54 THE ADMIRALTY METHOD OF TIDAL PREDICTION (N.P. 159)
A revised method of tidal prediction developed by the Hydrographic Department was introduced in the new version of N.P. 159 first published in 1975, and it is to this new Admiralty Method of Tidal Prediction (N.P. 159) that this section refers to The method calculated the diurnal and semidiurnal tides and combines them in a vectorial diagram (Form A) together with any appropriate shallow water corrections (Form C), from which the resultant tidal curve can be plotted on a graph (Form B).
The Admiralty Method of Tidal Prediction must be used by Navigators when: –
The duration of rise or fall of the tide is outside the limits of 5 to 7 hours at Standard Ports in Volumes 2 and 3 of Admiralty Tide Tables.
The letter ‘p’ is inserted instead of time differences at Secondary Ports in Volumes 2 and 3 of A.T.T.
The height of the tide at times other than HW or LW has required at all secondary ports in Volumes 2 and 3 of A.T.T.
Harmonic constants are given for all secondary ports, and some Standard Ports in Part II of A.T.T., while in Table V11 data is given to adjust the harmonic constants for the day and year to a standard place. The four constituents used are those already referred to, namely:
M2 the lunar semidiurnal force and the part of the tide due to it;
S2 the solar semidiurnal force and the part of the tide due to it;
K1 the combined average lunar and solar diurnal force and the tide due to it;
O1 is the principal variation in lunar diurnal force and the tide due to it.
The period of time taken by the four major constituents to complete their cycle of forces or harmonic curves is different in each case. M2 takes 12.4 hours, S2 takes 12.0 hrs., K1 24.48 hrs. and O1 13.6 days.
In order to combine the curves to predict a resultant tide at a particular instant of time, angles up to 360° are used to indicate the state of each constituent. Thus, 000° indicates the instant when a constituent has its maximum tide-raising force or the instant when its high water occurs, and 180° indicates the maximum tide-lowering force or the instant of low water of a constituent. The rate of change of angle per hour, or the speed, of a constituent, is given by dividing 360° by its period. For plotting purposes on Form A of N.P. 159, the speed of the combined semidiurnal constituents is taken to be 29° per hour, and that of the combined diurnal constituents 14.5° per hour.
A standard Force for each constituent is taken for a standard place on the meridian of Greenwich and all other forces are referred to this by using the data given in A.T.T. consisting of Angles, Factors and Harmonic Constants, as follows:
A Tidal Angle of a constituent denotes the instant on a particular day when the Standard Force is maximum on the Greenwich meridian.
A Tidal factor is the increase or decrease in the average magnitude of the Standard Force on a particular day.
The Harmonic Constant “g” of a constituent is the angle denoting the time lag between the maximum Standard Force on the Greenwich meridian and the high water of that constituent at a particular port.
The Harmonic Constant “H” of a constituent is the HW rise or the LW fall from Mean Sea Level at the secondary port due to an average Standard Force.
Mean Sea Level or Zo. is the average level of the sea above Chart Datum. This level often varies owing to meteorological causes and can be corrected from the Seasonal Corrections table.
Mo is the symbol to indicate the height of the mean (or half) tide level above the Chart Datum.
In order to predict the tide by harmonic constants it is necessary to have the Admiralty Tidal Prediction Form N.P. 159. This can be purchased from any Admiralty Chart Agent and contains thirteen sets of Form A (Plotting Circle) and Form B (Tidal Curve Graph), and nine sets of Form C (Shallow Water Corrections). The instruments required are parallel rulers, dividers, a pair of compasses and, of course, the appropriate volume of Admiralty Tide Tables for the port and year in question.
A sample set of Forms A and B is supplied for working on the tidal problem given in Self Assessment 11. The necessity to use Form C (Shallow Water Corrections is comparatively rare, there being only 186 ports (out of 5,100) in Volume 3 of A.T.T. which have Shallow water corrections.
However, in the example given on the following pages the use of Form C is included so that students can see how Shallow water corrections when appropriate are applied.
Plotting a tidal curve from the harmonic constants is a somewhat complex and involved process, but it is not difficult as the detailed step-by-step instructions supplied with N.P. 159. (and reproduced with the example on the following pages) are quite easy to follow. Students are given the opportunity of testing their ability to use the Admiralty Method of Tidal Prediction on form N.P. 159. in the Self Assessment at the end of the Study and will then have the satisfaction of knowing that they can predict the height of the tide at any time and for any place in the world from the three volumes of Admiralty Tide Tables.
It must be borne in mind, however, that however accurate the data in A.T.T., and however accurately the navigator calculates and plots, the resultant tidal curve remains a theoretical prediction. Meteorological conditions which differ from the average will cause corresponding differences between the predicted and the actual tide and all tidal predictions must therefore be treated with caution, particularly when navigating with small under-keel clearances.
