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Tidal calculations

The attraction of the sun and moon creates the tides. It is often thought that the tides are created by just the attraction of the moon pulling the water towards it. But if the tides arise only because of that, then there would be a tidal period only once every 24 hours. To explain the second tidal period, earth and moon must be seen as one system revolving around a common centre. That centre is in the earth, though not at the centre, but about 1,500 kilometres below the earth’s crust. The centrifugal force of the earth-moon system is greatest on the opposite sides of both celestial bodies. So at the “back” of the earth, as seen from the moon comes a second tidal wave. In fact, it is better to also consider the tidal wave on the side facing the moon in this way: in the orbiting earth-moon system, both celestial bodies are somewhat stretched into an ellipse. As the earth rotates in 24 hours, each point passes through a tide mountain twice a day and an ebb valley twice. So it is high water twice a day and low water twice a day.

threshold for the port

Spring tide and neap tide

When the earth, moon and sun are aligned, high tide is extremely high and, as a result, low water is extremely low. Thus, the range (difference between high and low water) is then extremely large. As a result of the large range, the tidal currents also reach high speeds. This situation is called spring tide.
When the sun, moon and earth are not in a line, high tide is less high and low tide less low. So the range is smaller and the tidal currents reach lower speeds. This situation is called neap tide.

spring tide and neap tide

You can tell from the moon whether it is spring tide or neap tide. In The Netherlands it is 2 days after full and 2 days after new moon (the moon cannot be seen) spring tide. 2 days after first quarter (the right side of the moon is visible) and 2 days after last quarter (the left side of the moon is visible) is neap tide. The month orbits the earth in 1 month and so we get to experience all four phases of the moon every month. So every month it is neaps twice and spring twice. Spring tide and neap tide occur only two days after the moon phase because the tide (which originates on the ocean) takes two days to reach the North Sea.

Before doing the calculations, always draw a timeline with dates, moon phases and neap tide or spring tide.

spring tide and neap tide

Definitions

Horizontal tidal movements are the currents and vertical tidal movements involve rising and falling tides.
The chart datum is used to define the depths, for example, LLWS, LAT, etc.
Height of Tide is the height of the water above the chart datum.
Depths above the chart datum are shown underlined on the chart.
The flood tide is the current that causes high water.
The ebb tide is the current that causes low water.
On for example the Waddenzee we often have to deal with shallow places where actually two tidal currents “meet”. Because theoretically there never is a strong current in those places, it is also shallower. Often, we can only sail across these shallow places in a certain time window.

Where is the range the greatest?

The greatest range occurs in “estuaries” that look like a funnel, such as The Channel between Dover and Calais or the Bristol Channel. At high tide, the “wave” that causes the high water “sloshes” into such a funnel and, of course, as the funnel gets narrower and narrower, the water rises enormously fast, because the mass of water has nowhere else to go. In the Netherlands, this is why there is a big range on the Westerschelde. There, the range is greater than in Scheveningen, where there is hardly any funnel. Hardly indeed, as the North Sea can also be seen as a funnel towards the narrow Channel. In the south of the Netherlands, the distance to England is much smaller than in the north. Therefore, in the south the range is greater than in the North. In Belgium, the range is even greater again than in the Netherlands.

Tidal current Diamonds

There are tidal current diamonds in the chart with a letter in them corresponding to the letters in the current table. To determine the current direction and current rate, we need a tide table to determine the time of HW on the day in question for the port used as a standard for the current table (Hoek van Holland below). Suppose we sail near diamond A at noon and it is high tide at Hoek Van Holland at 10.15 that day and it is spring tide, we look in the table column at wiebertje A, at 2 hours after high tide, in the left column (spring tide). The current at A at noon is 26 degrees and 2.7 knots.

stream wobblers

That current in the direction of 26 degrees and at the speed of 2.7 knots runs from 11.45 to 12.45. See the table below, which also reserves a column for the calculated current velocity, between spring tide and neap tide.

Time+/-fromtodirectionSpring tideNeapsCalculated
speed
04.15-603.4504.45203º3,82,5 
05.15-504.4505.45205º4,02,5 
06.15-405.4506.45204º3,11,8 
07.15-306.4507.45195º1,30,7 
08.15-207.4508.45043º1,10,5 
09.15-108.4509.45030º3,11,6 
10.15HW09.4510.45027º4,02,3 
11.15+110.4511.45026º3,82,2 
12.15+211.4512.45026º2,71,6 
13.15+312.4513.45024º1,30,9 
14.15+413.4514.45197º0,90,2 
15.15+514.4515.45201º2,91,3 
16.15+615.4516.45203º4,02,2 

It is often given in the exams that, for example, you are located at Waypoint A about 11.45 and are going to sail a trip to Waypoint B (it is spring tide) and the trip will take about an hour. So in that case we can use the current of HW +2 with the direction of 26 degrees and with the speed of 2.7 knots, because it runs from 11.45 to 12.45. So if we leave at 11.45, that works out just right.

