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that the directions of minimum intensity are not quite in the plane perpendicular to the line of impact, being nearer the side of the lighter ball.

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A result of some importance indicated by theory is that when one of the spheres is much heavier than the other, replacing the former by a still heavier sphere of the same diameter should not result in any important alteration in the distribution of the intensity of sound in different directions due to impact. This is clear from expression (16). For when U is much larger than U, any diminution in the value of U, should not appreciably affect the value of the expression. This indication of theory is in agreement with experiment. Several series of measurements have been made with various pairs of balls of the same size but of different densities, e. 9., wood and marble, wood and iron, billiard ball and iron ball, and so forth. Generally, similar results are obtained in all cases. It was noticed also that the form of the intensity distribution as shown by the ballistic phonometer was not altogether independent of the thickness of the mica disk used in the instrument. This is not surprising, as the behaviour of the mica disk before the pointer

attached to it ceases to touch the mirror of the indicator would no doubt depend, to some extent, on the relation between its natural frequency and the frequency of the sound-waves set up by the impact. The best results were obtained with a disk neither so thick as to be relatively insensitive nor so thin as to remain with its pointer in contact with the indicator longer than absolutely necessary.

4. The general case of spheres of any diameter
and density.

When the impinging spheres are both of different diameter and of different density, the result generally obtained is that the sound is a maximum on the line of impact in either direction, and a minimum which approaches zero in directions asymmetrically situated with reference thereto. Generally speaking, no maxima in lateral directions are noticed, that is, the curve consists of two nearly closed loops. The difference of the intensity of the sound in the two directions of the line of impact may sometimes be very considerable. As a typical case, the results obtained by the impact of a sphere of wood 3 inches in diameter with a brass sphere only 1 inch in diameter are shown in fig. 8. It is observed that the sound due to impact is actually of greater intensity on the side of the small brass ball. As a matter of fact, the result generally obtained is that the intensity is greater on the side of the ball of the denser material even if its diameter be the smaller.

The mathematical treatment of the general case is precisely on the same lines as in the two preceding sections. It is found in agreement with the experimental result that in practically all cases in which both the densities and the diameters are different, the zonal harmonic of the first order is of importance and that the intensity curve consists of two nearly closed loops, as in the case of two spheres of the same diameter but of different density.

5. Summary and Conclusion.

The investigation of the origin and characteristics of the sound due to the direct impact of two similar solid spheres which was described in the Phil. Mag. for July, 1916, has been extended in the present paper to the cases in which the impinging spheres are not both of the same diameter or

material. The relative intensities of the sound in different directions have been measured by the aid of the ballistic phonometer, and in order to exhibit the results in an effective manner, they have been plotted in polar coordinates, the point at which the spheres impinge being taken as the origin, and the line of collision as the axis of x. As might be expected, the curves thus drawn show marked asymmetry in respect of the plane perpendicular to the line of impact.

A detailed mathematical discussion of the nature of the results to be expected is possible by considering the analogous case of two rigid spheres nearly in contact which vibrate bodily along their line of centres. By choosing an appropriate wave-length for the resulting motion, intensity curves similar to those found experimentally for the case of impact are arrived at. A further confirmation is thus obtained of the hypothesis regarding the origin of the sound suggested by the work of Hertz and of Lord Rayleigh on the theory of elastic impact.

When the impinging spheres, though not equal in size, are of the same or nearly the same density, the intensity-curve drawn for the plane of observation shows the sound to be a maximum along the line of impact in either direction, and also along two directions making equal acute angles with this line. The sound is a minimum along four directions in the plane. In practically all other cases, that is when the spheres differ considerably either in density alone, or both in diameter and density, the intensity is found to be a maximum along the line of impact in either direction, and to be a minimum along directions which are nearly but not quite perpendicular to the line of impact. The form of the intensity curve is practically determined by the diameters and the masses of the spheres.

