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the damped wave-train is about half as large again as for the steady ray.
Utilizing these values of for calculating the intensity of the reflected ray, we get the pair of curves in fig. 9. The
Curves showing the difference between the intensities of the portions of damped and undamped wave-trains reflected from an infinitely thick plate of conductivity 01 X 10-9.
result is striking. The energy of the reflected waves for the
Experimental evidence for these results would be very
I am glad to have this opportunity of expressing my indebtedness to Professor Karl Pearson, to whom my best thanks are due for much advice and assistance.
TABLE of the principal Symbols used.
The suffix 1 refers to the dielectric; the suffix 2 to the plate. In our case, where all media are assumed to be non-magnetic, B-dielectric constant, a=4 x conductivity.
b=ratio of the reflected to the incident amplitude at the first surface of the plate.
e=the corresponding ratio at the second surface.
c ratio of the refracted to the incident amplitude at the first surface.
f=the corresponding ratio at the second surface. d-thickness of slab.
λ=wave-length in dielectric.
-the change of phase on reflexion at the first surface.
p=-p1+ip, is defined by (4).
n, x are defined by (9).
X gives the rate of damping of the wave-train, being 90° for a steady ray.
p, are defined by (10).
XXXIII. On the Objective Reality of Combination Tones. By A. W. RÜCKER, M.A., F.R.S., and E. EDSER, A.R.C.S.* THE question of the objective existence of combination
tones has been keenly disputed. At first they were regarded as produced within the ear itself. Von Helmholtz in part adopted this view, and gave a theoretical explanation of the way in which the construction of the ear might lend itself to such a result. (Sensations of Tone,' App. XII.) He also believed that they existed objectively when the
* Communicated by the Physical Society: read March 22, 1895. Phil. Mag. S. 5. Vol. 39. Nọ. 239. April 1895. 2 A.
amplitudes of the vibrations of the notes which give rise to them are so great that powers higher than the first have to be considered. He supported this view by mathematical demonstrations, and stated ( Sensations of Tone,' transl. by A. Ellis, p. 157) that he had proved their objective existence by making membranes and resonators to respond to combinational tones produced by the siren and harmonium. These views and statements have been adversely criticised by König, Bosanquet, and Preyer. A very lucid account of the controversy was given by the late Mr. Ellis in his translation of the Tonempfindungen, and the net result of the impression produced in his mind is shown by two notes on pages 156 and 157. He there states that the result of Mr. Bosanquet's and Prof. Preyer's experiments is to show that the combinational tones are produced in the ear itself, and that it is probable that the apparent reinforcement of the resonators noticed by Helmholtz arose from imperfect blocking of both ears when using them.
These statements were unqualified, and no condition was made as to the way in which the combination-tones were produced. Helmholtz, for reasons which we need not recount, regarded the siren as the best instrument for producing objective combination-tones; and we recently determined to submit the question of their existence, which seemed to be decided against him, to another experimental test.
In this paper we give the result of our investigations, as far as they have at present been carried out. We do not regard them as complete, but they at all events prove that when the conditions under which we experimented are fulfilled, there can be no doubt that difference and summation-tones are produced which are capable of disturbing resonating bodies.
The resonator employed in the first instance was a tuningfork. It is well known that this instrument is relatively difficult to excite by resonance, and it was therefore necessary to use an extremely delicate method of detecting whether it was set in motion. For this purpose a mirror attached to one of the prongs was made one of a system by which Michelson's interference-bands were produced. A movement of the prong amounting to half a wave-length of light (say 1/80,000 of an inch) would alter the length of the path of one of the interfering rays by a wave-length. A periodic vibration of this amplitude would cause the band to disappear.
It is therefore evident that an extremely minute movement could be detected. It was at first open to question whether the apparatus would not be so sensitive to accidental dis
turbances as to be untrustworthy. This difficulty has, however, been entirely overcome.
A plan of the apparatus is shown in fig. 1. The shank of the tuning-fork, F, is imbedded in a mass of lead. One of
the prongs carries a mirror, M. To the other is attached a square of wood, W, of larger area but of the same weight as the mirror. A Lissajous' figure (an ellipse) was formed by reflexion from the mirror and from a small square of silvered glass attached to one of König's large forks adjusted to give the C of 64 complete vibrations per second, and the pitch of F was adjusted until only one beat occurred in two minutes. The pitch was thus very accurately known. A double siren, S, was placed between a large König resonator tuned to 64 vibrations and a wooden cone or pyramid, C. The end of the cone was placed about half an inch from W, which was rather larger than the narrow end of the cone. The sensitiveness of the apparatus depended in part upon the distance between W and C. If the distance was too large, the sensitiveness was diminished. If it was too small, the instrument was unduly affected by chance puffs of air even when not periodic in character.
A source of light, L, was used to form a system of interference-bands by means of the half-silvered mirror M1, and the two mirrors M and M2. The two interfering rays travelled over the paths LM,M,M,B and LM,MM1B respectively. The distances MM, and M1M, were approximately equal; and since when soda-light was used a change in the length of either of the paths of × 589 μμ would cause a dark band to shift into the position previously occupied by the next bright band, and since, further, any movement of M altered the length of the path of the ray by twice its own magnitude, it is evident that a movement of x 589 μμ, or, say, of one hundred thousandth of an inch, could be easily detected.
The fork F and the mirrors M, M1, and M, rested on a square stone, which was suspended by wires and india-rubber
door-fasteners from a heavily weighted beam, which itsel rested on india-rubber balls placed at a convenient height on a double pair of wooden "steps." With these precautions it was found that the bands remained tolerably steady even by day when persons were moving about the building, and when the traffic on the frost-bound road produced considerable mechanical vibrations. Even under these conditions, we have satisfied all who have seen the apparatus of the reality of the phenomena. The experiments on which we rely, however, were made on several occasions between midnight and 2 or 3 a.m. The bands were then absolutely clear and steady. They were undisturbed for many minutes at a time when the bellows were being worked and the siren was sounding loudly. It was only when a note of 64 vibrations per second was directly or indirectly produced that they vanished, and there could be no possible doubt or mistake as to whether the disturbance was or was not produced by the sound or combination of sounds under investigation.
Up to the present we have used the fork above described only. It was chosen because it was fairly stiff, and as removable metal mirrors for the production of Lissajous' figures were attached to its prongs, it was possible to replace them by the glass mirror and square of wood without altering its pitch. It would be quite possible to use properly made forks of higher pitch as resonators, and the steadiness of the bands at night is so remarkable, that we believe that if the apparatus were set up in the country, on a stone isolated from the rest of the room, the degree of sensitiveness we have attained could be far surpassed.
Tuning the Siren.
Three methods were used for determining when the siren was producing the required notes. When one of these was fairly high, the beats given by it and a standard fork were noticed, and the note could thus be kept hovering about the required pitch for a considerable time. Although with the aid of König's large forks we could apply this method to vibration-frequencies of 48 per second and upwards, it was difficult when the notes were very low to recognize the beats with sufficient certainty.
The prongs of various tuning-forks were therefore furnished with two pieces of tin-foil, which opened and closed a slit made in them twice in every complete vibration. They were also compared with a standard by the aid of a revolving cylinder, and were adjusted by weights to the required frequencies. These were so selected as to make one of the