205. ADMIRALTY METHOD OF TIDAL PREDICTION – N.P. 159 FORMS A, B AND C.
In those places where the Shallow Water effect is known and is appreciable, the predicted tidal curve can be improved by the addition of corrections based on the quarter and sixth diurnal constituents, using Form C. In these cases, the appropriate angles f4 and f6 and the factors F4 and F6 are tabulated for the place concerned in Admiralty Tide Tables, part II. If no data for these angles and factors, Form C is not required.
On Forms A and B, all working and plotting should be done to 2 decimal places, to provide predictions correct to 1 decimal place. If the range of the tide to be predicted is either very large or very small, it may be desirable either to double or to halve the height scale of the Graph Sheet (Form B). If this is done, the same factor MUST be applied to the scale of the Plotting Circle on Form A.
On Form C, the working and plotting should be done to 3 decimal places to provide Shallow Water Corrections correct to 2 decimal places for entering on Form B. The scale of the Plotting Circle on this form is considerably greater than on Forms A and B; like the latter, it may be amended, if desired, for ease of plotting. When transferring the total Shallow Water Corrections at the foot of Form C to the Graph Sheet, the scale of Form B MUST be used.
The multiplication required on Forms A and C may be done long-hand, by slide-rule or by logarithms. A table of logarithms to a sufficient degree of accuracy is included inside the front and back covers of N.P. 159. Fig. 53-1.
The data used in the Example does not refer to any particular port.
The Admiralty Method of Tidal Prediction is also suitable for use on an electronic calculator. Details are published in the Instructions section of each volume of Admiralty Tide Tables.
206. FORM A INSTRUCTIONS & PLOTTING CIRCLE
Enter PLACE, DATE and ZONE TIME (from Admiralty Tide Tables, part II).
TABLE Row 3 From A.T.T. part II, for the place, enter g and H of M2, S2, K1 and 01 and Mean Level (Z0).
Row 4 From A.T.T. table V11, for the date, enter Tidal Angle (A) and Factor (F) for M2, S2, K1, 01.
From A.T.T., part II, enter Seasonal Correction to M.L.
Row 5 Add α°, g° and A° (subtracting 360° or 720° if necessary), to get m°, s°, k°, o°, Multiply H and F, to get M, S, K, and 0.
Add Z0 and Seasonal Correction to get A0
Construction of Plotting Circle:
From centre C, lay off length M in direction m°, to get point M2.
From point M2, lay off length S in direction s°, to get point H2.
Draw line ②——② through centre C and perpendicular to CH2
With centre C and radius CH2, describe a circle (Semi-Diurnal circle).
From centre C, lay off length K in direction k°, to get point K1
From point K1, lay off length 0 in direction o° to get point H1
Draw line ①——① through centre C and perpendicular to CH1
With centre C and radius CH1, describe a semi-circle to cut all the short (pecked) radii from 0b to 1B (Diurnal semi-circle).
Take off the angle h2 (= direction of H2 from C) and the length H2 (- CH2) and enter them in the appropriate spaces in the small panel to the right of the circle. These are required later, for use with Shallow Water Corrections (Form C), if these are appreciable.
207. FORM B – INSTRUCTIONS:
Enter PLACE, DATE and ZONE TIME (from Admiralty Tide Tables, part II).
Draw a line across the sheet at height A0.
From Form A take off, with dividers, the shortest (perpendicular) distance from the intersection of the Semi-Diurnal circle with the long (firm line) radius 06 to the line ②——②. Transfer this length to Form B, laying it off from the line A0 along the vertical marked 06 at the top, upwards (+ ve) if the intersection and H2 are on the same side of ②——② and downwards (— ve) if they are on opposite sides of ②——②
Repeat this process for each of the intersections of the S.D. circle with the long (firm line) radii 07 to I8, laying off the distances, with appropriate signs, from the line A0 along the verticals marked 07 to I8 at the top. Mark these points ⊡ if there are Shallow Water Corrections, otherwise ʘ.
At this stage, the Shallow Water Corrections (if any) should be applied. For the method of doing this, see the instructions for Form C.
After applying the Shallow Water Corrections, return to Form A. Take off, with dividers, the shortest (perpendicular) distance from the intersection of the Diurnal semi-circle with the short (pecked line) radius 06 to the line ①——① Transfer this length to Form B, laying it off from the point ʘ on the vertical marked 06 at the top, upwards (+ve) if the intersection and H1 are on the same side of ①——① and downwards (—ve) if they are on opposite sides of ①——①.
Repeat this process for each of the intersections of the D. semi-circle with the short (pecked line) radii 07 to 18, laying off the distances, with appropriate signs, from the points ʘ already plotted on the appropriate verticals 07 to 18 on Form B, marking these points +.
The result is a series of 13 points +. These points are joined by a smooth curve, which is the predicted tide curve from 0600 to 1800 on the date at the head of the form.
To obtain predictions for further periods of 12 hours on either side of these times, see instructions for extending the predictions.
208. FORM C – (SHALLOW WATER CORRECTIONS) INSTRUCTIONS:
Enter PLACE, DATE and ZONE TIME (from Admiralty Tide Tables, part II).
