Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/454

Rh 428 WAVE T H E O R Y The peculiarity to be explained appears to depend upon the cur vature of the surfaces bounding the plate. For simplicity suppose that the lower surface is plane (/ = 0), and that the approximate equation of the upper surface is y^=a + bx-, a being thus the least distance between the plates. The black of the n ih order for wave length A occurs when an.l thus the width (&e) at this place of the band is given by ^ = 2bxSx (13), or If the glasses be in contact, as is usually supposed in the theory of Newton s rings, = 0. and 5.raA 5, or the width of the band of the i th order varies as the square root of the wave-length, instead of as the first power. Even in this case the overlapping and subse quent obliteration of the bands is greatly retarded by the use of the prism, but the full development of the phenomenon requires that a should be finite. Let us inquire what is the condition in order that the width of the band of the ?i th order may be stationary, as A varies. By (14) it is necessary that the variation of &quot;/(fyi - a) should vanish. Hence a = , so that the interval between the surfaces at the place where the ii th band is formed should be half due to curvature and half to imperfect contact at the place of closest approach. If this condition be satisfied, the achromatism of the ?i th band, elFected by the prism, carries with it the achromatism of a large number of neighbouring bands, and thus gives rise to the remarkable effects described by Newton. 9. Nc ivtons Diffusion Rings, In the fourth part of the second book of his Optics Newton investigates another series of rings, usually (though not very appropriately) known as the colours of thick plates. The funda mental experiment is as follows. At the centre of curvature of a concave looking-glass, quicksilvered behind, is placed an opaque card, perforated by a small hole through which sunlight is admitted. The main body of the light returns through the aperture ; but a series of concentric rings are seen upon the card, the formation of which was proved by Newton to require the co-operation of the two surfaces of the mirror. Thus the diameters of the rings depend upon the thickness of the glass, and none are formed when the glass is replaced by a metallic speculum. The brilliancy of the rings depends upon imperfect polish of the anterior surface of the glass, and may be augmented by a coat of diluted milk, a device used by the Due de Chaulnes. The rings may also be well observed without a screen in the manner recommended by Stokes. For this purpose all that is required is to place a small flame at the centre of curvature of the prepared glass, so as to concide with its image. The rings are then seen surrounding the flame and occupying a definite position in space. The explanation of the rings, suggested by Young, and developed by Herschel, refers them to interference between one portion of light scattered or diffracted by a particle of dust, and then regularly refracted and reflected, and another portion first regularly refracted and reflected and then diffracted at emergence by the same Stokes s particle. It has been shown by Stokes 1 that no regular inter- prin- ference is to be expected between portions of light diffracted by ciple. different particles of dust. In the memoir of Stokes will be found a very complete discus sion of the whole subject, and to this the reader must be referred who desires a fuller knowledge. Our limits will not allow us to do more than touch upon one or two points. The condition of fixity of the rings when observed in air, and of distinctness when a screen is used, is that the systems due to all parts of the diffusing surface should concide ; and it is fulfilled only when, as in Newton s experiments, the source and screen are in the plane passing through the centre of curvature of the glass. Plane As the simplest for actual calculation, we will consider a little mirror further the case where the glass is plane and parallel, of thickness and lens, t and index /u., and is supplemented by a lens at whose focus the source of light is placed. This lens acts both as collimator and as object-glass, so that the combination of lens and plane mirror replaces the concave mirror of Newton s experiment. The retardation is calcu lated in the same way as for thin plates. In fig. 2 the diffracting particle is situated at B, and we have to find the relative retardation of the two rays which emerge finally at inclina tion 6, the one diffracted at emergence following the path ABDBIE, and the other diffracted at entrance and following the path ABFGH. Fig. 2. The retardation of the former from B to I is 2/j.t + Bl, and of the latter from B to the equivalent place G is 2^BF. Now FB = 2sec0, 8 being the angle Trans.. Is. p. 147, 1801. of refraction ; BI = 2tan0 sinfl ; so that the relative retardation R is given by R, = 2^ { 1 + (i- l tan ff sin - sec 6 } = Ijd (I - cos 6 ) . If 6,9 be small, we may take as sufficiently approximate. The condition of distinctness is here satisfied, since R, is the same for every ray emergent parallel to a given one. The rays of one parallel system are collected by the lens to a focus at a definite point in the neighbourhood of the original source. The formula (1) was discussed by Herschel, and shown to agree with Newton s measures. The law of formation of the rings follows immediately from the expression for the retardation, the radius of the ring of n th order being proportional to n and to the square root of the wave-length. 