314 DEEP WATER WAVES, ETC. . [393
The two alternatives indicated in.(50) correspond to the particular cases already considered. In the first (PP' + QQ' = 0) we have a purely progressive wave and in the second a purely stationary one.
When the condition (50) does not hold good, it is impossible to satisfy our equations as before with constant values of n, P, Q, P', Q'; and it is perhaps hardly worth while to pursue the -more complicated questions which then arise. It may suffice to remark that an approximately stationary wave can never pass into an approximately progressive wave, nor vice versa. The progressive wave has momentum, while the stationary wave -has none, and momentum is necessarily conserved.
When ft, ft', 7, 7', are not zero, additional terms enter. Equations (26), (30), show that the additions to b, b', vary as the sine and cosine of V(2#) • t, and represent waves which might exist in the complete absence of the principal wave.
The additions to c, c, are more complicated. As regards the parts depending in (31), (32), on dy/dt, dy'/dt, they are proportional to the sine and cosine of v'(3#) • t, and represent waves, which might exist alone. But besides these there are other parts, analogous to the combination-tones of Acoustics, resulting from the interaction of the /3-waves with the principal wave. These vary as the sine and cosine of \/g. {V2 ± 1} t, thus possessing frequencies differing from the former frequencies. Similar terms will enter into the expression for nz as determined from (28), (29). '
In the particular case of ft, ft', vanishing, even though 7, 7' (assumed still to be of the third order) remain, we recover most of the former simplicity, the only difference being the occurrence in c, c', of terms in \f(S<y). t, such as might exist alone.hange of dashed and undashed letters, and thus it serves equally for the equation in a'.os if + £ sin t', a' = A' cos if. + J3' sin t', with &' = -aa', &=£(a'2-a2), c' = 0, c = 0.