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14                                              HYDRODYNAMICAL NOTES
that are not infinitely small.    For the stream-function of the .waves r< as progressive, we have, as in (1),
•& = — &e~y cos (x — ct] + terms in a4, so that
+ (d-/dY = a*cr*« + terms in a5.
Thus the mean kinetic energy per length x measured in the direc propagation is
where y is the ordinate of the surface.    And by (3)
r y2dx = {Ł (a2 + \ a4) + -Ja^} x.
Hence correct to a4,
K.E. = Again, for the potential energy
or since g = 1 — of,
P.E. =
The kinetic energy thus exceeds the potential energy, when a4 is retai
Tide Races.
It is, I believe, generally recognized that seas are apt to be excepi heavy when the tide runs against the wind.    An obvious explanation founded upon the fact that the relative motion of air and water greater than if the latter were not running, but it seems doubtful v this explanation is adequate.
It has occurred to me that the cause may be rather in the motion stream relatively to itself, e.g. in the more rapid movement of the upper Stokes' theory of the highest possible wave shows that in non-rotating the angle at the crest is 120° and the height only moderate. In sucl the surface strata have a mean motion forwards. On the other h Gerstner and Rankine's waves the fluid particles retain a mean positi here there is rotation of such a character that (in the absence of wa\ surface strata have a relative motion backwards, i&. against the direc propagation*. It seems possible that waves moving against the tic approximate more or less to the Gerstner type and thus be cap; acquiring a greater height and a sharper angle than would other\ expected. Needless to say, it is the steepness of waves, rather tha:
* Lamb's Hydrodynamics, § 247.f, additional to thtwu c!Xpr«KHtjtl in (Hi).    Tluwe are