You can also sketch a timeline to determine how many hours before or after HW you are:

10.15 11.15 12.15
-6 -5 -4 -3 -2 -1 HW +1 +2 +3 +4 +5 +6

Tidal Squares

The map also contains squares with a letter in them. These squares correspond to the table with vertical tide data. They show the height of tide at HWS (high water spring tide), HWN (high water neap tide), LWS (low water spring tide) and LWD (low water neap tide), so not for days between NT and ST. We can use the 1/7th rule to calculate the height of tide at HW or LW for days between ST and DT. First calculate the difference in Height of tide at HWS and HWD. The rise at HWD + (the number of days after neap tide / 7 ) x the difference between HWN and HWS, is the value for the day you are looking for.

tide squares

HP33

In books such as the HP33, it is possible to read the height of tide per day and even per hour. The chart shows only the maximum and minimum values at spring and neap tides.

HP33

Difference 1/7 rule and 1/12 rule

The 7 of the 1/7th rule comes from the 7 days between spring and neap tides. We use it to calculate the rise (or current rate) when we only have the data from spring tide and neap tide, but not from the days in between. This is the case, for example, for the squares in the map showing the rise at HWS, HWN, LWN, LWS. This is also the case with the current rate at the arrows. So suppose we still want to know the height of tide or current rate on a day between spring and neap tides? Then we can interpolate linearly using the 1/7th rule.

There are 12 hours between HW and the next HW. We use this 1/12th rule of thumb if we only have the data of HW and LW for that day, but we want to calculate the rise at a time in between HW and LW. With the HP33, we do have the hourly data, so then we don’t need the 1/12th rule. If we have a tidal curve, we use the graphical method, because it is more accurate than the 1/12th (fist) rule, which assumes an average.

If we would not have a tide table of a particular area, but we have to make do with the squares in the nautical chart (averaging HWS, HWN, LWN and LWS) and we know how many days before or after spring tide we are in, then we first have to use the 1/7th rule to calculate what the rise is at HW and LW on the day in question, before we can start applying the 1/12th rule.

1/7th line

In the table with squares in the sea chart, the height of tide is given for spring tide and for neap tide. But if we want to know the rise on a day between spring and neap tides, we will have to calculate it using the 1/7 rule. The 1/7 formula can be used to calculate the following:

  1. Height of tide at high water (HW) between high spring tide (HWST) and high water neap tide (HWNT)
  2. Height of tide at Low water (LW) between low water spring tide (LWST) and low water neaps (LWNT)
  3. Current rate on a day between spring tide (ST) and neap tide (NT).

Height of tide at High Water:

A.) Height of tide at HW = HWNT + (the number of days to NT / 7) x the difference between HWST and HWNT

B.) Height of tide at HW = HWST – (the number of days to ST / 7) x the difference between HWST and HWNT

Height of tide at Low Water:

A.) Height of tide at LW = LWST + (the number of days to ST / 7) x the difference between LWST and LWNT

B.) Height of tider at LW = LWNT – (the number of days to NT / 7) x the difference between LWST and LWNT

Current rate:

In the current table, the current rate is given for spring tides and neap tides and therefore not for the intermediate days. We can also calculate these using the 1/7th rule:

A.) Current rate = Current at neap tide + (the number of days to NT / 7) x the difference in current velocity between ST and DT

B.) Current rate = Current at ST – (the number of days to ST / 7) x the difference in current rate between ST and NT

1/12

The water rises from low tide to high tide in 6 hours. In the first hour the water rises 1/12 of the total range, in the second hour 2/12, in the third hour 3/12, in the fourth hour 3/12, in the fifth hour 2/12 and in the last hour 1/12. So in total, the water rises in 6 hours 12/12 x range, or 100%. The map shows squares containing a letter corresponding to the data in the tide table. This table only gives the rise for high water and low water (but not for intermediate hours). If we still want to calculate the rise of an intermediate hour, we calculate it as follows:

  1. First calculate the range, which is the difference between high and low water on 1 day.
  2. Then calculate how many hours after/before HW or LW we are.
  3. Multiply the number of twelfths applicable by the range.
  4. Calculate:

High water – (../12 x gradient) = the height of tide

Or

Low water + (../12 x decay) = the height of tide

Exact interpolation method / Graphical method

Using the exact interpolation method, the rise can be read in a tidal curve. For example: What is the height of tide on 19 April (is printed in red so spring tide) at 14.45 UT?
In the Almanac, we read:

HW Dover (6.6 m) at 11.41 UT
LW Dover (0.8 m) at 19.10 UT

1. 14.45 ut lies between HW Dover at 11.41 UT and LW Dover 19.10 UT.
2. Enter the relevant times after HW on the bottom axis of the tidal curve.
3. Draw a line from HW 6.6 to LW 0.8 to the left of the curve.
4. Draw a vertical line from 14.45 to the spring tide curve.
5. Draw a horizontal line to the diagonal to the left of the curve.
6. Draw a vertical line upwards and read rise: 4.3 metres.

exact interpolation method

Note that the spring tide graph is always below the neap tide graph.

