250 DR. HAROLD JEFFREYS ON TIDAL FRICTION IN SHALLOW SEAS. 
account of its narrowness or for some other reason, there is little transverse motion, 
the velocity along the channel is related to the pressure gradient across it according 
to the equation 
2 oopu = — dp/dy, 
where p is the density of the water, 
w is the component of the earth’s angular velocity of rotation about the 
vertical at the point considered, 
p is the pressure, and 
y is the distance measured across the channel. 
In the Yellow Sea the entrance is not narrow, but there seems reason to believe 
that the velocity across it is small, which is all that is required for the truth of the 
above equation. If now g denote the intensity of gravity, and >/ the elevation of the 
surface of the water above its mean position, then at any fixed point, at whatever 
depth, the variation of p is equal to that of gp >/. Again, if Q be the earth’s angular 
velocity, and X the latitude of the place, 
w = Q sin X, 
and we have on putting X = 35°; Q = 7'3xl0 _5 /l sec.; g = 981 cm./sec. 2 ; 
u = -lT7x 10 7 ~ 
oy 
where C.G.S. units must now be used. 
On the coast of Korea the tide has an amplitude of about 10 feet, or 300 cm. 
The velocity of the inward current is about 4 knots, or 200 cm./sec. Now suppose if 
possible that the current remained constant right into the middle of the entrance; 
then the above formula shows that at a distance of 176 km. from the side there 
would be little vertical movement of the surface, and further away still a huge tide 
with an amplitude of some 30 feet would exist. It is not reasonable that the tide in 
the middle should be greater than that at the side, though it may easily be smaller. 
The alternative hypothesis is therefore that the current decreases as we approach 
the middle, and is very small over most of the sea. This will be adopted in the 
forthcoming discussion. We shall suppose that the current at distance y from the 
coast is in the same phase as that at the coast, and is a linear function of y. Then put 
u = (200 — ky) cosy t. 
At the shore ^ is equal to 300 cos (yt — a), where a is the difference in phase 
between the tide height and the current strength. For the semi-durnal tide it is 
twice the angle moved through by the moon relatively to the earth in one hour, 
or 29 degrees. In general 
t] — 300 cos [yt — a) — 8'6 x 10 -8 cos yt (200 y — ^ky 3 ). 
