Page:EB1911 - Volume 08.djvu/271

of polarized light are executed in a direction perpendicular to the plane of polarization.

The factor (1 + cos θ) shows in what manner the secondary disturbance depends upon the direction in which it is propagated with respect to the front of the primary wave.

If, as suffices for all practical purposes, we limit the application of the formulae to points in advance of the plane at which the wave is supposed to be broken up, we may use simpler methods of resolution than that above considered. It appears indeed that the purely mathematical question has no definite answer. In illustration of this the analogous problem for sound may be referred to. Imagine a flexible lamina to be introduced so as to coincide with the plane at which resolution is to be effected. The introduction of the lamina (supposed to be devoid of inertia) will make no difference to the propagation of plane parallel sonorous waves through the position which it occupies. At every point the motion of the lamina will be the same as would have occurred in its absence, the pressure of the waves impinging from behind being just what is required to generate the waves in front. Now it is evident that the aerial motion in front of the lamina is determined by what happens at the lamina without regard to the cause of the motion there existing. Whether the necessary forces are due to aerial pressures acting on the rear, or to forces directly impressed from without, is a matter of indifference. The conception of the lamina leads immediately to two schemes, according to which a primary wave may be supposed to be broken up. In the first of these the element dS, the effect of which is to be estimated, is supposed to execute its actual motion, while every other element of the plane lamina is maintained at rest. The resulting aerial motion in front is readily calculated (see Rayleigh, Theory of Sound, § 278); it is symmetrical with respect to the origin, i.e. independent of θ. When the secondary disturbance thus obtained is integrated with respect to dS over the entire plane of the lamina, the result is necessarily the same as would have been obtained had the primary wave been supposed to pass on without resolution, for this is precisely the motion generated when every element of the lamina vibrates with a common motion, equal to that attributed to dS. The only assumption here involved is the evidently legitimate one that, when two systems of variously distributed motion at the lamina are superposed, the corresponding motions in front are superposed also.

The method of resolution just described is the simplest, but it is only one of an indefinite number that might be proposed, and which are all equally legitimate, so long as the question is regarded 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. In this case the motion in different directions varies as cosθ, 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.

In order to apply these ideas to the investigation of the secondary wave of light, we require the solution of a problem, first treated by Stokes, 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 D dx dy dz is

DZ dx dy dz,

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 ξ, η, ζ, δ expressing the resulting motion, are to be supposed proportional to e^int, where i = √(-1), and n = 2π/τ, τ being the periodic time. Under these circumstances the double differentiation with respect to t of any quantity is equivalent to multiplication by the factor -n², and thus our equations take the form

	$(b^{2}\nabla ^{2}+n^{2})\xi +(a^{2}-b^{2}){\frac {d\delta }{dx}}=0$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$	(7).
	$(b^{2}\nabla ^{2}+n^{2})\eta +(a^{2}-b^{2}){\frac {d\delta }{dy}}=0$
	$(b^{2}\nabla ^{2}+n^{2})\zeta +(a^{2}-b^{2}){\frac {d\delta }{dz}}=-\mathrm {Z}$

It will now be convenient to introduce the quantities.ϖ₁, ϖ₂, ϖ₃ which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations

$\varpi _{3}={\frac {d\xi }{dy}}-{\frac {d\eta }{dx^{\prime }}},\qquad \varpi _{1}={\frac {d\eta }{dz}}-{\frac {d\zeta }{dy^{\prime }}},\qquad \varpi _{2}{\frac {d\zeta }{dx}}-{\frac {d\xi }{dz^{\prime }}}$

(8).

In terms of these we obtain from (7), by differentiation and subtraction,

	(b2Δ2 + n2) ϖ3 = 0	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$	(9).
	(b2Δ2 + n2) ϖ1 = dZ/dy
	(b2Δ2 + n2) ϖ2 = −dZ/dx

The first of equations (9) gives

ϖ3＝0

(10).

