satisfies the ordinary equation of continuity, and we must find the variation of the density p by means of the equation

or

Thus

др

δε

+p{-cke-ky cos k(x+ct) + mek2e-ky cos k (x+ct)}

To find the variation of density in time at a fixed point we omit the last two terms, which are in any case small quantities

of the second order, since u', v', Thus we have

до др
да ду

are all supposed small.

At the barrier, where x=0, y=0, this becomes

where X is the wave-length, and c the velocity of propagation. Thus if n/ or /c2 do not exceed a certain amount, the cohesive forces may suffice to prevent rupture, but if this amount be exceeded, the tendency to discontinuity will prevail, and the wave will break.

Waves in Shallow Water.

When the water is of depth h the velocity potential for a train of waves

n=sin k(x-ct)

so that the tendency to rupture now depends on n coth kh/^ or n/c2. Since coth kh> 1 we see that, for a given wavelength, waves break more readily in shallow water than in deep.

An

If we examine the formulae, we find that they agree fairly well with the behaviour of waves breaking against a seawall or pier. On a quiet day, the waves meeting the wall are reflected in the ordinary way, but when the waves are large enough to exceed the critical value of 7, they break. advancing wave causes the water to rise smoothly against the wall up to a certain point, when, for no apparent reason, it breaks suddenly into spray. This is what the formula lead us to expect, for when once the critical value of ʼn is reached, p/p increases as an exponential function of 7, so that the transition from smooth motion to spray is rapid. When the trough of the wave reaches the wall, the water sinks quietly to a certain point, and then begins suddenly to seethe, the critical value of ʼn being again passed.

Since y is measured downwards it is negative at the crests of the waves. Now m-1 is negative, hence p>p, at the crests, and the spraying is due to the rapid rise of pressure consequent on the inability of the water to contract in accordance with the formula. The seething in the troughs is similarly due to water running down the slopes of the waves to supply the deficiency due to the inability of the water to expand, as required by the fact that p <P: The slope of the exponential curve being greater for positive

than for negative values of the exponent, the action at the crests is more violent and sudden than in the troughs.

Oblique Reflexion.

When the waves are incident in a direction not at right angles to the barrier, the water has a motion parallel to the obstacle in a horizontal direction. This will also be subject to the drag of the obstacle, and the effect can be investigated on the principles laid down.

Taking the line of the obstacle as axis of z, suppose the waves incident at an angle a, so that the ordinary reflected wave is reflected at the same angle on the other side of the axis of x. The advancing train is represented by

The ordinary reflected wave is given by

'ce-ky cos k(x cos a-z sin a+ct). Thus the velocity system to be investigated is

u' = —ck cos a e―ky sin k(x cos a − z sin a+ct),
v' = -mck e-ky cos k(x cos x-z sin a +ct),
w' =nck sin a e-ky sin k(x cos a-≈ sin a+ct),

where m and n are both nearly equal to unity. The dilatation Ju' av' dw'

+

+

dic ду

ละ

at the origin is thus

ch2(m-n sin2 a-cos2 a) cos ket,

and we have at the origin

1 др

Pat

Thus the quantity m-1 which occurs in connexion with

direct incidence is here replaced by m-n sin2 a-cos2 a. Since m and n both represent the effect of the drag due to the wall, they may be taken to be equal, and the quantity now becomes (m-1) cos2 a. Thus

p=p1e(m−1) cos2 a. 2#nλ

η

Consequently the wave will break if n cos2 a exceed a certain value. For a given wave-length the critical amplitude will vary as the square of the secant of the angle of incidence.

1.

XXXVI. On Rotatory Polarization in Biaxial Crystals.
By H. C. POCKLINGTON, M.A., D.Sc.*

THE A

HE only doubly refracting crystals in which a rotation of the plane of polarization of the transmitted light has been observed are uniaxials, e. g. quartz. It is, however, clearly possible for a biaxial crystal to produce this rotation, for quartz subject to a uniform stress perpendicular to the axis must act as a biaxial crystal without losing its rotatory power, and a biaxial crystal must acquire rotatory power in a magnetic field. The object of this paper is to investigate from a theoretical standpoint the phenomena in the second case, assuming that the action of the magnetic field may be attributed to a Hall effect into the specification of which the anisotropy of the crystal does not enter. The formulæ are afterwards (§ 7) extended to the more important case of crystals where the rotation is due to the mode of arrangement, or to the structure, of the molecules. The general equations are found in § 2, and the equation of the index-surface in §3. The shape of the wave-surface near an axis is discussed in § 4, together with the modifications produced by the rotatory power in the conical refraction. In $5 it is proved that the two waves propagated in any direction are elliptically polarized, and that the ellipses are similar but with corresponding axes perpendicular. These ellipses are completely determined, and also the relative retardation of the two waves.

The case of a plate in convergent polarized light, and also that of a pair of such plates of opposite rotatory powers (Airy's spirals) is investigated in § 6. Finally, § 8 contains the results of some experiments made with plates cut from sugar crystals, which confirm the mathematical investigation. 2. Let the magnetic force be 7+70, where 7 is the variable force due to the light-waves, and 7, is the constant field in which the crystal lies. Let the electric displacement be σ, Communicated by the Author.

and let the electric force be do when To=0. Then when T is not zero, the electric force is the sum of two terms, one being that given above, and the other that due to the Hall effect, viz. V (T+7)σ=cV7σ if we neglect 7 in comparison with 7 in this small term.

The equations of the electromagnetic field are now

V.VT=0,

V.v(po+cV7,0)=—†,
Svo=0, SVT=0.

Eliminating these give

ö+V.vVv(po+eVτ ̧0)=0.

3. In order to discuss the case of plane waves, let

σ =μ exp @p(t+Sλp),

(1) where is the scalar quantity (−1), and p is the vector of any point, so that A is the vector of wave-slowness, and the locus of its extremity is the index-surface. The calculations will be carried out with this imaginary value of σ, and at the end we shall, in virtue of the linearity of the differential equations, reject the imaginary parts of the various expressions found for the electromagnetic forces.

The operators and d/dt when the above expression is the operand are equivalent to multipliers -wpλ and op respectively, hence the vector differential equation satisfied by o • Cf. Basset, Treatise on Physical Optics,' p. 394.