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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 See also:

cos nt (17), the disturbance expressed by ,=TZsin¢ cos(nt-kr) 4abr (18) The occurrence of See also: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 See also:law of secondary disturbance, when a See also:primary See also:wave =sin(nt-kx) . . . . (19) is supposed to be broken up in passing the See also:plane x= o. The first step is to calculate the force which represents the reaction between the parts of the See also:medium separated by x=o. The force operative upon the See also:positive See also:half is parallel to OZ, and of amount per unit of See also:area equal to -b2D di/dx=b2kD cos nt; and to this force acting over the whole of the plane the actual See also:motion on the positive See also:side may be conceived to be due. The DZ dx dy dz, . . (13). . . r(14). . . (15), . . (16), secondary disturbance corresponding to the See also: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 See also:

act upon the positive half only of the medium, produces just See also: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 See also: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 See also: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 See also:common to the See also:present law and to that given by See also:Stokes, but here there is no dependence upon the angle 8 between the primary and secondary rays. The occurrence of the See also:factor (Xr)-r, and the See also:necessity of supposing the phase of the secondary wave accelerated by a See also:quarter of an undulation, were' first established by See also:Archibald See also:Smith, as the result of a comparison between the primary wave, supposed to pass on without See also: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 See also: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 See also:physical considerations respecting the mode of See also:action of obstacles. Thus, to refer again to the acoustical analogue in which plane waves are incident upon a perforated rigid See also:screen, the circumstances of the See also: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 See also:aperture is very small, this method gives the correct result, See also:save as to a See also:constant factor. In like manner our present law (2o) would apply to the See also:kind of obstruction that would be caused by an actual physical See also: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. See also:

Sommerfeld (Math. See also:Ann., 1895, 47, p. 317), with See also:great mathematical skill, has solved the problem of the See also:shadow thrown by a semi-See also:infinite plane screen. A simplified exposition has been given by See also:Horace See also:Lamb (Prot. Lond. Math. See also:Soc.,1906, 4, p.

190). It appears that See also:

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 See also:reason to suppose that Fresnel's results would be departed from materially.

End of Article: DTZ

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