For ϖ₁, we have

$\varpi _{1}={\frac {1}{4\pi b^{2}}}\iiint {\frac {d\mathrm {Z} }{dy}}{\frac {e^{-ikr}}{r}}dxdydz$

(11),

where r is the distance between the element dx dy dz and the point where ϖ₁ is estimated, and

k＝n/b＝2π/λ

(12),

λ being the wave-length.

(This solution may be verified in the same manner as Poisson’s theorem, in which k = 0.)

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 co-ordinates that the rotations are to be estimated. Integrating by parts in (11), we get

$\int {\frac {e^{-ikr}}{r}}{\frac {d\mathrm {Z} }{dy}}dy=\left[\mathrm {Z} {\frac {e^{-ikr}}{r}}\right]-\int \mathrm {Z} {\frac {d}{dy}}\left({\frac {e^{-ikr}}{r}}\right)dy,$

in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus

$\varpi _{1}={\frac {1}{4\pi b^{2}}}\iiint \mathrm {Z} {\frac {d}{dy}}\left({\frac {e^{-ikr}}{r}}\right)dxdydz.$

Since the dimensions of T are supposed to be very small in comparison with λ, the factor ${\frac {d}{dy}}\left({\frac {e^{-ikr}}{r}}\right)$ is sensibly constant; so that, if Z stand for the mean value of Z over the volume T, we may write

$\varpi _{1}={\frac {\mathrm {TZ} }{4\pi b^{2}}}\cdot {\frac {y}{r}}\cdot {\frac {d}{dr}}\left({\frac {e^{-ikr}}{r}}\right)$

(13).

In like manner we find

$\varpi _{2}=-{\frac {\mathrm {TZ} }{4\pi b^{2}}}\cdot {\frac {x}{r}}\cdot {\frac {d}{dr}}\left({\frac {e^{-ikr}}{r}}\right)$

(14).

From (10), (13), (14) we see that, as might have been expected, the rotation at any point is about an axis perpendicular both to the direction of the force and to the line joining the point to the source of disturbance. If the resultant rotation be ω, we have

$\varpi ={\frac {\mathrm {TZ} }{4\pi b^{2}}}\cdot {\frac {{\sqrt {}}(x^{2}+y^{2})}{r}}\cdot {\frac {d}{dr}}\left({\frac {e^{-ikr}}{r}}\right)={\frac {\mathrm {TZ} \sin \phi }{4\pi b^{2}}}{\frac {d}{dr}}\left({\frac {e^{-ikr}}{r}}\right),$

φ denoting the angle between r and z. In differentiating e^−ikr/r with respect to r, we may neglect the term divided by r² as altogether insensible, kr being an exceedingly great quantity at any moderate distance from the origin of disturbance. Thus

$\varpi ={\frac {-ik\cdot \mathrm {TZ} \sin \phi }{4\pi b^{2}}}\cdot {\frac {e^{-ikr}}{r}}$

(15),

which completely determines the rotation at any point. For a disturbing force of given integral magnitude it is seen to be everywhere about an axis perpendicular to r and the direction of the force, and in magnitude dependent only upon the angle (φ) 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 ζ′, is in the plane containing z and r, and perpendicular to the latter. Its connexion with ϖ is expressed by ϖ = dζ′/dr; so that

$\zeta ^{\prime }={\frac {\mathrm {TZ} \sin \phi }{4\pi b^{2}}}\cdot {\frac {e^{\prime (at-kr)}}{r}}$

(16),

where the factor e^int is restored.

Retaining only the real part of (16), we find, as the result of a local application of force equal to

DTZ cos nt

(17),

the disturbance expressed by

$\zeta ^{\prime }={\frac {\mathrm {TZ} \sin \phi }{4\pi b^{2}}}\cdot {\frac {\cos(nt-kr)}{r}}$

(18).

The occurrence of sin φ 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

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