10. Hiujfjcns s Principle. Theory of Shadoivs. The objection most frequently brought against the undulatory theory in its infancy was the difficulty of explaining in accordance with it the existence of shadows. Thanks to Fresno! and his fol lowers, this department of optics is now precisely the one in which the theory has secured its greatest triumphs. The principle employed in these investigations is due to Huygens, and may be thus formulated. If round the origin of waves an ideal closed surface be drawn, the whole action of the waves in the region beyond may be regarded as due to the motion continually propagated across the various elements of this surface. The wave motion due to any element of the surface is called a secondary wave, and in estimating the total effect regard must be paid to the phases as well as the amplitudes of the components. It is usually convenient to choose as the surface of resolution a wave-front, i.e., a surface at which the primary vibrations are in one phase. Any obscurity that may hang over Huygens s principle is due mainly to the indefiniteness of thought and expression which we must be content to put up with if we wish to avoid pledging our selves as to the character of the vibrations. In the application to sound, where we know what we are dealing with, the matter is simple enough in principle, although mathematical difficulties would often stand in the way of the calculations we might wish to make. The ideal surface of resolution may be there regarded as a flexible lamina ; and we know that, if by forces locally applied every element of the lamina be made to move normally to itself exactly as the air at that place does, the external aerial motion is fully determined. By the principle of superposition the whole effect may be found by integration of the partial effects due to each ele ment of the surface, the other elements remaining at rest. We will now consider in detail the important case in which uniform plane waves are resolved at a surface coincident with a wave-front (OQ). We imagine the wave-front divided into element ary rings or zones, called Huygens s zones, by spheres described round P (the point at which the aggregate effect is to be estimated), the first sphere, touching the plane at 0, with a radius equal to PO, and the succeeding spheres with radii increasing at each step by ^A. There are thus marked out a series of circles, whose radii x are given by ar-t-r 2 =(r + ^?iA) 2, or x 2 = nr. L-*_.-_ nearly ; so that the rings are at first of nearly equal I / area. Now the effect upon P of each element of the plane is proportional to its area ; but it depends also upon the distance from P, and possibly upon the inclination of the secondary ray to the direc tion of vibration and to the wave-front. These Huy- gens s principle Plane primary wave. Huy gens s zones. Fg. questions will be further considered in connexion with the dyna mical theory ; but under all ordinary circumstances the result is independent of the precise answer that may be given. All that it is necessary to assume is that the effects of the successive zones gradually diminish, whether from the increasing obliquity of the secondary ray or because (on account of the limitation of the region of integration) the zones become at last more and more incomplete. The component vibrations at P due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to /ero. In such a series each term may be regarded as very nearly indeed destroyed by the halves of its immediate neighbours, and thus the sum of the whole series is represented by naif the first term, which stands over uncompensated. The question is thus reduced to that of finding the effect of the first zone, or central circle, of which the area is itXr. We have seen that the problem before us is independent of the law of the secondary wave as regards obliquity; but the result of the integration necessarily involves the law of the intensity and phase of a secondary wave as a function of r, the distance from the origin. And we may in fact, as was done by A. Smith,&quot; determine - Cainh. Jfnt/i. Juur., iii. p. 40, 1843.
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