440 WAVE THEORY as to suit the resolution of an infinite uniform primary wave, we have, as the effect of QQ , dp sinit(at-p) . (I). /v^// VVp / ) The development of this expression for the operation of a linear source would take us too far. 1 We must content ourselves with the limiting form assumed when KT is great, as it would almost always be in optics. Under these circumstances the denominator may be simplified by writing so that (1) becomes Now /SlTLKlldli f COSKlldu f TT ~T i!T~ = / / = A. / ( 5~ ) V A, f u Jo V z< *V .2ic/ and thus we obtain dx ~ r ~ COS Kat ~ r > which gives the effect of a linear source at a great distance. The occurrence of the factor r~* is a consequence of the cylindrical ex pansion of the waves. The whole effect is retarded one-eighth of a period in comparison with that of the central element, instead of one-quarter of a period as in the case of a uniform wave extending over the whole plane. Plane The effect of the uniform plane wave can be recovered by inte- unit orm grating (2) with respect to x from - <x> to + oo , on the supposition wave. that nr is great. We have dx _ rdr _ /r.d(r-z) fr~ /r.x~ ^J(r~+z). /(r-z) and in this, since the only elements which contribute sensibly to the integral are those for which (r - 2) is small, we may write The integral can then be evaluated by the same formula as before, and we get finally COSK (at-z), the same as if the primary wave were supposed to advance without resolution. The recom position of the primary wave by integration with rectangular coordinates is thus verified, but only under the limitation, not really required by the nature of the case, that the point at which the effect is to be estimated is distant by a very great number of wave-lengths from the plane of resolution. Variable We will now suppose that the amplitude and phase of the pri- ainpli- mary wave at the plane of resolution 2 = are no longer constants, tude and but periodic functions of x. Instead of cosnat simply, we should phase. have to take in general A cos (px +/)cos KCit + B cos (px + g} sin tat ; but it will be sufficient for our purpose to consider the first term only, in which we may further put for simplicity A = 1, /= 0. The effect of the linear element at x, 0, upon a point at |, z, will be, according to (2), -cospx sTiK(at-r- |A), where r is the distance, expressed by r 2 = z 2 + (a;-) 2 . Thus, if we write o;=| + o, the whole effect is sin (K ~ **" ~ KT +pa} + s(Kat-pt-$iT- Kr-pa}} . . (3), where r s =z 2 + a~. In the two terms of this integral the elements are in general of rapidly fluctuating sign ; and the only important part of the range of integration in (for example) the first term is in the neigh bourhood of the place where pa. - KT is stationary in value, or where 2)da icdr Q (4). In general a da - r dr = 0, so that if the values of o and r corre sponding to (4) be called a , r , we have (5). P K J(lC-p~) Now, in the neighbourhood of these values, if a = a,
- i.Yi . r
2r 1- 1 Theory of Sound, 341. in which by (5) the term of the first order vanishes. in (3), we get for the first term -~ - -*-Kr Q +^a ) cos h ai ~ Using this - cos (teat +p - %ir - KT O +pa ) s where for brevity h is written for The integration is effected by means of the formula /coshu-du = / siiihu-du = // x J*> and we find -^ 5- cos (nat +p% - ZJ(K" p") The other term in (3) gives in like manner K so that the complete value is When p = 0, we fall back on the uniform plane wave travellin with velocity a. In general the velocity is not a, but Velocity of pro- The wave represented by (6) is one in which the amplitude at various points of a wave-front is proportional to cospf, or cos^c; and, beyond the reversals of phase herein implied, the phase is con stant, so that the wave-surfaces are given by 2= constant. The wave thus described moves forward at the velocity given by (7), and with type unchanged. The above investigation may be regarded as applicable to gratings which give spectra of the first order only. Although vary, there is no separation of colours. Such a separation requires either a limitation in the width of the grating (here supposed to be infinite), or the use of a focusing lens. It is important to remark that p has been assumed to be less than K, or cr greater than A; otherwise no part of the range of integration in (3) is exempt from rapid fluctuation of sign, and the result must be considered to be zero. The principle that irregularities in a wave-front of periods less than A cannot be pro pagated is of great consequence. Further light will be thrown upon it by a different investigation to be given presently. The possibility of the wave represented by (6) is perhaps sufficiently established by the preceding method, but the occur rence of the factor K//(K" ~2)") shows that the laws of the secondary wave (determined originally from a consideration of uniform plane waves) was not rightly assumed. The correct law applicable in any case may be investigated as Cor- follows. Let us assume that the expression for the wave of given rected periodic time is law of , y)dxdy; (8), second ary and let us inquire what the value of F (x, y) must be in order that the wave - application of Haygens s principle may give a correct result. From (8) dx and We propose now to find the limiting value of dfyjdz when z is very small. The value of the integral will depend upon those elements only for which x and y are very small, so that we replace F(a;, y) in the limit by F(0, 0). Also, in the limit, so that
f/~-z
dxdy=// dxdy= - 2-n ; p tip P -f : d l/ iKCtt Limit = - 2irc dz i.at i F(0,0). The proper value of e lKa F(x, y) is therefore that of -dtyjdz at the same point (x, y, 0) divided by ITT, and we have in general iKft dxdy .... (9). In the case of the uniform plane wave, iK(at z) . i K (atz) so that 7/ c agreeing with what we have already found for the secondary wave
in this case.