On the other hand, as regards the tuning of the intervals of consonant chords, it must be remembered that the actual criterion is usually the obliteration of beats, so that on this point the judgment is not disturbed by a small subjective uncertainty in the estimation of pitch. I have also convinced myself that, when harmonic upper partials are present, the sense of tonality is largely dependent upon them; and harmonics from their higher pitch and generally feebler intensity will suffer less subjective repression of pitch than the fundamental.
II. Objective Demonstration of Combination-Tones.
14. When two notes differing in frequency are powerfully sounded in the neighbourhood of one another, secondary tones are produced, of which the most prominent have frequencies respectively equal to the sum and difference of the frequencies of the parent tones. It has been maintained by some writers that these secondary tones are purely subjective, and have no existence external to the ear, and the following experiment was designed to show that in some cases at least the first difference-tone has a real physical existence. If the vibration of the air external to the ear has really a component of the frequency in question, we must suppose this component to have arisen from a failure of the principle of superposition, that is, from the circumstance that as the parent vibrations
*For example, Lord Rayleigh mentions (Theory of Sound,' 2nd ed. vol. i. § 67) that "tuning-forks rise a little, though very little, in pitch as the vibration dies away."
are of considerable amplitude, the equations of motion cannot be regarded as sensibly linear; and accordingly it is to be expected that if two sources of sound are brought closer together, the intensity of the difference-tone will increase. I have used two stopped organ-pipes of white metal giving the notes e and g', the difference-tone being consequently C. The pipes are connected to a well-weighted organ-bellows by flexible rubber tubes, and the distance between them can be varied at will from a few feet to a couple of inches (the walls of the two pipes being then in contact). It is best for the listener to be stationed in a distant room, so that the sound which reaches him is only of moderate intensity, while the comparatively small distances through which the pipes are moved does not appreciably affect the sounds which they individually send to his ear. The bellows being filled, the sounding pipes are held alternately far apart and near together, and each time, as they nearly approach one another, the difference-tone C1 is heard to boom out with greatly increased intensity. From this, I would suggest, we are to infer that the difference-tone has a real objective existence.
(Further experiments have not confirmed this result. Even Prof. Rücker's very sensitive arrangement of interferencebands failed to show any objective difference-tone from the two organ-pipes, suitably tuned. An attempt will be made to deal with the subject more fully in a subsequent paper.)
To the Editors of the Philosophical Magazine.
10 Charnwood Street, Derby, April 3, 1895.
GENTLEMEN, N 1888 I communicated to the February No. of the Phil. Mag. a short paper on glacier motion. Since then I have had an opportunity of examining, in company with Mr. G. Fletcher, F.G.S., the structure of some of the largest Swiss and Norwegian glaciers. The results of our investigation are given in a paper printed in the April number of the Geological Magazine for the current year.
In it we show that the small granular particles of ice resulting from the partial melting and refreezing of the snow, and also the minute crystal particles collected during frosty sunless days, in course of time, when they have become welded by pressure into a compact mass, undergo a striking change. The greater number of the granules and crystalline particles disappear, the molecules composing them going to build up a comparatively small number of large crystalline Phil. Mag. S. 5. Vol. 39. No. 240. May 1895.
particles. In our paper we remark: "Although it is not quite clear why some of the grains should increase in size and others disappear, the transference of the molecules from crystal to crystal offers no difficulty. . . . ."
Since this was written I have been led to think that the known laws regulating surface-tension phenomena perhaps afford an explanation for the disappearance of the smaller granules and the growth of the larger ones.
Glacier-ice consists of a kind of conglomerate formed of glacier-grains. These grains have very irregular outlines Fig. 1.
(4) Horizontal slice from the Eismeer of Untergrindelwald Glacier.
() Vertical section from Mer de Glace Glacier.
and fit closely together. Figs. 1 and 2 are drawings of slices of glacier-ice as seen under polarized light.
Fig. 2 is a sample of veined ice, the crystalline particles having been sheared by the motion of the glacier.
Geol. Mag. April 1895, p. 155.