300 REPORTS ON THE STATE OF SCIENCE, ETC. 
Meteorological Tides. 
4. The second hypothesis mentioned in $1 deals with the effects of the varying 
distribution of atmospheric pressure, both statically and by the operation of wind, 
on sea level and tides. A paper is being written by the Secretary on this subject, to 
be published elsewhere; the main results are recorded in the Journal of Section A 
and also in the Tidal Institute Reports for 1922 and 1923. A great deal of numerical 
work has been done and the Newlyn residues have been of very great value ; in fact. 
much of the work would have been impossible without them. 
Definite evidence of perturbations of tidal motion has been obtained, but the law of 
operation is somewhat complex. It is clear, however, that these effects are second- 
order effects, and are governed by the product terms in the equations of motion and 
continuity in just the same way as are the shallow water effects discussed in 1921. 
The product terms involve currents as well as heights, and since the meteorological 
current and height are probably fairly well represented by linear combinations of 
atmospheric pressure and its gradients in two directions, it is obvious that the product 
terms will give a rather complex law for meteorological tides. This law has not yet 
been obtained. It is, however, possible that contributions of importance arise from 
the long period changes in the distribution of atmospheric pressure, in which case the 
law will be simplified. The effect will show itself as a perturbation of harmonic 
‘ constants,’ or, alternatively, as new harmonic constituents, and this may be the 
explanation of a part of the residual semi-diurnal oscillation at Newlyn. It may 
possibly account for a perturbation of 0-5 foot and period of half a year in the M, 
constituent at Liverpool for the year 1918. Further analyses, however, will be 
necessary before this matter can be adequately considered. 
Shallow Water Effects. 
5. It was shown in the Report for 1921 that a simple relation exists between the 
quarter-diurnal tide and the square of the semi-diurnal tide. The sixth-diurnal tide, 
according to the theory of motion in a shallow canal, should be proportional to the 
sixth-diurnal part of the cube of the semi-diurnal tide, with a constant factor and 
constant phase shift, while tests on tidal records at Liverpool confirm this law and 
also a similar law for the eighth-diurnal tide. We can investigate the shallow water 
effects, therefore, as follows. 
Taking the time-origin at High Water of the semi-diurnal tide and writing C,, for the 
nth diurnal tide, we have the compound tide given by 
C€=6461+C.+ Cet Acie ie We P 
with C.= RB cos ot 
a= c, R? cos (26¢ + 4) (1) 
Z,= ¢; B® cos (3ot + 75) 
Ca= Cs R4 cos (40¢ + 7s) 
where ¢,, 7,, are constants. 
High Water of the compound tide will occur when 
sin ot = — 2c, R sin (2ot + 74) — 3c, R* sin (Sat + 7) 
— 4c, R’ sin (4ct + 75) — - . Perky) 
The constants c,, 7, can be determined from the harmonic constants for the principal 
lunar constituents, for when R is equal to the amplitude of M, then the tide is 
correctly represented by M,, M,, M,, M,, . 
The data for Liverpool are 
9 ie} 
M, ; Hs 9:975 ft. «= 320:7 
M,: H= -691ft. «=—211 10°,—-695 y,= 70 
M,: H= -196f. «=—331 10’,=-197 ¥,— 272 
M,: H= -068ft. «=255  10!c,—-069 y,— 308 
whence we can construct the following table : 
R 5 10 15 
eR? 174 695 1-564 
c,R® 025 -197 665 
c,R! 004 -069 “350 
2¢,R 070 -139 -209 
3c,R2 O15 059 133 
4c,R® 004 028 -093 
pana 
