252 ON THE THEORY OF LONG WAVES AND BORES [385
In a small positive progressive wave, the relation between the particle-velocity u at any point (now reckoned relatively to the parts outside the wave) and the elevation h is
If this relation be violated anywhere, a wave will emerge, travelling in the negative direction. In applying (7) to a wave of finite height, the appropriate form of (7) is
where / is a known function of I + h, or on integration
To this particle- velocity is to be added the wave-velocity
VRJ + A)/}, ................................. (10)
making altogether for the velocity of, e.g., the crest of a wave relative to still water
Thus if/ be constant, say g, (9) gives De Morgan's formula
«- 2 V$r{(* + *)*-**), .......
and (11) becomes
(13)
If, again, /. .................... .............. (14)
.(11) gives as the velocity of a crest
/, I A /i I _ /ffft /-IK\
l + h +l + h~^(flL)> .......... ' ................ ( '
which is independent of h, thus confirming what was found before for this law of force.
, As regards the question of a bore, we consider it as the transition from a uniform velocity u and depth I to a uniform velocity uf and depth I', I' being greater than I. The first relation between these fotir quantities is that given •by continuity, viz., "
lu^l'u' ...... ............................ (16)
'Th'e second relation arises from a consideration * of momentum. It maybe convenient to take first the usual case of a constant gravity g. The mean pressures at the two sections are %gl, %gl', and thus the equation of momentum is , ..'... , , '
P) ........................... -.(17)o the undisturbed water lying beyond it on the two sides, and it is implied that the uniform levels on the two sides are the same. But the theory of long waves is not thus limited, and we may apply it to the case where the uniform levels on the two sides of the variable region are different, as, for example, to bores. This 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.