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Let us now suppose that the principal terms represent a progressive wave. In accordance with (9) we may take
a = A cost',         a! — A s'mt', ..................... (17)
where t' = \fg . t.    Then if /3, /3', 7, 7', do not appear, c, c', are zero, and b = £ A2 (sin2 if - cos2 1'),   V = - A* cos t' sin if ; so that
representing a permanent wave-form propagated with velocity V<7- So far as it goes, this agrees with (1). But now in addition to these terms we may have others, for which b, b' need only to satisfy
(d*/dt"- + 2)(b, 60 = 0, ........................... (19)
and c, c' need only to satisfy
(cZ2/^/2 + 3)(c, c') = 0 ............................ (20)
The corresponding terms in y represent merely such waves, propagated in either direction, and of wave-lengths equal to an aliquot part of the principal wave-length, as might exist alone of infinitesimal height, when there is no primary wave at all. When these are included, the aggregate, even though it be all propagated in the same direction, loses its character of possessing a permanent wave-shape, and further it has no tendency to acquire such a character as time advances.
If the principal wave is stationary we may take
a = Acost',       a' = 0 If /3, £',-7> 7) vanish,
b = -%a?,    6' = 0,   c = 0,    c' = 0, and                           y = A cos ae . cos t' - % A2 cos 2« . cos2 If ................ (22)
According to (22) the surface comes to its zero position everywhere when cos t' — 0, and the displacement is a maximum when cos £' = + 1. Then
y — ±A cos as— ^A2coB 20, ........................ (23)
so that at this moment the wave-form is the same as for the progressive wave (18). Since y is measured downwards, the maximum elevation above the mean level exceeds numerically the maximum depression below it.
In the more general case (still with /3, etc., evanescent) we may write
a = A cos if + £ sin t',    a' = A' cos if. + J3' sin t', with                &' = -aa',    &=£(a'2-a2),    c' = 0,    c = 0.
When /3, ft, 7, 7', are finite, waves such as, might exist alone, of lengths equal to aliquot parts of the principal wave-length and of corresponding frequencies, are superposed. In these waves the amplitude and phase are arbitrary.ith the same notation as before we have