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82                                     ON THE PROPAGATION OF WAVES                                 [<
If B=±A, reduction ensues to the familiar positive or negative ]_: gressive wave. But if £ be not equal to ± A, (55), taking the form
<b = | (A + B) cos (pt - lex) + \ (A — B) cos (pt + lex),
clearly does not represent a progressive wave. The mere possibility reduction to the form (57) proves little, without an examination of • character of H and 0.
It may be of interest to consider for a moment the character of 0 in (( If B/A, or, say, m, is positive, 0 may be identified with lex at the quadra but  elsewhere they differ, unless m = l.   Introducing the imaginary pressions for tangents, we find
6 = Tex + M sin 2kx + $M* sin 4kx + %M3 sin 6kx + . . . , ...... (61
where                                        M= — ^ ................................. (62
•m + 1
When k is constant, one of the solutions of (53) makes </> proportional e~ikx. Acting on this suggestion, and following out optical ideas, let assume in general
^=,r,e-^adx, ............................... (63;
where the amplitude rj and a are real functions of x, which, for the purp of approximations, may be supposed to vary slowly. Substituting in (i we find
^ + (*«-a')i7-2»a*^(aS) = 0 ................... (84
For a first approximation, we neglect cPy/dx-.    Hence
7c = a,       k^Tj = C, ................................. (65
so that                             <j> = Ck-*eipte-ilkdxl ........................... (66
or in real form,                <f> = Ck~% cos (pt — fkda) ......................... (67
If we hold rigorously to the suppositions expressed in (65), the sa faction of (64) requires that d2r}fdxz = Q, or dzk~^/dxz = 0. With omiss of arbitrary constants affecting merely the origin and the scale of on, t makes k* — x~4, corresponding to the differential equation
72,
whose accurate solution is accordingly
^GW^-1^...............................(69
In (69) the imaginary part may be rejected.    The solution (69) is. course, easily verified.    In all other cases (67) is only an approximation.We may illustrate this even from the case of the uniform medium where ^ (x) = cos lex, % (as) = sin kx. In this case (56) becomes