The investigation was carried out in the Physical Laboratory of the Indian Association for the Cultivation of Science. It is hoped when a suitable opportunity arises to study also the case of oblique impact. The writer has much pleasure in acknowledging the helpful interest taken by Prof. C. V. Raman in the progress of the work described in the present.

paper.

Calcutta, 15th June, 1917.

XI. On the Asymmetry of the Illumination - Curves in Oblique Diffraction. By SISIR KUMAR MITRA, M.Sc., Sir Rashbehary Ghosh Research Scholar in the University of Calcutta.

[Plate V.] Introduction.

N the Phil. Mag. for May 1911, C. V. Raman has given the results of a photometric study of the unsymmetrical diffraction-bands due to an obliquely held rectangular reflecting surface previously observed by him†. The measurements showed a very marked asymmetry in the distribution of intensity in the diffraction pattern, the theoretical explanation of which is discussed in the papers quoted. The following were the principal conclusions arrived at by Raman as the result of the quantitative experimental study of the

case:

(a) The illumination at the points of minimum intensity in the diffraction pattern is zero at all angles of incidence, and the positions of the minima are accurately given by the formula

8=±π, ±2π, ±3π, &c.,

where &= (sini-sin ), a being the width of the aperture,

πά
λ

λ the wave-length, and i, the angles of incidence and diffraction respectively; the fringes are wider on the side on which >i and their number is limited on that side, as

π

cannot be greater than 2'

(b) The formula of the usual type (I=sin2 8/82) for the illumination in the pattern fails to represent the observed intensity-curves at oblique incidences except in regard to the position of the minima (8=+π, +2π, &c.). The intensities at corresponding points on either side of the central fringe for which the values of & are numerically the same are not equal.

(c) The observed distribution of intensity was found to fit in with the theoretical formula, if the latter is multiplied by a factor proportional to the square of the cosine of the

*Communicated by Prof. C. V. Raman.

† C. V. Raman, M.A., "On the Unsymmetrical Diffraction Bands due to a Rectangular Aperture," Phil. Mag, Nov. 1906. See also Phil. Mag.

Jan. 1909.

obliquity, which, of course, is not the same at all points in the diffraction pattern. In other words, the ordinates of the illumination curve were found to be proportional to the expression cos sin2 8/82.

The question arises whether these results, practically those indicated in (b) and (c) above, are peculiar to the case of a surface of rectangular form, or whether similar phenomena might be expected with other forms of surface as well. The cases which it seemed of particular interest to examine are those in which the reflecting surface is not a single individual area but consists of two, three, or more parallel elements lying in the same plane. A satisfactory surface of this kind which can be used at very oblique incidences may be prepared by etching out deep grooves on the optically plane surface of a thick plate of glass with hydrofluoric acid, the edges of the reflecting strips left on the surface being subsequently ground so as to be sharp, straight, and parallel. I have prepared several such surfaces containing two and three equidistant reflecting strips respectively. By placing one of these on the table of a spectrometer, the diffraction pattern produced by reflexion at very oblique incidences may be readily observed through the telescope of the instrument. The present paper describes the results of the quantitative study of the phenomena thus obtained. Incidentally the opportunity has also been taken of testing the results obtained by Raman for the case of a single aperture using improved optical and photographic appliances. The experiments and determinations have throughout been made using monochromatic light. This was secured by illuminating the slit of the spectrometer with light of a definite wave-length isolated by a monochromator from sunlight or arc light.

Unsymmetrical Interference-fringes due to two
parallel apertures.

Fig. I (Pl. V.) reproduces a photograph of the diffraction pattern due to a surface containing two reflecting elements each of width 0.48 cm., and 3.60 cm. apart. The direct image of the slit of the spectrometer also appears in the figures to the right of the diffraction-pattern. The photograph is reproduced from a dense negative taken to show the perfect blackness of the minima of illumination, and the progressive increase (from left to right) in the width of the interference-fringes of the light diffracted by the two reflecting elements. It was obtained by replacing the telescope Phil. Mag. S. 6. Vol. 35. No. 205. Jan. 1918. I

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