Let A, A', A, be the amplitudes of the incident, reflected and refracted waves, then we may write

The vibration in the second medium is a forced vibration produced and maintained by the incident waves, hence the coefficients of t must be the same in all three waves. Also the coefficients of y must be the same, since the traces of all three waves on the plane x =

0 move together. Hence KV = K1V1, KM = K1M1⁄2.............

.(8).

The first of these equations is the well-known law of sines, whilst the second expresses the condition, that the period of the refracted wave must be equal to that of the incident.

Substituting from (6) and (7) in (4) and (5), we obtain

If I, I, I, be the square roots of the intensities, it follows from (1) that

Up to the present time, we have not assumed that any relation exists between n and n1. If however we assume with Green, that the rigidities are the same in all isotropic media, and

HYPOTHESIS OF NEUMANN AND MAC CULLAGH. 201

that refraction is consequently due to a difference of density, we must put n=n, and we obtain

which are the same as Fresnel's formulæ.

If on the other hand we adopt the hypothesis of Neumann and MacCullagh, that the density of the ether is the same in all media, and that refraction is consequently due to a difference of rigidity,

These expressions are the same as Fresnel's formulæ for the intensity of light polarized perpendicularly to the plane of incidence.

194. The preceding formulæ do not enable us to decide the question, whether the vibrations of polarized light are in or perpendicular to the plane of polarization; or whether the hypothesis of Green on the one hand, or of Neumann and MacCullagh on the other, is the best representative of the facts. For the present we shall adopt Green's view, and shall proceed to calculate the change of phase which occurs, when the angle of incidence exceeds the critical angle.

hence if p1> P: μ>1, and r is always real; but if p1<p, μ< 1, and the angle of refraction becomes imaginary, when the angle of incidence exceeds the critical angle.

When p1 <p, we shall write μ for μ, so that μ denotes the index of refraction from the rarer into the denser medium, and

Since the expressions for the amplitudes of the reflected and refracted waves become complex, we must write

whence

îî ̧l1 = (2π/λμ) (μ2 sin2 i − 1) = κα1 (say),

the lower sign being taken, because x is negative in the second medium.

The boundary conditions give

A+ a + iẞ=α1 + iẞ1

(A-a-iẞ) cos i = (a + iẞ,) u.

Equating the real and imaginary parts, we obtain

From these equations we see, that when the angle of incidence. exceeds the critical angle, the reflection is total and is accompanied by a change of phase, whose value is given by (19).

Since the refracted wave involves an exponential term in the amplitude, it becomes insensible at a distance from the surface which is equal to a few wave-lengths.

All the foregoing results are in agreement with Fresnel's formulæ.

VIBRATIONS IN THE PLANE OF INCIDENCE.

203

195. The investigation of the problem, when the light is polarized perpendicularly to the plane of incidence is more difficult.

In this case w=0, and by (12) of § 187, the equations of motion in the upper medium are

d'u

ddu dv
(k + ‡n) dx dx
+
dx dx dy

= (k + {n) dy

ddu dv
+

d2v
+ dv)+n
dx + dy)

with similar equations for the lower medium. In these equations k is the resistance to compression, which Green supposes to be very large compared with n.

of which the first two express the conditions of continuity of displacement, and the last two the conditions of continuity of

stress.

We have therefore four equations to determine two unknown quantities. Now we have already shown, that elastic media are capable of propagating waves of two distinct types; viz. dilatational waves, which involve condensation and rarefaction, and distortional waves, which involve change of shape without change of volume. When the vibrations of the incident wave are not parallel to the reflecting surface, there will be a dilatational as well as a distortional reflected and refracted wave, whose amplitudes must be determined by (22) and (23); accordingly we have four unknown quantities and four equations to determine them. When the resistance to compression is very large, & or du/dx + dv/dy is very small, but we are not at liberty to treat the latter quantity as zero, because kd is finite; we must therefore introduce the dilatational waves.

so that U and V are the velocities of the dilatational and distortional waves respectively. Then

with similar equations for the lower medium.

Let us now assume that in the first medium

..(26),

......(27),

In these equations the coefficients of y and t must be the same in all the waves, whence

also since the dilatational wave is propagated with velocity U, it follows that its wave-length is equal to UX/V which is therefore very large compared with X.

Substituting from the second of (27) in the first of (26), we

obtain

V2 = (a2 + m2) U.............

..(30),

and since VU is very small, we shall have to a sufficient approximation

lα=- -m;

the lower sign being taken, because a is positive in the upper medium.

Similarly from the second of (28) and (26), we shall obtain