Abstract
In the theory of long waves in two dimensions, which we may suppose to be reduced to a "steady" motion, it is assumed that the length is so great in proportion to the depth of the water that the velocity in a vertical direction can be neglected, and that the horizontal velocity is uniform across each section of the canal. This, it should be observed, is perfectly distinct from any supposition as to the height of the wave. If
l
be the undisturbed depth, and
h
the elevation of the water at any point of the wave,
u
0
,
u
the velocities corresponding to
l
,
l
+
h
respectively, we have, as the equation of continuity,
u
=
lu
0
/
l
+
h
. (1) By the the principles of hydrodyamics, the increase of pressure due to retardation will be ½
ρ
(
u
0
2
-
u
2
) =
ρu
0
2
/2. 2
lh
+
h
2
/(
l
+
h
)
2
. (2)
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