240 
DR. HAROLD JEFFREYS ON TIDAL FRICTION IN SHALLOW SEAS. 
wave of tidal type to cross. This time is known to be independent of the period of 
the disturbance, depending only on the width and depth. Such a waye is reflected 
from one side to the other again and again before it reaches the other end. It is also 
known that the transverse velocity at the sides is zero, since water cannot cross a 
rigid boundary. Thus if we compare two points on opposite sides of the channel, we 
know that the times of arrival of the wave at them differ by a small fraction of a 
period; and since the transverse velocity at both is zero, it cannot be great at any 
intermediate point, for that would contradict the hypothesis that the wave-length is 
much greater than the width of the channel. The transverse velocity may accord¬ 
ingly be neglected in problems of this class. 
If the period of the entering wave is of the same order of magnitude as the time 
needed to cross the channel, we can no longer infer that the transverse velocity is 
much less than the longitudinal one. This case seldom or never arises. The velocity 
of a tidal wave is (i/D)* where g is the intensity of gravity and D is the depth ; and if 
the water was only 20 fathoms deep a tidal wave would in 12 hours travel 700 km., 
which is far greater than the width of almost any channel whose length is much 
greater than its width. Where the width is greater, the depth also is always greater, 
so that the above argument always holds in long channels of whatever size. 
If now x be the distance of a point from the entrance to the channel, y the distance 
from the side, u the longitudinal velocity of a particle there, and the height of 
the free surface above its undisturbed position, the equations of motion of the particle 
are 
SjU (7 3 W j J t n • i • 
— = — 7 —— a term due to motion 
at ox 
2 usii = —g 
3 1] 
Zy 
where w is the component of the earth’s angular velocity of rotation about the vertical 
at the point. The equation of continuity is 
_3 
dx 
(D «)=-N. 
v ’ 3 1 
From this and the second equation of motion we deduce at once 
dt] 
dx 
3>/° . 2u) j l/3>; 
dx g JD \3£ 
where ?/ 0 is the value of >/ at the side. In this we see from the conditions that the 
channel is narrow and the depth slowly varying along it that the first term is much 
greater than the others. Accordingly ~ is the same for all particles in the same 
OX 
cross-section of the channel, and the first equation of motion then shows that the 
