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DTZ

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Originally appearing in Volume V08, Page 255 of the 1911 Encyclopedia Britannica.
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DTZ cos nt (17), the disturbance expressed by ,=TZsin¢ cos(nt-kr) 4abr (18) The occurrence of sin 4, shows that there is no disturbance radiated in the direction of the force, a feature which might have been anticipated from considerations of symmetry. We will now apply (18) to the investigation of a law of secondary disturbance, when a primary wave =sin(nt-kx) . . . . (19) is supposed to be broken up in passing the plane x= o. The first step is to calculate the force which represents the reaction between the parts of the medium separated by x=o. The force operative upon the positive half is parallel to OZ, and of amount per unit of area equal to -b2D di/dx=b2kD cos nt; and to this force acting over the whole of the plane the actual motion on the positive side may be conceived to be due. The DZ dx dy dz, . . (13). . . r(14). . . (15), . . (16), secondary disturbance corresponding to the element dS of the plane may be supposed to be that caused by a force of the above magnitude acting over dS and vanishing elsewhere ; and it only remains to examine what the result of such a force would be. Now it is evident that the force in question, supposed to act upon the positive half only of the medium, produces just double of the effect that would be caused by the same force if the medium were undivided, and on the latter supposition (being also localized at a point) it comes under the head already considered. According to (18), the effect of the force acting at dS parallel to OZ, and of amount equal to 2b2kD dS cos nt, will be a disturbance dS sin ¢ cos (nt—kr) . . . . (20), regard being had to (12). This therefore expresses the secondary disturbance at a distance r and in a direction making an angle 4, with OZ (the direction of primary vibration) due to the element dS of the wave-front. The proportionality of the secondary disturbance to sin is common to the present law and to that given by Stokes, but here there is no dependence upon the angle 8 between the primary and secondary rays. The occurrence of the factor (Xr)-r, and the necessity of supposing the phase of the secondary wave accelerated by a quarter of an undulation, were' first established by Archibald Smith, as the result of a comparison between the primary wave, supposed to pass on without resolution, and the integrated effect of all the secondary waves (§ 2). The occurrence of factors such as sin 4i, or 1(r+cos 8), in the expression of the secondary wave has no influence upon the result of the integration, the effects of all the elements for which the factors differ appreciably from unity being destroyed by mutual interference. The- choice between various methods of resolution, all mathematically admissible, would be guided by physical considerations respecting the mode of action of obstacles. Thus, to refer again to the acoustical analogue in which plane waves are incident upon a perforated rigid screen, the circumstances of the case are best represented by the first method of resolution, leading to symmetrical secondary waves, in which the normal motion is supposed to be zero over the unperforated parts. Indeed, if the aperture is very small, this method gives the correct result, save as to a constant factor. In like manner our present law (2o) would apply to the kind of obstruction that would be caused by an actual physical division of the elastic medium, extending over the whole of the area supposed to be occupied by the intercepting screen, but of course not extending to the parts supposed to be perforated. On the electromagnetic theory, the problem of diffraction becomes definite when the properties of the obstacle are laid down. The simplest supposition is that the material composing the obstacle is perfectly conducting, i.e. perfectly reflecting. On this basis A. J. W. Sommerfeld (Math. Ann., 1895, 47, p. 317), with great mathematical skill, has solved the problem of the shadow thrown by a semi-infinite plane screen. A simplified exposition has been given by Horace Lamb (Prot. Lond. Math. Soc.,1906, 4, p. 190). It appears that Fresnel's results, although based on an imperfect theory, require only insignificant corrections. Problems not limited to two dimensions, such for example as the shadow of a circular disk, present great difficulties, and have not hitherto been treated by a rigorous method ; but there is no reason to suppose that Fresnel's results would be departed from materially. (R.)
End of Article: DTZ
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