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fringes seen in the field are due only to the interference of the direct and reflected wave-trains. The phenomena noticed as the focal plane is advanced towards the source of light, represent a gradual transition from this stage to one in which Airy's theory becomes fully applicable. In the transition-stages the field of illumination is a continuous whole, of which, however, the different parts present distinct characteristics. First, within the geometrical edge of the shadow, we have a finite number of fringes (one, two, or more according to the position of the plane of observation, but not an indefinitely large number as contemplated by Airy's theory); these may be regarded as the interferencefringes in the neighbourhood of the caustic due to the reflected light alone. Following these we have a long train of fringes due to the interference of the direct and the reflected pencils. The first few of these should evidently be modified by the diffraction which the direct rays suffer at the edge C before they reach the observing microscope. Finally, we may also have a part of the field in which the illumination is due only to the direct pencil, the reflected rays not entering the objective of the microscope owing to their obliquity. This part of the field should appear less brightly illuminated than the rest.
19. A complete theoretical treatment of the transitionstages described in the preceding paragraph is somewhat difficult, and has to be deferred to some future occasion. There is no difficulty, however, in calculating the positions of the fringes due to the interference of the direct and the reflected pencils when the focal plane is in advance of the edge, provided the diffraction-effect due to the edge is neglected. It is easily shown that the path-difference between the direct and reflected rays at a point a' is given by
where is measured from C' and d=CC'. By putting S'nλ and eliminating 0, the positions of the minima of illumination may be calculated. A complete agreement of the results thus obtained with those found in experiment cannot, however, be expected, as the fringes are narrow and the modifications due to diffraction are not negligible. As regards the fringes alongside the caustic due to the reflected rays, we cannot expect to find a complete agreement between their widths and those found from Airy's theory, so long as the latter is not fully applicable. The divergence, if any,
should be most marked when the region of the caustic under observation is nearest the edge of the cylinder, and for the fringes which are farthest from the caustic.
20. The foregoing conclusions have been tested by a series of measurements made with the focal plane in various positions in advance of the edge. To prove that the boundary of the field within the shadow is the caustic and not the surface of the cylinder, measurements were made of the length C'P1, the rays incident on the cylinder being a parallel pencil.
Observed value of C'P1.
The following shows the widths of the fringes observed in the neighbourhood of the caustic when the focal plane was 1.6 mm. in advance of the edge, and the widths calculated from Airy's theory.
Observed Calculated... 155, (Airy's theory.)
Width of fringes in cm. x 10-5.
The agreement in both cases is satisfactory.
21. Table IV. shows the results of measurements made of the fringes in the transition-stages when the focal plane was only a little in advance of the edge, and Airy's theory is not fully applicable. The observed results are in general agreement with the indications of theory set out in paragraph 19. It will be seen that the fringes farthest within the region of the shadow show a fair agreement with Airy's theory, and the others are more nearly in agreement with the widths calculated from formula (B).
Summary and Conclusion.
22. C. F. Brush has recently published some observations of considerable interest on the diffraction of light by cylindrical edges. The views put forward by him to explain the phenomena, however, present serious difficulties and are open to objection. My attention was drawn to this subject by Prof. C. V. Raman, at whose suggestion the present work was undertaken by me in order to find the true explanation of the effects, and to develop a mathematical theory which would stand a quantitative test in experiment. This has now
been done, and in the course of the investigation various features of importance overlooked by Brush have come to light. The following are the principal conclusions arrived at: (a) The fringes seen in the plane at which the incident light grazes the cylinder are due to the simple interference of the direct and the reflected rays, the positions of the dark bands being given by the formula x=3. (2a)}. (nλ)3 ; (b) the fringes in a plane further removed from the source of light than the cylinder are due to diffraction at the edge grazed by the incident rays but modified by interference with the light reflected from the surface of the cylinder. The positions of the dark bands in these fringes are (to a close approximation) given by the formulæ * x=2d0+3a02/2, and nλ=2d02+2a03, from which 0 is to be eliminated; (c) when the focal plane of the observing microscope is on the side of the cylinder towards the light, the direct and reflected rays do not both cover exactly the same part of the field, and by putting the focal plane sufficiently forward towards the light,
* This formula is subject to a small correction which is of importance only when d is large.
