262 
DR. HAROLD JEFFREYS ON TIDAL FRICTION IN SHALLOW SEAS. 
for the periods of the two tides are not very different, so that inertia will affect their 
amplitudes in the same ratio. In shallow areas, however, the frictional force is not 
proportional to the velocity but to its square, and accordingly friction has more 
relative effect in reducing the tides when they are great than when they are smaller. 
The ratio of the solar to the lunar tide, as found from observations, is accordingly less 
than the theoretical value, since the ratio of the ranges at springs and neaps is 
reduced. This ratio is stated in the ‘Admiralty Tide Tables for 1920 ’ to be 1 : 273 on 
an average. If now 0 be the phase of the lunar tide, let (l— r) 0 be the phase of the 
solar tide, so that r is l/29. Let A be the amplitude of the lunar tidal current and 
Av that of the solar tidal current. The total current is 
Afcos 6 + v cos (l— r)d} = A (l + 2v cos r 0 + v 2 Y cos ( 6— tan -1 — ^ — 
\ l + i/ cos r 6 
which is now expressed as a simple harmonic motion with a slowly varying amplitude 
and period. The amplitude at springs isA(l + j/). The dissipation is proportional to 
the cube of the current, and therefore to the cube of the amplitude. The ratio of 
the mean dissipation to the dissipation at springs is therefore the average of 
(1 + 2j/ cos r6 + j/ 2 )V(l + r) 3 , If v 6 be neglected, the numerator of this is l+f 
Assuming that the ratio of the velocities is the same as that of the vertical ranges, 
we find that this fraction is equal to 0'51. Applying this correction to the spring 
tide dissipation, we find that the average dissipation is l'l x 10 19 ergs per second, 80 
per cent, of what is required. It would give a secular acceleration of the moon 
of 7" per century per century. 
The agreement between the dissipation in shallow seas and that necessary to 
account for the lunar secular acceleration is much closer than the data would entitle 
us to expect. Two-thirds of that found takes place in the Bering Sea, the estimate 
for which may be incorrect by half its amount. What we are entitled to assert, how¬ 
ever, is that this dissipation is certainly enough to account for a large fraction of the 
secular acceleration, and that there is nothing to prove that it is incapable of 
accounting for the whole of it. 
It is uncertain whether the dissipation in any other coastal regions is notable in 
comparison with those already considered. The only partly enclosed areas not treated 
here that are of considerable size are some of those in the North-west Passage. There 
is an extensive shallow region off the coast of Patagonia, but it is in no way enclosed, 
being perfectly open to the Atlantic. Thus it is difficult to make any reliable inference 
about the currents in it. Many records of tidal currents along the coast are given, 
some reaching several knots, but all of them seem to refer to currents up rivers or near 
their mouths, where the general currents must be magnified, or to currents over bars 
and shoals; there seem to be no data about the currents more than a few miles out to 
sea. 
The dissipation over local shallows like shoals and bars and in narrow bays and 
