84 ON THE PBOPAGATION OF WAVES
and thus if we suppose 77 to pass from one constant value to another thrc a finite transitional layer, the transition is also from one uniform /< another; and (73) shows that there is no reflection back into the medium. If the terminal values of r/ and therefore of 7c2 be given, and transitional layer be thick enough, it will always be possible, and that i infinite number of ways, to avoid a negative &2, and thus to secure com] transmission without reflection back; but if with given terminal values layer be too much reduced, A'2 must become negative. In this case reflet cannot be obviated.
It may appear at first sight as if this argument proved too much, and there should be no reflection in any case so long as kz is positive througl But although a constant 77 requires a constant k", it does not follow versely that a constant kz requires a constant 77, and, in fact, this is not One solution of (72), when & is constant, certainly is ?)'2—G2/k; but complete solution necessarily includes two arbitrary constants, of which not one. From (60) it may be anticipated that a solution of (72) may b
i72 = An- cos2 ken + & sin2 kas = $ (A* + J52) + \ (A- - B") cos 2fe»... .(7 From this we find on differentiation
and thus (72) is satisfied, provided that
k*A2B2 = C2 ................................. (7
It appears then that (77) subject to (78) is a solution of (72). second arbitrary constant evidently takes the form of an arbitrary add to oc, and 17 will not be constant unless A2 = B-.
On the supposition that 77 and a are slowly varying functions. approximations of (65) may be pursued. We find
The retardation, as usually reckoned in optics, is fkdtv. The addit retardation according to (80) is
dv* - 2 [ dx J *H d*~J •"" As applied to the transition from one uniform medium to another retardation is less than according to the first approximation by
doche changes are gradual enough, a may be identified with k, and then i) oc & , as represented in (67).