Tides...

Tidal movements

The tide is the vertical rise and fall of the sea level surface caused primarily by the change in gravitational attraction of the moon, and to a lesser extent the sun.
As the earth spins on its axis the centrifugal force results in slightly deeper water near the equator as opposed to shallower water at the poles. In fact it causes a flow from the poles to the equator.
The earth is also in orbit around the sun (one revolution in one year) creating not only another centrifugal force but also a gravitational interaction. These two yield a bulge on the night site (centrifugal) and a bulge on the day site (gravitational) both of them moving as the world turns. Therefore, a certain place on this world will experience two high and two low tides each day.
With these forces alone, we would not have spring tides and neap tides. Spring tides have higher high tides and lower low tides whereas neap tides have lower high tides and higher low tides. Hence, the range (difference in water level between high and low tide) is much larger in a spring tide than in a low tide.

This animation shows how the tide changes during the lunar cycle. When the sun, moon and earth are aligned : spring tide.
When at right angles the forces are not aligned:
neap tide.
The time between spring and neap is approximately 7 days.

These differences in range can be explained if we include the moon into our earth-sun system. The moon and the earth orbit each other around a point (called the barycenter or baricenter) 2000 odd kilometres inside the earth, creating a centrifugal and a gravitational bulge. Moreover, despite the sun's immensely larger mass, the moon exerts a 2.25 times larger gravitational attraction, since the moon is much closer to our earth.
It is the combined effect of the sun and moon that creates spring and neap tides. In the animation the gravitational forces of both the sun and the moon are taken into account. When aligned with the earth they combine their attraction and otherwise they counteract their attraction. The sun is located in the corner right below, far outside this picture (note the eclipse) while the moon is revolving round the earth. One full circle corresponds to one lunar cycle (about 28 days).

The figure below shows the ideal sinusoids of both spring and neap tides. Vertically the water height is shown versus horizontally the time. Ideally, the time between a low and a successive high is somewhat more than 6 hours.
The sinoidal curve of the Tide
The time difference between spring tide and neap tide is normally 7 days and is in accordance with the phases of the moon. Yet, water has mass and therefore momentum. Moreover, it is a viscous fluid that generates friction if moved. Therefore, the actual spring tide lags a day or so behind a full moon or new moon occurrence.

So, tidal movements are intrinsically periodical, resulting in a Tidal day of 24 hours and 50 minutes containing one tidal cycle, namely two highs and two lows. This basic pattern may be distorted by the effects of landmasses, constrained waterways, friction, the Coriolis effect, or other factors. Hence, predictions are possible and we expect the the next day's high tide to come about 50 minutes later.
However, a closer look at the orbit of the moon reveals that the moon is not always in the equatorial plane, resulting in three types of tides:

	Semi-diurnal tide: Featuring two highs and two lows each day, with minimal variation in the height of successive high or low waters. This type is more likely to occur when the moon is over the equator.
	Diurnal tide: Only a single high and a single low during each tidal day; successive high and low waters do not vary by a great deal. This tends to occur in certain areas when the moon is at its furthest from the equator.
	Mixed tide: Characterized by wide variations in heights of successive high and low waters, and by longer tidal cycles than those of the semi-diurnal cycle. These tides also tend to occur as the moon moves furthest north or south of the equator.

Chart Datums

The depths and heights in the chart need a plane of reference: the Chart Datum (see interactive figure below). Depths are usually described with respect to low water reference planes (yielding lower charted depths, which are safer) and heights are shown with respect to high water reference planes (again, yielding lower vertical clearances on the chart, which are safer). As such, the chance that the observed depth or vertical clearance beneath a bridge is smaller than the charted depth or height is rather small.



In this example the Charted Depths are related to LAT. The Observed Depth or Drying Height is a combination of Tidal Height & Charted Depth.	This example shows the various spring and neap tides around mean water level. Note that spring low water is the lowest. Both ranges are indicated.	In this example the light elevation is reduced to high water. Also a clearance under a bridge is charted in that way. The 'height' refers to the building itself. On land yet another CD can be in use.

Some Chart Datums and their abbreviations:

	MHWS : Mean High Water Spring
	HW : High Water
	MHWN : Mean High Water Neap
	ML : Mean Level
	MLWN : Mean Low Water Neap
	MLWS : Mean Low Water Spring
	LAT : Low Astronomical Tide

Tide Predictions...

