The square of this expression correctly represents the observed intensity of the reflected beam, if the incident beam is polarized in the plane of incidence. We conclude that the electric force is at right angles to the plane of polarization, a result in accordance with the conclusion arrived at in the study of double refraction.

If we take the incident beam to be polarized at right angles to the plane of incidence, R, a, ẞ vanish, and the surface conditions become KP=K1P1, Q = Q1)

The last equation secures the continuity of y. But the form of our assumed disturbance shows that dy/dy = iby and hence if γ is continuous so is also dy/dy and vice versa. Also according to (18) KdP/dt = dy/dy, when ẞ= 0: the first and last surface conditions are therefore identical and we may disregard the latter.

W

F W'

F

If WF (Fig. 165) be the incident wave-front, the displacement is now in the plane of the paper and parallel to the wave-front. Let the direction indicated by the arrow be that in which the displacements are taken to be positive. W'F' represents the reflected wave-front, and we may again arbitrarily fix that direction for which we shall take the displacements to be positive.

F

Fig. 165.

It is obvious that for normal incidence there is no distinction between this case and the one already considered when the displacement is at right angles to the plane of incidence. It is therefore convenient to take that direction as positive which agrees with that of the incident wave when the incidence is normal. The arrow indicates the direction. Similarly for the transmitted wave W1F1. Taking the amplitude of the incident beam again to be unit amplitude, and resolving along OX and OY, we may put in the upper medium P: sin ei (ax+by-ct) + r sin Oei (−ax+by-ct),

Q = cos 0 ¿¿ (ax+by−ct) + r cos 0 ̧1 (−ax+by−ct),

These are the only conditions that need be satisfied.

This again is a formula agreeing with observation, at any rate as a first approximation. The application of the equations (34) and (36) to the cases of oblique polarization or unpolarized light has already been discussed in Art. 27 as well as the observed departures from (36).

It has often been suggested that the experimental deviations from the tangent law may be due to the fact that the transition between the two media is not sudden but takes place within a layer comparable in thickness with the length of a wave. L. Lorenz* first investigated the question and showed that a thickness of from the tenth to the hundredth part of a wave-length is sufficient to cause the observed effect. Drude †, treating the same subject from the standpoint of the electromagnetic theory, has arrived at similar results, a thickness of the transition layer of 0175 being found to be sufficient in the case of flint-glass to account for the elliptic polarization observed near the polarizing angle.

141. Reflexion in the elastic solid theory. In elastic solids the conditions at the boundary are obtained by the consideration that as a tearing of the medium can only take place under application of forces which exceed the limits of elasticity, the displacements on both sides of the boundary must be the same, while the medium is performing oscillations under the conditions of perfect elasticity.

A second condition is imposed by the third law of motion. The stresses must be continuous. The continuity of stress together with that of displacement satisfies also the requirements of the law of conservation of energy, as the work done across any surface is the product of stress and rate of change of displacement.

The components of displacement which we had previously called a, ß, y, shall, in order to distinguish them from the magnetic forces for which we have introduced the same letters, now be designated by έ, n, L.

* Pogg. Ann. cxI. p. 460 (1860) and cxiv. p. 238 (1861).

+ Lehrbuch der Optik, p. 266.

where the right-hand sides refer to the lower medium.

The stresses on a surface normal to the axis of x are, by Art. 131,

Writing m k+n, we obtain for the conditions of continuity of

=

where m1, n1, define the elastic properties of the second medium.

Let the plane of xy be the plane of incidence, and the vibrations of a plane wave be at right angles to that plane. All displacements vanish except , and is independent of z. Hence the equations of continuity reduce to

dł

= ni

dx

dx

The equation of motion in the upper medium is, according to (9),

For the velocity of wave propagation in an elastic solid, we have v2=n/p. Different wave velocities in different media may either be

due to differences in the rigidity or to differences in density. Hence we must distinguish the two cases.

This agrees with the result obtained in the electromagnetic theory if the displacements are made to correspond to electric force.

This is the equation for the reflected light when the incident wave is polarized at right angles to the plane of incidence. Hence if different media differ by their rigidities, the reflexion of light vibrating at right angles to the plane of incidence can only be accounted for by supposing that the plane of polarization contains the vibration.

To work out completely the more complicated case that the vibration lies in the plane of incidence, we must transform the equations of motion.

η

In equations (9) alter the notation, put = 0, and let έ and be independent of z, this being the condition that the wave normal lies in the plane of xy. The equations then become

=

From (35) it appears that the displacements έ and ʼn, due to changes of 4, are at right angles to the surface constant, while the displacements due to changes in lie in the surface = constant. If we adopt the same form of solution for 4 and 4, it is the latter function which gives the motion which we require for the propagation of light in which the displacements are in the wave-front. We put therefore for the incident wave (ax+by-ct), and assume for the form of solution generally,

In the upper medium:

=

↓ = ei (ax+by−ct) + rei (−ax

$ = pei (a'x+by−ct)

In the lower medium:

¥1 = sei (ax+by−ct)

P1 = get (ax+by-ct))

Substituting these values in equations (41) we obtain

...(42).

..(43).

· (α22 + b2) =

From the first two equalities we obtain as before the law of refraction, but as m and m1 are indefinitely great the last equalities give

This shows that the motion due to is that of an incompressible liquid. As represents the velocity potential, the motion is irrotational. Also by substitution of (45) into (42) and (43) retaining only the real parts:

=pe-bx cos (by — ct), 1 = qe cos (by - ct).

-bx

The displacements in so far as they are due to are