CHAPTER IV. DIFFRACTION. 35. WHEN light after passing through an aperture, whose dimensions are comparable with the ordinary standards of measurement, is received upon a screen, the boundary of the luminous area is well defined; similarly if an obstacle of sufficient size is placed in the path of the incident light, a well-defined shadow of the obstacle is cast upon the screen. It thus appears that, as long as the apertures or obstacles with which we are dealing are of moderate dimensions, light travels in straight lines. Now it is well known that sound does not in all cases travel in straight lines; for if a band is playing a piece of music out of doors, a person seated in a room with an open window can hear the music distinctly, even though his position may be such as to prevent him seeing any of the musicians. The objection was therefore raised against the undulatory theory in its infancy, that inasmuch as sound is known to be due to aerial waves, and that such waves are able to bend round corners, a theory which seeks to explain optical phenomena by means of the vibrations of a medium, ought to lead to the conclusion that light as well as sound is capable of bending round corners, which is contrary to ordinary experience. The reason of this apparent discrepancy between observation, and what was at first supposed to be the result of the undulatory theory, arises from the fact that the wave-length of light is exceedingly small compared with the linear dimensions of such apertures and obstacles as are ordinarily met with, whilst the wave-lengths of audible sounds are not small compared with them'. In fact it requires as extreme conditions to produce a shadow in the case of sound, as it does to avoid producing one in the case of 1 The wave-length of the middle c of a pianoforte is about 4.2 feet. light. At the same time it is quite possible for a sound-shadow to be produced. Thus :-"Some few years ago a powder-hulk exploded on the river Mersey. Just opposite the spot, there is an opening of some size in the high ground which forms the watershed between the Mersey and the Dee. The noise of the explosion was heard through this opening for many miles, and great damage was done. Places quite close to the hulk, but behind the low hills through which the opening passes, were completely protected, the noise was hardly heard, and no damage to glass and such like happened. The opening was large compared with the wave-length of the sound1." On the other hand it is not difficult to produce a soundshadow with an obstacle of small dimensions, by means of a sensitive flame and a tuning-fork, which yields a note whose wavelength is so short as to be inaudible; for although the vibrations of the air produced by the tuning-fork are incapable of affecting the ear, yet they are capable of producing a well-marked disturbance of the sensitive flame, by means of which the existence or non-existence of the sound is made manifest. And if an obstacle be held between the tuning-fork and the flame, it is observed that the oscillations of the latter either cease altogether or appreciably diminish, which shows that a sound-shadow has been produced. 36. When light passes through an aperture, such as a narrow slit, whose dimensions are comparable with the wave-length of light, and is received on a screen, it is found that a well-defined shadow of the boundary of the aperture is no longer produced. If homogeneous light be employed, a series of bright and dark bands is observed on those portions of the screen, which are quite dark when the dimensions of the aperture are large in comparison with the wave-length of light; and if white light be employed, a series of coloured bands is produced. Experiments with small apertures thus show, that light is capable of bending round corners under precisely the same conditions as sound; and thus the objections which were formerly advanced against the undulatory theory fall to the ground. These phenomena are usually known by the name of Diffraction, the object of the present 1 Glazebrook, Physical Optics, p. 149. 2 For further information on the Diffraction of Sound, see Lord Rayleigh's Theory of Sound, ch. xiv. and Proc. Roy. Inst. Jan. 20, 1888. chapter is to show, that they are capable of being accounted for by means of the undulatory theory. 37. Let us suppose that plane waves of light are passing through an aperture in a screen, whose plane is parallel to that of the wave-fronts. Each wave upon its arrival at the aperture may be conceived to be divided into small elements dS. If O be any point at a distance from the screen, it is clear that every element dS must contribute something to the disturbance which exists at O. When we consider the dynamical theory of diffraction', it will be shown that, if we suppose that the disturbance existing in that portion of the wave which passes through the aperture, is the same as if the screen in which the aperture exists were not present, or that the wave passed on undisturbed; the vibration at O produced by an element dS of the primary wave, would be represented by the expression cdS (1 + cos 0) sin & cos 2π (Vt-r).........(1), where is the distance of O from dS, 0 and are the angles which makes with the normal to ds drawn outwards and with the direction of vibration respectively, and c sin 27 Vt is the displacement of the primary wave at the plane of resolution. In all cases of diffraction, the illumination is insensible unless the inclination of r to the screen is small, which requires that should be small and nearly equal to π; we may therefore as a sufficient approximation put cos = sin = 1, and the formula becomes and the resultant vibration at O will be obtained by integrating this expression over the area of the aperture. The formula (1), which is due to Stokes, will be hereafter rigorously deduced by means of a mathematical investigation, which is based on the assumption that the equations of motion of the luminiferous ether are of the same form as those of an elastic solid, whose power of resisting compression is very large in comparison with its power of resisting distortion. It will not 1 Stokes, "On the Dynamical Theory of Diffraction;" Trans. Camb. Phil. Soc. vol. ix. p. 1; and Math. and Phys. Papers, vol. 1. p. 243. however be necessary at present to enter upon any investigations of this character, since all the leading phenomena of diffraction may be explained by means of the Principle of Huygens'. 38. Let O be any point towards which a plane wave is advancing; draw OP=r perpendicular to the front of the wave, and with O as a centre describe a series of concentric spheres whose radii are r + ±λ, ... r + {nλ. These spheres will divide the wave-front into a series of circular annuli, which are called Now PM2= (r+nλ)2 — r2, and therefore, if Huygens' zones. A be neglected, PM, nrλ, and the area of each zone is equal to πηλ. = 2 Let cos (Vt-r), where κ = 2π/, be the displacement at O due to the original wave. Then it might be thought, that the displacement at O due to an element at M, would be An cos x (Vt-r — Inλ); this however is not the case, inasmuch, as we shall presently see, that it is necessary to suppose the phases of successive elements to be different from that of the primary wave. Let e be this difference of phase; then since the amplitude of the zone is proportional to its area, the displacement produced by the nth zone will be TrλA COS K (Vt − r — {nλ + e) = πrλ (-)" A, cos x (Vt-r+e). cos − : Accordingly the total displacement at O is πrλ {A ̧ − A1 + A2- ... + (-)"An} cos x (Vt − r + e). 1 2 Now the amplitude of the vibration produced at O by any 1 The so-called Principle of Huygens is not a very satisfactory or rigorous method of dealing with the question of the resolution of waves. The reader may therefore, if he pleases, assume for the present the truth of Stokes' law. See Ch. XIII. and also, Proc. Lond. Math. Soc. vol. xxп. p. 317. zone, is inversely proportional to its distance from 0; we may therefore write An= Bn/(r + {nλ), and the series becomes also since Ban, B+1, Ban+2 are very nearly equal whence every term of the series is approximately neutralized by half the sum of the terms which immediately precede and succeed it; accordingly the effect of the wave upon a distant point O is almost entirely confined to half that of the central portion PM, which remains over uncompensated. It therefore follows that the displacement at O is equal to Since this expression must be equal to the displacement produced at O by the primary wave, we must have Bλ = 1, e = {λ. We thus obtain the important theorem that the displacement produced at O by any element dS of the primary wave This result may also be obtained, as was done by Archibald Smith', by integrating over the whole wave-front, for |