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 (20) 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 (Proc. 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.)
[1] The descending series for J0(z) appears to have been first given by Sir W. Hamilton in a memoir on “Fluctuating Functions,” Roy. Irish Trans., 1840.
[2] Airy, loc. cit. “Thus the magnitude of the central spot is diminished, and the brightness of the rings increased, by covering the central parts of the object-glass.”
[3] ”Man kann daraus schliessen, was moglicher Weise durch Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B, dessen Durchmesser = (λ) ist, und der aus zwei Theilen besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt werden. Dieses zeigt uns eine Grenze des Sehvermogens durch Mikroskope” (Gilbert’s Ann. 74, 337). Lord Rayleigh has recorded that he was himself convinced by Fraunhofer’s reasoning at a date antecedent to the writings of Helmholtz and Abbe.
[4] The last sentence is repeated from the writer’s article “Wave Theory” in the 9th edition of this work, but A. A. Michelson’s ingenious échelon grating constitutes a realization in an unexpected manner of what was thought to be impracticable.—[R.]
[5] Compare also F. F. Lippich, Pogg. Ann. cxxxix. p. 465, 1870; Rayleigh, Nature (October 2, 1873).
[6] The power of a grating to construct light of nearly definite wave-length is well illustrated by Young’s comparison with the production of a musical note by reflection of a sudden sound from a row of palings. The objection raised by Herschel (Light, § 703) to this comparison depends on a misconception.