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1914] ON THE THEORY OF LONG WAVES AND BORES 253
By these equations w.and u' are determined in terms ofl,l':
u* = %g(l + I') . I'll, u'* = %g(l + V) . Ill' ............. (18)
We have now to consider the question of energy. The difference of work done by the pressure at the two ends (reckoned per unit of time and per unit of breadth) is lu (\gl — ^gl'). And the difference between the kinetic energies entering and leaving" the region is lu(^u2 — ^u'2), the density being taken as unity. But this is not all. The potential energies of the liquid leaving and entering the region are different. The centre of gravity rises through a height \ (If — I), and the gain of potential energy is therefore hi. kg (I'— I). The whole loss'' of energy is accordingly
-l)} = lu gl-ffl'
This is much smaller than the value formerly given, but it remains of the same sign. "That there should be a loss of energy constitutes no difficulty, at least in the presence of viscosity ; but the impossibility of a gain of energy shows that the motions here contemplated cannot be reversed."
We now suppose that the constant gravity is replaced by a force/, which is a function of y, the distance from the bottom. The pressures p, pf at the two sections are also functions of y, such that
(20) The equation of momentum replacing (17) is now
lu (u - u') = p' dy -\pdy~
rl "IP (™ ~]l rl'
the integrated terms vanishing at the limits. This includes, of course, all special cases, such as /= constant, or /oc y~~*.
As regards the reckoning of energy, the first two terms on the left of (19) are replaced by
(i r* ir1'}
lu H pdy-j, p'dy\ ...................... (22)
l^ JO f> J Q )
The third and fourth terms representing kinetic energy remain as before. For the potential energy we have to consider that a length u and depth I is converted into a length u' and depth I'. If we reckon from the bottom, the potential energy is in the first case
ri rv dy\ fdy,
Jo Jois is a problem which I considered briefly on a former occasionf, when it appeared that the condition of conservation of energy could not be satisfied with a constant gravity. But in the calculation of the loss of energy a term was omitted, rendering the result erroneous, although the general conclusions are not affected. The error became apparent in applying the method to the case above considered of a gravity varying as the inverse cube of the depth. But, before proceeding to the calculation of energy, it may be well to give the generalised form of the relation between velocity and height which must be satisfied in a progressive wave}, whether or not the type be permanent.