they may be entirely separated. When this is the case, the fringes of the ordinary Fresnel type due to the edge of the cylinder may be observed, and inside the shadow we have also an entirely separate system of fringes due to the reflected rays, the first and principal maximum of which lies alongside the virtual caustic formed by oblique reflexion; the distribution of intensity in this system can be found from the well-known integral due to Airy; (d) but when the focal plane is only a little in advance of the edge, the caustic and the reflecting surface are nearly in contact, and Airy's investigation of the intensity in the neighbourhood of a caustic requires modification. It is then found that only a finite number of bands (one, two, three, or more according to the position of the plane of observation) is formed within the limits of the shadow, and not an indefinitely large number as contemplated by Airy's theory. The rest of the fringes seen in the field are due to the interference of the direct and reflected rays, but modified by diffraction at the edge of the cylinder.
The Indian Association for the Cultivation of Science, Calcutta, 8th May, 1917.
X. On Aerial Waves generated by Impact. Part II. By SUDHANSUKUMAR BANERJI, M.Sc., Assistant Professor of Applied Mathematics, University of Calcutta*.
HE origin and characteristics of the sound produced by the collision of two solid spheres were discussed by me at some length in the first paper under the same title that was published in the Philosophical Magazine for July, 1916. It was shown in that paper that the sound is not due to the vibrations set up in the spheres, which in any ordinary material are both too high in pitch to be audible and too faint in intensity, but to aerial waves set up by the reversal of the motion of the spheres as a whole. The intensity of the sound in different directions for the case in which the two spheres were of the same material and diameter, was investigated by the aid of a new instrument which will be referred to as
* Communicated by Prof. C. V. Raman. Phil. Mag. S. 6. Vol. 35. No. 205. Jan. 1918.
"the ballistic phonometer *." The intensity was found to be a maximum along the line of collision, falling off gradually in other directions to a value which is practically zero on the surface of a cone of semi-vertical angle 67°, and rising again to a second but feebler maximum in a plane at right angles to the line of collision.
In view of the interesting results obtained for the case of two equal spheres, it was arranged to continue the investigation and to measure the distribution of intensity when the colliding spheres were not both of the same radius or material. A mathematical investigation of the nature of the results to be expected in these cases was also undertaken. In order to exhibit the results of the measurements and of the theoretical calculation, a plan has now been adopted which is much more suitable than the one used in the first paper. This will be best understood by reference to fig. 1 (Pl. IV.), which refers to the case of two spheres of the same material and diameter. The figure has been drawn by taking the point at which the spheres impinge as origin, and the line of collision as the axis of a, and setting off the indications of the ballistic phonometer as radii vectores at the respective angles which the directions in which the sound is measured make with the line of collision. The curve thus represents the distribution of intensity round the colliding spheres in polar coordinates, the points at which the intensity of the sound is measured being assumed to be all at the same distance from the spheres. The results are brought much more vividly before the eye by a diagram of this kind than by plotting the results on squared paper.
2. Case of two spheres of the same material but of
Fig. 2, which shows the observed distribution of intensity when two spheres of wood of diameters 3 inches and 24 inches collide with each other, is typical of the results obtained when the impinging spheres are nearly of the same density and are of different diameters. There is a distinct asymmetry about a plane perpendicular to the line of impact. In addition to the maxima of intensity in the two directions of the line of collision, we have the maxima in lateral directions, which are not at right angles to this line. The
*This name was suggested by Prof. E. H. Barton, D.Sc., F.R.S., writing in the Science Abstracts,' p. 399, Sept. 1916.