WAVE THEORY 453 buted motion at tlie lamina are superposed, the corresponding motions in front are superposed also. irious The method of resolution just described is the simplest, but it ithods is only one of an indefinite number that might be proposed, and reso- which are all equally legitimate, so long as the question is regarded ,ion. as a merely mathematical one, without reference to the physical properties of actual screens. If, instead of supposing the motion at dS to be that of the primary wave, and to be zero elsewhere, we suppose the force operative over the element dS of the lamina to be that corresponding to the primary wave, and to vanish elsewhere, we obtain a secondary wave following quite a different law. 1 In this case the motion in different directions varies as cos 6, vanishing at right angles to the direction of propagation of the primary wave. Here again, on integration over the entire lamina, the aggregate effect of the secondary waves is necessarily the same as that of the primary. ipressed In order to apply these ideas to the investigation of the secondary ~ce. wave of light, we require the solution of a problem, first treated by Stokes, 2 viz., the determination of the motion in an infinitely extended elastic solid due to a locally applied periodic force. If we suppose that the force impressed upon the element of mass Ddxdi/dzis ,-, , , . T>Zdxdydz, being everywhere parallel to the axis of Z, the only change required in our equations (1), (2) is the addition of the term Z to the second member of the third equation (2). In the forced vibration, now under consideration, Z, and the quantities |, TJ, 8 expressing the resulting motion, are to be supposed proportional to c int , where i = V (~ 1), an< i n 2Tr/T, r being the periodic time. Under these circumstances the double differentiation with respect to t of any quantity is equivalent to multiplication by the factor - 7i 2 , and thus our equations take the form (7). It will now be convenient to introduce the quantities -a^ &, 3 , which express the rotations of the elements of the medium round axes parallel to those of coordinates, in accordance with the equations _dj_ d-n _dv dC d( rf &3 ~dy dx> ^~dz~dy> * = dx~dz (8) " In terms of these we obtain from (7), by differentiation and subtrac tion, (9). The first of equations (9) gives For -a-^ we have (10). - (n) 3 where r is the distance between the element dxdydz and the point whore Wj is estimated, and k = n/b = 2ir/ ....... (12), A. being the wave length. We will now introduce the supposition that the force Z acts only within a small space of volume T, situated at (x, y, z), and for simplicity suppose that it is at the origin of coordinates that the rotations are to be estimated. Integrating by parts in (11), we get / f -ikr . r -iki i /- f j / f -ikr -/z( e -- r J J dy r dy L r J J dy in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus d fe-*** . . . T ( -- }dx dy dz. dy r J Since the dimensions of T are supposed to be very small in com- d / e -ikr parison with A, the factor ( - ) is sensibly constant ; so that, dy r ) if Z stand for the mean value of Z over the volume T, we may write TZ y_ In like manner we find TZ (14). From (10), (13), (14) we sec that, as might have been expected, the rotation at any point is about an axis perpendicular both to 1 Loc. (it., equation (10). 2 3 This solution may bo verified in the same m which i=0. 2 Loc. ct/., 27-30. lanner as 1 oisson s theorem, in the direction of the force and to the line joining the point to the source of disturbance. If the resultant rotation be w, we have =.. !L Z _ . V^ + Z/ 3 ) . i c ikr = TZsin0 d_f dr r J 4-rrl 2 dr denoting the angle between r and z. In differentiating e-^/r with respect to r, we may neglect the term divided by r 2 as altogether insensible, kr being an exceedingly great quantity at any moderate (1 1 tif^TlfP Trnm +llP nviciriii rif rlic-f ni-lio-nnn r Pl-mci distance from the origin of disturbance. - ik . TZ sin Thus (15), which completely determines the rotation at any point. For a dis turbing force of given integral magnitude it is seen to be every where about an axis perpendicular to r and the direction of the force, and in magnitude dependent only upon the angle (<) between these two directions and upon the distance (r). The intensity of light is, however, more usually expressed in terms of the actual displacement in the plane of the wave. This displacement, which we may denote by ( , is in the plane containing z and r, and perpendicular to the latter. Its connexion with -ar is expressed by -u = dgjdr ; so that TZsin0 e (" -hQ 1 47r& 2 r where the factor c int is restored. Retaining only the real part of (16), we find, as the result of a local application of force equal to DTZcosn^ ....... (17), the disturbance expressed by ., _ TZsinji cos (nt - kr) f ~ 4,r& 2 ~^~ The occurrence of sin< shows that there is no disturbance radi ated 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 C=-shi(nt-kx) ...... (19) is supposed to be broken up in passing the plane a: = 0. The first step is to calculate the force which represents the reaction between the parts of the medium separated by = 0. The force operative upon the positive half is parallel to OZ, and of amount per unit of area equal to = 1-kD cos nt ; Resulting disturb- 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 secondary disturbance corresponding to the element rfSof the plane may be supposed to be that caused by a force of the above magni tude 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 Law of upon the positive half only of the medium, produces just double of secondary the effect that would be caused by the same force if the medium wave. 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 will be a disturbance rfS sin (20), regard being had to (12). This therefore expresses the secondary disturbance at a distance r and in a direction making an angle with OZ (the direction of primary vibration) due to the clement dS of the wave-front. The proportionality of the secondary disturbance to sin< is Compari- common to the present law and to that given by Stokes, but here son with there is no dependence upon the angle 6 between the primary and Stokes s secondary rays. The occurrence of the factor (Ar)" 1 , and the law. 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 ( 10). The occurrence of factors such as sin$, or ^(1 + cos 6), 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 mathe matically 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 symme
trical secondary waves, in which the normal motion is supposed to