ζ = sin(nt − kx) (19)
is supposed to be broken up in passing the plane x = 0. The first step is to calculate the force which represents the reaction between the parts of the medium separated by x = 0. The force operative upon the positive half is parallel to OZ, and of amount per unit of area equal to
−b²D dζ/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 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
| ζ′ = | dS sin φ | cos (nt − kr) (20), |
| λr |
regard being had to (12). This therefore expresses the secondary disturbance at a distance r and in a direction making an angle φ with OZ (the direction of primary vibration) due to the element dS of the wave-front.
The proportionality of the secondary disturbance to sin φ is common to the present law and to that given by Stokes, but here there is no dependence upon the angle θ between the primary and secondary rays. The occurrence of the factor λr−1, 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 φ, or ½(1 + cos θ), 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.