Transfer the values of h2 and H2 from the small panel on Form A to the appropriate spaces in the Table at the top of the form. Hence evaluate, and enter in the appropriate spaces, 2h2 and 3h2 (subtracting 360° or 720° if necessary), H22 and H23.
From A.T.T. part ll, for the place, obtain the angles f4 and f6 and factors F4 and F6 and enter them in the appropriate spaces in the Table.
Evaluate, and enter in the appropriate spaces in the Table, h4 (=f4+2h2), h6, (=f6+3h2) (subtracting 360° if necessary), H4 (=F4 x H22) and H6 (=F6 x H23). If no data is given for F6, then H6 is assumed to be zero.
On the Plotting Circle, lay off distance H4 in direction h4 from centre C to get point H4. Draw the line ④——④ through centre C and perpendicular to CH4. With centre C and radius CH4, describe the Quarter-Diurnal semi-circle to cut the continuous radii A and P to U.
Take off the shortest (perpendicular) distance from the intersection of the Q.D. semi-circle with the radius A to the line ④——④ and enter it in the Q.D. row of the 1200 column of the Table at the foot of the Form, calling it positive (+Ve) if the intersection and H4 are on the same side of ④——④ and negative (—Ve) if they are on opposite sides.
Repeat this process for the intersections of the Q.D. semi-circle with the radii P, Q, R, S, T and U, entering the values, with appropriate signs, in the 06, 07, 08, 13, 14 and 1500 columns of the Table respectively, as indicated therein.
Complete the Q.D. row of the Table by entering P, Q, R, S, T and U, with OPPOSITE signs, in the 09, 10, 11, 16, 17 and 1800 columns respectively, again as indicated therein.
Repeat step 5, with H6 and h6 replacing H4 and h4, to get point H6, the line ⑥——⑥ and the Sixth-Diurnal semi-circle cutting the continuous radius A and the pecked radii V to Z.
Take off the shortest (perpendicular) distance from the intersection of the 6-D. semi-circle with the continuous radius A to the line ⑥——⑥ and enter it, with an appropriate sign, in the 6-D. a row of the 1200 column of the Table.
Repeat this process for the intersections of the 6-D. semi-circle with the pecked radii V, W, X, Y and Z, entering the values, with appropriate signs, in the 06, 07, 11, 15 and 1600 columns of the Table respectively, as indicated therein.
Complete the 6-D. a row of the Table by entering V, W, A, X, A, Y and Z, with OPPOSITE signs, in the 08, 09, 10, 13, 14, 17 and 1800 columns respectively, again as indicated therein.
Sum, algebraically, the Q.D. and 6-D. values in each of the 13 columns and enter the results in the bottom line of the Table, to obtain the total Shallow Water Correction for each hour.
Using the scale on the Graph Sheet (Form B), apply the total S.W.C. for each hour to appropriate S.D. points □ already plotted on the Graph Sheet, applying them upwards if the correction is positive (+VE) and downwards if it is negative (-VE). Mark these points ʘ.
209. EXTENDING THE PREDICTIONS:
The preceding instructions enable the predicted tide curve from 0600 to 1800 on the date to be drawn, using the time scale at the top and bottom of the main framework of Form B. This tide curve can be simply extended for a further 12 hours in both directions, giving predictions from about 1800 on the day before to about 0600 on the day after the date at the head of the form. These extended predictions are not as accurate as those for the central 12 hours but are adequate for most purposes. Previously, the diurnal tide at, say, hour 06 has been taken from the Plotting Circle on Form A with dividers and laid off in the appropriate direction along the 06 hour vertical from the already plotted point ʘ (the combined S.D. and S.W.C. point, or, if there is no S.W.C., the SD. point). This gives the height of the whole tide at 0600 on the date. Now lay off the same length from the same point ʘ in the OPPOSITE direction along the 06-hour vertical, marking it x. This point gives the approximate height of the whole tide 12½ hours both before and after 0600 on the date – i.e. at 1730 on the day before and at 1830 on the date, as indicated on the lower time scales. Repeat this process for the hours 07 to l8; the result will be 13 points x. Join these points by a smooth curve — a pecked line to distinguish it from the continuous line curve already drawn. This pecked curve, read in conjunction with the appropriate lower time scale, extends the original continuous tide curve for a further 12 hours in both directions. If H1, the diurnal tide, is negligibly small, then the curve obtained by joining the points ʘ is the predicted tide curve from 0600 to 1800 on the date at the head of the form, using the time scale at the top and bottom of the main framework of Form B. It is also the predicted tide curve for the 12 preceding and the 12 subsequent hours, using the appropriate one of the two-time scales beneath the graph. It may sometimes be desirable to extend the continuous curve to the edges of the Graph Sheet, at 0500 and 1900 on the date (e.g. when high or low waters occur near 0600 and 1800). The height of the tide at 0500 on the date appears on the pecked curve 1½ hours from the right-hand edge of the Graph Sheet; this height is transferred to the 05-hour vertical on the left-hand edge, to give another point on the continuous curve. Similarly, a further point can be obtained at 1900 on the right-hand edge, enabling the continuous curve to be drawn from 0500 to 1900 on the date.