Sailing over a sill with the Reeds

The exact interpolation method, or graphical method, is ideally suited for calculating the time window in which you can sail over a sill. In many harbours in areas with large ranges, such as Brittany, there is a sill in front of the harbour entrance. That sill ensures that the harbour does not run dry at low tide. In that case, you actually calculate in the reverse order of the above example.

  1. First, calculate the minimum amount of heaving required taking into account the chart depth (which may also be above LAT in the case of a sill), the ship’s draught and the minimum under keel clearance (UKC).
  2. Draw in 2 diagonal lines in the tidal curve. 1 diagonal for the period between low and next high water and 1 for the period between high and the next low water. Those 2 low waters may in fact differ.
  3. Then draw a vertical line from the horizontal axis from the minimum required rise to the 1st diagonal and read from which time you can sail over the threshold.
  4. Then draw a vertical line from the horizontal axis from the minimum required rise to the 2nd diagonal and read off to what time you can sail over the threshold.
  5. So you have found the time window in which you can cross the threshold. An X number of hours before HW to an X number of hours after HW.

Sailing across a shallow bank with the HP33

Using the HP33, we can also calculate in what time frame we can sail over a shallow sand bank. Imagine we want to sail across a shallow part flatbottomboat with a draft of 60cm and we want to maintain an under keel clearance of 40cm. The shallowest point is 40cm above LAT, see the underlined value 04 on the map on the green coloured part. So that means we need 60cm + 40cm + 40cm = 1.40cm rise, or 14dm.

Imagine that day is 1 March 2011. So then you can cross Zuidoostrak in the time window from 05.00 to 09.00 and again from 17.00 to 23.00.

hp33

Navionics

Nowadays, navigation Apps such as Navionics are increasingly used. This also allows us to quickly read the time window in which we can sail over the mooring tide. We use the same boat data as in the previous issue. Draught of 60cm. Under keel clearance of 40cm. So 14dm height of tide is needed. Furthermore, in this issue, it is 21 June 2018.

What you can do in the Navionics App is to use your fingers to put the cross on the nearest “tide station”, which in this example is Kornwerderzand. Then the tide graph with timeline will appear below the map. You can slide that timeline so that the rise indicates at least 1.4m. So that is from 02.01h see below.

And that then lasts until 05.19, see below. After that, the rise is not enough to sail across the tidal flats.

Case running aground

Suppose the tide table shows the following at Flushing on a certain day:

10.00 4,8m
16.10 0,3m
22.20 5,2m

You run aground at noon with a ship 2 metres deep the chart depth is unknown.

1. Calculate the chart depth.
2. Calculate what time you will be floating again.

Answer case

question 1

Chart depth + height of tide = 2 metres

The rise at 12.00 is 3.6m, see chart below.

Chart depth + 3.6 = 2 metres

Chart depth is thus -1.6 metres (dry above the chart datum, that is)

The range is 4.8 – 0.3 = 4.5. In the rectangle at the top right of the graph below, we see that a decay of 4.5 occurs at spring tide.
So we use the continuous spring tide graph.

task dry and loose

Elaboration of question 2

The height of tide at 22.20 is 5.2m so hence we need to draw a second diagonal line. As shown below, we float again 1 hour and 50 minutes before the high tide of 22.20. So at 20.30.

task dry and loose

Questions & Answers

Question 1: Where will it be high tide sooner?

a: In IJmuiden
b: In Scheveningen
c: On Texel

Question 2: Where is the greatest range?

a: in Vlissingen
b: in Scheveningen
c: in Texel

Question 3: What do we calculate with the 1/7th rule?

a: Height of tide between high and low tide
b: Current
c: Height of tide between spring and neap tide


Question 4: What do the diamonds in the chart indicate?

a: Rise
b: Current
c: Range

Question 5: The 1/12th rule is intended to calculate the height of tide:

a: Between high and low tide.
b: At high or low tide.
c: Between spring tide and neap tide.

Question 6: Dutch summer time corresponds to:

a: UTC
b: UTC+1
c: UTC+2

Question 7: The exact interpolation method uses:

a: Linear interpolation
b: the 1/12th line
c: the tidal curve

Question 8: What is height of tide?

a: The water depth
b: The difference between high and low tide
c: The distance between chart depth and the water level

Question 9: What is the range?

a: The difference between high and low tide
b: The distance between chart depth and the water level
c: The water depth

Question 10: When is it spring tide?

a: After full and half moon
b: After first and last quarter
c: After Full and New Moon