THE standard method of determining the height of tide at any period between successive times of high and low water consists of drawing a circle and dividing the circumference proportionately to the high water interval. The projection of these intervals on the diameter of the circle then gives the relative height of tide for the time chosen. This method is based upon the fact that for the period between successive high and low water points the motion of the tide may be assumed sinusoidal. It is of course well known that the tide may be expressed by Fourier’s series, and the approximation involved above depends upon the fact that the particular harmonic value producing daily changes in tide predominates. Therefore, this approximation is satisfactory where a uniform tide is experienced, but will not hold where peculiar conditions prevail as with some islands in the Pacific, the Bay of Fundy, and the like.
The circumference of the circle must be divided up differently for each tide interval or else still further approximations must be involved by assuming an average interval.
Theory for square chart. The object of this article is to describe a different form of chart which may be universally used under the same approximations and limitations as hold for the circle. The chart consists of two Z-type alignment charts placed together forming a square with a common diagonal. The scales are based upon the following relation:
Chart for determining height of tide.
h = R/2[1 ±cos(t?/D)]
Where: t = Hours elapsed since last high or low tide.
D = Tide interval.
R = Range of tide.
h = Height of tide at the time t. The sign is plus for ebb and minus for flood tides. These values are shown graphically in Fig. 1.
[FIG. 1.: APPROXIMATE TIDAL MOTION FROM LOW TO HIGH WATER. Defining symbols used in chart.]
Construction of square chart. In the main chart the left-hand edge of the square is divided uniformly in values giving range of tide in feet. The right hand edge of the square is divided uniformly in values of height above the last low water, or distance below the last high water. In other words, this edge of the square gives the change in height of water from the first slack water tide chosen.
[FIG. 2. CHART FOR DETERMINING HEIGHT OF TIDE
Example:—Given D =6h16mR = 10.8 ft.
Draw AB from D to time desired—,i.e., 3 hrs. here.
Read 48 per cent on diagonal. Draw MN from R through 48 per cent.
Answer:—Change of tide:—H = 5 ft.]
The top of the square is uniformly divided in hours past the first tide chosen. The bottom of the square is uniformly divided in values giving the interval between low water and high water or the reverse.
The upper part of the diagonal is divided in a non-linear manner so that it relates the time elapsed to the tide interval reading in per cent. The lower half of the diagonal repeats these same percentage figures spaced in accordance with a special function scale so that it relates the height of tide h to the range in accordance with the formula above.
Method of use. To use this chart the time for the low- or high-water period just past and the high- or low-water period being approached is taken from tide tables for the locality. The range of tide between these two values is also noted. The time at which knowledge of the height of tide is desired is then expressed as hours and minutes past the previous time of slack water. A ruler or any straight line is then drawn through the elapsed time on the top of the chart to the interval at the bottom of the chart and the per cent of interval elapsed noted on the diagonal. The straight edge or line is then moved to the range on the left-hand side of the chart so that it passes through the same per cent of elapsed interval upon the lower half of the diagonal. Its intersection with the right-hand side of the chart then gives the change in height of tide from the reference value chosen.
Example 1. The tide at Boston August 15, 1928, gave the following values:
Low tide 4h 37m a.m. — 1.2 ft.
High tide 10h 53m a.m. + 9.6 ft.
D = 6h 16m R = 10.8 ft.
- What will tide be at 7h 37m a.m.?
t = 3h 0m
%D = .48 h = 5 ft.
Answer.—5 ft. above low water or 5—1.2 = 3.8 ft. above mean low water.
- When will tide = mean low water, i.e., h = 1.2?
%D = 0.21 t = 1h 20m
Answer. Mean low water at 4h 37m + 1h 20m = 5h 57m.
- When will tide give 6 ft. above mean low water, i.e., h = 6 + 1.2 = 7.2?
%D = 0.6 t = 3h 45m
Answer. 6 ft. at 4h 37m + 3h 45m = 8h 22m.
Example 2. Boston, October 24, 1928.
High tide 7h 08m a.m. + 7.9 ft.
Low tide 1h 14m p.m. + 1.5 ft.
D = 6h 06m R = 6.4 ft.
A. What will tide be at 11h 08m a.m.?
t = 4h 0m %D = 0.65 h = 4.6 ft.
Answer. 4.6 ft. below high water
= 3.3 ft. above low water
+ 1.5
4.8 ft. above mean low water
B. When will tide give 6 ft. above mean low water?
6 ft. above mean low water = 4.5 ft. above low water =1.9 ft. below high water.
%D = .36 t = 2h 15m
Answer. 7h 08m + 2h 15m = 9h 23m a.m.
Note: The chart has dotted lines showing values for Example 1A. The accuracy will vary with the characteristic of the tide, configuration of the locality, and other such features. For large open harbors where the tidal currents are not constrained by the configuration of the bottom and for times of year when successive high-water intervals are not materially different, the accuracy should be quite high, the errors probably being of the same order of magnitude as those due to the effects of wind and other such variable conditions.
When the successive high-water intervals have different values it has the effect of tipping the axis of the curve shown in Fig. 1. Under these conditions the error due to this cause will be minimum half way between low water and high water. Since exact values are known at low and high water, the method of construction therefore minimizes error. As stated above where the tide for particular reasons does not follow a uniform curve the chart is not applicable.