1 - Information from the chart

Most often the chart presents succinct tide tables for certain positions. These positions are marked with the 'square'. The table below shows us an example for two different positions. The first refers to Cowes (UK), the second to a position south of Cowes.

Position	Heights above LAT
	Mean HW		Mean LW
	Spring	Neap	Spring	Neap
Cowes	1.7 m	1.5 m	0.2 m	0.4 m
	5.2 m	4.3 m	0.4 m	1.2 m

This data only provides us with average high and low waters heights. Moreover, it is merely valid at spring or neap tides. To use it we need to first find out how many hours we are from high water. Secondly, we need to know if it is spring or neap or sometime in between at that particular moment. We shall use this table to solve two types of problems. Finding height of tide at a particular location at a particular time:

	To get over a shoal.
	To pass under a bridge.

Almanacs and many other nautical publications contain predictions of the times of high and low tides at many major standard ports. Also listed are differences in times of tides from these ports for additional secondary ports. To work with this succinct data we need two extra tools:

To interpolate between high and low water heights we use the Rule of Twelve. We assume the tidal curve to be a perfect sinusoid with a period of 12 hours. The height changes over the full range in the six hours between HW and LW.

	During first hour after HW the water drops 1/12^th of the full range.
	During the second hour an additional 2/12^th.
	During the third hour an additional 3/12^th.
	During the fourth hour an additional 3/12^th.
	During the fifth hour an additional 2/12^th.
	During the sixth hour an additional 1/12^th.

Hence, two hours after the HW the water has fallen 3/12 of the full range.

To interpolate between spring and neap tides we use the Rule of Seven. Since the change from spring range to neap range can be assumed linear (instead of sinusoid), each day the range changes with 1/7th of difference between the spring and neap ranges.
Hence, the daily change in range is (spring range - neap range)/7.

Shoal problem:
Our shoal near Cowes has a charted depth of 1 meter and we would like to cross it at about 15:00 hours with our yacht (draft 1,5 m).

From any nautical almanac we find that HW occurs at 03:18 15:53 and LW occurs at 09:45 22:03at a standard port nearby. We also find that at our location HW occurs one hour later and that spring tide is due in two days. Hence, we have a HW around 17:00.

	Via the rule of seven we find out that today the range is: spring range - 2 x ( (spring range - neap range)/7 ) <=> 4,8 - 2 x ( ( 4,8 - 3,1)/7 ) <=> 4,8 - 2 x 0,25 = 4,3 m.
	We also need today's HW height: which is Spring HW - 2 days x ( (5,2 -4,3)/7 ) = 5,0 m.
	Via the rule of twelve we find out that at two hours before high water the height is: 5,0 - 3/12 x 4,3 = height at 15:00 hours = 3,9 m.

So, after three interpolations we derive the water height at 1500 hours. Considering the charted depth leads to an observed depth of 4,9 meters, enough for our draft of 1,5 meters.

Bridge problem:
An overhanging rock, power lines or bridges have their clearances charted with respect to another chart datum than LAT. Normally, 'high water' or 'MHW spring' are used as reference planes.

An example:
Above our shoal hangs the 'Cowes bridge'. At 15:00 hours we would like to pass this bridge, which has a charted height of 20 meters to HW. Our mast is 23 meters high. In the example above we found that the water height was 1,1 meters below HW level at that time. Obviously, we will have to wait!
So, at what time will we be able to pass under this bridge?
The water height must be 3 meters lower than HW level (5,0 m). That is almost 9/12 of the range (4,3 m) indicating four hours after HW. Conclusion, we will have to wait at least six hours in total.

2 - Information from tide tables

Instead of mere averages, a tide table Detailed Tide Table. provides us each day with the times of high and low water for a particular place. Basically, it is same table like the one we found in the chart, but is extended for every day in a year. By using this method we get more accurate water heights since it involves less interpolation. The example shows us a part of a very detailed tide table, which even includes heights for every hour.

3 - Information from tidal curves

In most tables the tides can also be characterized by a tidal curve. This method substitutes the rule of twelve providing more accurate heights. The left side contains the water height information with the lowest heights to the left where also the chart datum is indicated. The low water height will be marked at the bottom and the high water height will be marked at the top.
Tidal Curve.
The area under the curve will be marked with the time information.
To find the water height at a specific time we need to know first how many hours before or after the HW this is. Then

Tidal Curve:
Finding Heights.

Tidal Curve:
Finding Time with Height.
Often this is done when the curve is not sinusoid and the rule of twelve is rendered useless.