TIDAL DATUM PLANES 71 
From Figure 34 it is seen that despite the relatively large changes in sea level from 
day to day, the relation of sea level to half-tide level at Boston remains very nearly 
constant. To be sure, this relation changes somewhat from day to day, but these 
changes are relatively small. Sea level here is, almost without exception, above half- 
tide level, on the average by 0.12 foot. 
It is not difficult to see why the relation of sea level to half-tide level is not constant 
from day to day. In the first place the fact that the tidal cycle has a period of very 
nearly 25 hours, and not 24 hours, introduces slight variations; and in the second place 
it is obvious that changes in wind and weather must vary that relationship. For 
example, suppose that at any given place we take two days during which the high 
waters, and likewise the, low waters were exactly similar. Half-tide level for the two 
days would therefore be exactly the same. And if the weather conditions during the 
two days were similar, sea level likewise would be the same for the two days. 
Suppose, however, that weather conditions on the second of the two days were 
the same as on the first day only until the occurrence of the last high or low water of 
the day (which, for the sake of illustration, we may assume to have occurred about 6 
p. m.). Suppose that from that time to the end of the day the direction or velocity 
of the wind was different. Obviously, the half-tide level for that day would not be 
changed since the last high or low water used in deriving it has already occurred. But 
the hourly heights of the tide for the remainder of the day would differ from the cor- 
responding heights on the first day, and hence, although half-tide level for the two days 
would still be the same, the sea levels would differ. 
If monthly heights of sea level and half-tide level are compared, the relation 
between the two is found to be less variable than in the case of the daily levels. Figure 
35 shows the monthly heights of sea level and half-tide level at Boston for the two 
years, 1947, and 1948. Without exception, monthly sea level is seen to be above 
half-tide level. For those two years sea level averaged higher than half-tide level by 
0.10 foot, the least difference between monthly values being 0.05 foot and the greatest 
0.15 foot. 
A comparison of yearly heights of sea level and half-tide level shows a more nearly 
constant relation than between monthly values. For the 27 years of observations 
available at Boston from 1922 through 1948 the yearly values of sea level are above 
half-tide level, averaging 0.12 foot higher, the greatest yearly difference being 0.14 foot 
and the least 0.09 foot. 
The relation between half-tide level and sea level at any place depends upon the 
amplitude and phase relations between the various constituents of the tide at that 
place. For tides of the semidaily and mixed types, the relation between half-tide level 
and sea level is given approximately, in the harmonic notation, by the formula 
HTL=SL+M, cos (2M3—M2) a 
2 
cos (Ms—K:—03), 
in which HTL stands for half-tide level, SL for sea level, and the other terms have their 
usual significance in the harmonic notation. 
. Since the amplitudes of the various components vary somewhat from year to year, 
it follows that the relation between sea level and half tide level may differ from year to 
year. Furthermore, the cosine of (2M:—Msz) and also of (Mz—Ki—Oi:) may be 
