EQUATIONS OF MOTION. 255 246. Let us now consider a portion of a crystalline medium, which is bounded by a plane; and let plane waves whose vibrations are transversal, be incident normally upon the medium. The incident wave will produce a train of waves within the medium, which, as will presently be shown, will involve dilatation and distortion, unless certain relations exist between the coefficients. But since the disturbance which constitutes light, consists of a vector quantity, whose direction is perpendicular to the direction of propagation of the wave, it follows that the medium must be one, which is capable of propagating waves of transversal vibrations unaccompanied by waves of longitudinal vibrations. Green therefore assumed that the medium possessed this property, and investigated the relations which must exist between the coefficients, in order that this might be possible. 247. The equations of motion of the medium are with two similar equations, where P=dW/de &c. Substituting the values of P, Q..., the equations of motion become d3u d3v dow Differentiate with respect to x, y, z and add, and we obtain +(2A + E') dz3 If in this equation we put E=F=G=μ Hence the relations between the coefficients which are given by (4), are the conditions that a longitudinal wave may be capable of being propagated through the medium, unaccompanied by transversal waves; and therefore if these conditions are satisfied, longitudinal waves will be propagated through the medium with a velocity (μ/p). By means of (4), the equations of motion may now be written .(7), ..(8), 2 W = μ (e + ƒ + g)2 + A (a2 − 4ƒg) + B (b2 − 4ge) + C' (c2 − 4eƒ).....(9). The stresses are given by the equations P = μd - 2 (Cƒ + Bg) Q=μd-2 (Ag + Ce) R = μd-2 (Be + Af) μδ S=Aa, T= Bb, UCc .(10). 248. Equations (6) and (7) show, that the special kind of aolotropic medium considered by Green, is capable of propagating two distinct types of waves, viz. dilatational waves, whose velocity of propagation has been shown to be equal to (μ/p), and distortional waves, whose velocity of propagation is determined by (7). We shall presently show, that the velocity of propagation of the distortional waves, is determined by the same quadratic equation as in Fresnel's theory; but previously to doing this, it will be desirable to consider a little more closely the properties of the medium. PROPERTIES OF THE MEDIUM. 257 249. In a crystalline medium, which possesses three rectangular planes of symmetry, the shearing stress across any plane which is not a principal plane, will in general be a function of the extensions as well as of the shearing strain parallel to that plane. It is however possible for a medium to be symmetrical, as regards rigidity, with respect to each of the three principal axes:- -in other words, the medium may be such, that if any plane be drawn parallel to one of the principal axes (say a), and S1, a1 be the shearing stress and strain parallel to that plane and perpendicular to the axis of a, then S1 = Aa,. We shall now show, that when a medium possesses this property, the relations (4) must exist between the coefficients. Let Ox, Oy, Oz be the axes of crystalline symmetry; and let BC be the intersection of any plane parallel to Ox with the plane yz; and consider a portion of the medium, which is bounded by the plane BC and two fixed rigid planes perpendicular to Ox. Draw Oy1, Oz, respectively perpendicular and parallel to BC, and let the suffixed letters denote the values of corresponding quantities referred to Ox, Oy1, Oz, as axes. If be the angle which Oy, makes with Oy, then Also, since the medium is supposed to be bounded by two rigid planes perpendicular to Ox, there can be no extension nor contraction parallel to Ox, whence accordingly, also Q=Ff+Eg, R= E'ƒ+Gg; S1 = Aa cos 20 + } { (E' − F) ƒ + (G – E) g} sin 20.....(11). But, if m = cos 0, n = sin 0, = a, cos 20 + (ƒ1 − g1) sin 20. = ( n d dy + m B. O. 17 Substituting in (11), we obtain S1 = {a, A cos2 20 + † (G + F − 2E') sin2 20} + } (fi− g1) {A −1 (G+F-2E)) sin 40 It therefore follows that if In a similar way it can be shown, that in order that T, = Bb,, and U, Cc,, we must have = E=F=G=μ, 2A=μ- E', 2B = μ-F", 2C = μ — G', which are equivalent to (4). If therefore a portion of the medium considered by Green, which is bounded by two fixed planes perpendicular to any one of the principal axes, be subjected to a shearing stress whose direction is perpendicular to that axis, and which lies in any plane parallel to that axis, the ratio of the shearing stress to the shearing strain is equal to the principal rigidity corresponding to that axis. Moreover a crystalline medium which possesses this property, also possesses the property of being able to transmit waves of transversal vibrations unaccompanied by waves of longitudinal vibrations. Hence the relations which Green supposed to exist between the nine constants, are not mere adventitious relations, which were assumed for the purpose of obtaining a particular analytical result, but correspond to and specify a particular physical property of the medium. 250. We shall now show, that the velocity of propagation of the distortional waves is determined by Fresnel's law. To satisfy (7) let u, v, w be those portions of the displacements upon which distortion depends; let l, m, n be the direction cosines of the wave-front, and X, μ, v those of the direction of vibration. Then we may assume that when u = Sλ, v=Sμ, w = Sv, S = ε (lx+my+nz− Vt) ̧ From these equations combined with (7) of § 187, we obtain ξ = ικ (ην - ημ), η = ικ (κλ - lv)S, ζ=ικ (Ιμ - ηλ) 5. If we put VELOCITY OF PROPAGATION. 259 λ' = mν - ημ, μ' = ηλ - lv, v=lu-mλ......(12), so that X', u', v' are the direction cosines of the rotation, and then substitute the values of , 7, in (7), we obtain (pV-A) X' + (Alλ' + Bmμ' + Cnv')l =0' + (AlX' + Bmμ' + Cnv′) m = 0 ..(13), (pV2 – C) v' + (Alx' + Bmμ' + C'nv') n = It follows from (14), that the velocities of propagation of the two waves within the crystal are determined by the same quadratic as in Fresnel's theory, and that the wave surface is Fresnel's. 251. From (12) it follows, that the direction of displacement and rotation are in the front of the wave, and also that these directions are at right angles to one another. Multiplying (13) by X', u', v' and adding, we obtain which shows that the velocity of propagation of either wave, is inversely proportional to the length of that radius vector of the ellipsoid Ax2+By2+C22 = 1, which is parallel to the direction of rotation. We also obtain from (13) (pV2 – A) X'/l = (p V2 – B) μ' /m = (p V2 — C) v′/n..............(17). 252. Writing a2 = A/p, b2 = B/p, c2= C/p, we see from (19) of § 109, that if x, y, z be the coordinates of the point of contact of the tangent plane to the wave surface, which is parallel to the wavefront, then x = lV (r2 − A/p)/(V2 – A/p), y = mV (r2 – B/p)/(V2 – B/p), z = nV (r2 − C/p)/(V2 — C'/p), and therefore by (17) - (r2 − A/p) X'/x = (r2 – B/p) μ' /y = (r2 — C/p) v′/z..............(18), |