DR. HAROLD JEFFREYS ON TIDAL FRICTION IN SHALLOW SEAS. 
241 
same is true of u if the friction is small. Thus in such cases the velocity is the same 
at all points in the same cross-section, so that observations made at the side will be 
correct for points in the middle. If friction is great this result must be modified, 
since for the same velocity the frictional force is independent of the depth of the 
water, whereas the mass affected is proportional to the depth. Thus friction has 
more influence in reducing the velocities in shallow water than in deep water. Hence 
when the channel is shallow at the sides and deep in the middle the velocity will be 
least at the sides. The ratio of the frictional term to the first term is 0'002 u/2D'Q, 
where D' is the depth at the shallow part. When this is 10 fathoms and the 
velocity 1 knot the fraction is 0'25, so that this effect is then appreciable ; and when 
the depths are less and the velocities greater, the influence of friction may increase to 
such an extent as to dominate the whole character of the motion. When this occurs, 
the velocity will always be in the direction of decreasing pressure, and inertia may 
be neglected. 
The last result may appear to contradict the general principle that “ still waters 
run deep.” There is no real contradiction, however, for the problems referred to are 
different. The above argument deals with the differences between the velocities at 
places in the same cross-section of the channel, whereas the proverb concerns rivers 
whose depth varies along them, and in such cases the motion is naturally slowest 
where the depth is greatest, since the amount of water crossing any section in a given 
time must be the same. It also has an important and well-known application in bays 
of varying depth and width, such as the Bay of Fundy. If, for instance, the bay is 
very long, and these quantities change only by small fractions of themselves per wave¬ 
length, it can be shown* that the height of the wave at any point is proportional 
to D - *, and the velocity of the water to where b is the width at the 
surface. The rate of dissipation of energy across any section is proportional 
to bu 3 or to It therefore increases slowly as the channel becomes narrower 
and much more quickly as it becomes shallower. When the depth and width vary 
much within a wave-length these results cease to be useful approximations, but the 
tendency for the height of the tide and the velocity of the tidal current to increase as 
the channel becomes narrower and shallower remains. Thus in such places we often 
find very high tides and strong tidal currents. Apparently, however, their limited 
area prevents the dissipation in them from being as great as that in larger places with 
less violent currents (at least, if the Bay of Fundy may be regarded as typical 
of them). 
The widths of most actual bays are, however, comparable with their lengths, and 
m these it is generally a matter of some difficulty to settle whether we can treat the 
recorded currents as a fair sample of the whole. The amplitude of the tide in mid¬ 
ocean is only about a foot, but in the shallow water around the coasts it is magnified 
to several feet, and the tidal currents are increased correspondingly. Where the 
* Lamb, ‘ Hydrodynamics,’ p. 258. 
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