469 



TJNDULATOHY THEOKY OF LIGHT. 



UNDULATOHY THEO11Y OF LIGHT. 



470 



of measuring the length of a wave of light, which is found to vary from 

 about 2ii6 to 1B7 ten-millionths of an inch, in passing from the extreme 

 red to the extreme violet. We infer, therefore, from the known 

 velocity of propagation, that from about 458 to 727 millions of millions 

 of vibrations must take place in one second. 



The locus of the particles, which at a given instant are in the Bame 

 state or phase of their motion, is called the front nf a wart. When 

 light di merges in the first instance in vacuum, or any singly refracting 

 medium, from an element of a self-luminous body, the front of a wave 

 ia of course spherical, or, at a sufficient distance from the body, when 

 we have to consider a small portion only of the front, sensibly plane, 

 but after reflection or refraction it may be of any other form. 



We now come to a principle of constant application in the theory of 

 undulations, which ia called Huygens's principle. It may be thus 

 stated : The front of a wave of light, either at a given instant, or as 

 the parts of it arrive in succession at a given surface, may be divided 

 into elements, of which each is conceived to be the centre of an ele- 

 mentary disturbance which spreads in all directions with the velocity 

 appropriate to the medium in which it is propagated. The disturbance 

 in front of the primary wave may be regarded as the aggregate of the 

 elementary disturbances due to these secondary waves ; and as these 

 are insensible when taken separately, the aggregate disturbance will 

 sensible except in the immediate neighbourhood of the enve- 

 lope, or surface of ultimate intersection, of the secondary waves. If I 

 we confine our attention to a small portion of the primary wave, we 

 shall have a corresponding small portion of the envelope, or general 

 wave, in its advanced position, in the neighbourhood of which alone 

 the disturbance due to the small element of the first wave is sensible. 

 Hence the conrse of a ray from any point of the first wave is the line 

 along which the point of ultimate intersection of the secondary waves 

 which start from the neighbourhood of that point of the first wave is 

 displaced. This is evidently the straight line joining the point lost 

 mentioned with the point of ultimate intersection at a given instant ; 

 or, again, the straight line joining the point in question of the first 

 wave with the [mint in which the secondary wave thence diverging is 

 touehed by the general envelope. 



The legitimacy of thus conceiving a wave to be broken up i.i a direct 

 consequence of the general dynamical principle of the co-existence of 

 small motion*. The secondary waves will have an envelope behind as 

 well as in front of the primary wave ; yet we must not infer that the 

 latter is pro] kwards as well as forwards. The explanation 



of the non-propagation in a backward direction, notwithstanding the 

 legitimacy of thus supposing a wave broken up, belongs to the dyna- 

 mical theory of diffraction, and is much too difficult a subject to be 

 entered upon here. (See a paper in the 'Cambridge Philosophical 

 Transactions,' vol. ix., p. 1.) 



The complete explanation of the existence of rays requires this 

 principle to be combined with the principle of interference, but for 

 the present we shall confine ourselves to its application to the demon- 

 stration, according to the undulatory theory, of the laws of reflection 

 and refraction. We may notice, however, in the mean time some 

 features of the propagation of light in a uniform medium. 



Conceive, then , a surface of any form to be at a given instant the 

 front of a wave propagated in a uniform medium, and first suppose the 

 velocity of propagation the same in all directions. If we conceive 

 the wave broken up as above explained, the secondary waves, which 

 will be all of the same magnitude, will by our supposition be spherical ; 

 and by very simple geometry we .arrive at the following construction 

 for determining the form of the wave in its onward course. At all 

 Iioints of the front of the wave at the time I draw normals, at the side 

 towards which the wave is travelling, equal in length to v f ; the locus 

 of the extremities of these normals will be the front of the wave at the 

 time t + t", and the normals themselves will be the courses of the rays. 

 We learn also that it is not any arbitrarily chosen system of straight 

 lines, the equations of which contain two, arbitrary parameter.-!, that 

 can represent a possible system of rays ; the lines" must admit of being 

 cut orthogonally by a system of surfaces. This geometrical property, 

 so readily suggested by the theory of undulations, may of course be 

 demonstrated as a consequence of the geometrical laws of n: 

 and refraction independently of any theory, the system being supposed 

 to consist of rays which, having originally emanated from a point, have 

 undergone any number of reflections and refractions. If the in<><l;ii;u 

 in which the wave is propagated be uniform, but differently consti- 

 tuted in different directions, as in the case of Iceland spar, for example, 

 the velocity of propagation will vary with the direction, the secondary 

 waves will no longer be spherical, and the course of a ray, while still 

 rectilinear, will no longer be perpendicular to the front of the wave. 



We come now to the explanation of ordinary reflection and refraction 

 on the undulatory theory, according to the principles laid down by 

 Huygens. We shall in the first instance, for the sake of simplicity, 

 suppose the reflecting surface to be plane, and the incident waves to be 

 plain: likewise. Let the plane of the paper be the plane of incidence, 

 and therefore perpendicular at the same time to the reflecting surface 

 and to the planes of the waves. Let it cut the reflecting surface along 

 A B ; and let M .v, p tj, u s, lying in the plane of the paper, be three 

 normals to the incident waves, representing therefore the con: s of 

 incident ray^. If there were no obstacle, the wave .it one instant at s 

 1, after the lapre of time required to describe the perpend iuul IT 



distance of ts plane from s, come into the position represented in 

 section by s^n, perpendicular to BS, i'Q2, andMNn. In order to 





examine the effect of the interruption, imagine each portion of the 

 wave as it arrives at x s to become a centre of disturbance, from 

 whence diverge hemispherical waves into the two media respectively, 

 with the velocity appropriate to each. Consider for the present the 

 first medium only. The times which have elapsed since the wave, now 

 supposed to be at s, was at the points N, Q, , are those required to 

 describe N ri, Q q, , and therefore the radii s n ', Q 7', , of the hemi- 

 spherical waves which diverged from N, Q - - -, are equal to x n, Qq ---- 

 Tlir m,'h s draw a plane perpendicular to the plane of I he paper, touch- 

 ing in ' the hemisphere whose centre is at 1C. It ia evident that the 

 point ' will lie in the plane of the paper, and that the plane s ' will 

 touch all the secondary waves, and therefore will be the front of the 

 reflected wave. Moreover N ', which is perpendicular to this plane, will 

 represent the reflected ray corresponding to the incident ray of M if. 

 Hence the reflected ray lies hi the plane of incidence ; and on account 

 of the equality of the triangles s N n, s N ', the augl.-s sxw, SN', 

 which are the complements of the anvil's of incident and reflection, 

 are eqttal, ami therefore the aii.'li'.i of incidence and reflection are 

 themselves equal. 



Consider now the second medium. Everything will be the same as 

 before, except that the radii K n', Q '/, of the secondary waves 



diverging from N, Q, instead of being equal, will only be propartimtal, 

 . Q q, bearing to them the ratio of r' to r, where r is the velo- 

 city of propagation in the second medium. The semicircles in which 

 the hemispheres diverging from the various points of N s are cut by 

 the plane of the paper will form a system of curves for which s is a 

 centre of similitude; and it will be readily seen that if through s be 

 drawn a plane perpendicular to the plane of the paper, and touching in 

 ' the hemisphere diverging from s, it will be the envelope of the 

 secondary waves in the second medium, the point n' will lie in the 

 plane of the paper, and N ' will be the course of the ray refracted at N, 

 which will therefore lie hi the plane of incidence. Also the angles 

 x s n, y s n', which are the complements of the angle.; s N , s x ', are 

 equal to the angled of incidence and refraction respectively, and sine 

 of incidence : sine of refraction : : NHN s : NM'-^-NS : : N : N n' : : v :v', 

 a ratio which is constant, that is, independent of the angle of incidence; 

 which gives the law of refraction. 



These laws may be easily extended to the general case in which the 

 incident waves and the reflecting or refracting surface are of any form. 

 ; For let P Q in either of the above figures represent a ray incident at ij, 

 and therefore normal to the incident wave, ami through Q imagine two 

 ! tangent planes drawn, one to the incident wave, and the other to the 

 j reflecting or refracting surface. Confining our attention to the secondary 

 j waves which start in either medium from the immediate neighbourhood 

 of the point q, if we suppose them to start wheu a wave coinciding 

 | with the tangent plane to the wave, instead of the actual wave itself, 

 ] arrives at the surface, and again, if we replace the actual surface by the 

 tangent plane to it drawn through <}, we shall only commit an error on 

 the position of the centre and magnitude of the radius of a secondary 

 wave which is a small quantity of the second order, the distance of its 

 centre from the point ii being deemed a small quantity of the first 

 ! order. Hence the line of intersection of any two such secondary waves 

 will only be rendered erroneous by a small quantity of the first order, 

 which vanishes in the limit, and therefore the point of ultimate inter- 

 section of the secondary waves will not be affected. Hence the laws of 

 reflection and refraction will remain the name as before, the normal to 

 the surface at the point of incidence taking the place of the perpen- 

 dicular to the plane. 



