Tidal Theory
The Mathematics of Sea Tides, or how to calculate the sea level for any time of the day.
In science, we have a very famous phenomenon which we call Simple Harmonic Motion. It is a simple oscillatory movement of a point, up and down, or from side to side, in such a way that the point moves at its slowest at the ends of the oscillation, and at its fastest in the middle of the oscillation.
Imagine you're standing at the end of your street, and way off at the other end there's a child running round a lamp post, in a perfect circle, with a lantern in his hand. The child is so far away, that you have no general perception of when the lantern is at its nearest to you and when its at its farthest... it just seems to move, or oscillate, from side to side as the child runs round. Further, the more perceptive observer will notice that the lantern seems to move more slowly near the ends of the oscillation, as it is actually moving towards or away from you, and more quickly in the middle of the oscillation, as it is traveling at 90 degrees to your line of sight. This movement of the lantern is a perfect example of Simple Harmonic Motion.
Now, if we take this picture, and turn it up on its end, so that the lantern is now performing the same motion but in the vertical, up and down instead of from side to side, then we have a perfect model of how the sea behaves, as its level rises and falls vertically twice a day with the tides.
Now lets fit a mathematical equation around this. It might be a bit easier if we go back to the child with his lantern. Well first, if we say that the child runs round the lamp post in a time T, then T is what we call the period of the oscillation. And if the child turns through so many degrees of angle in a second, as he goes in his circular path round the lamp post, then we say that his angular velocity is w. Actually, w is measured in radians per second, not in degrees per second. (1 radian = 57.3 degrees).
Now there are actually 2 equations we can use, one if we want to imagine the oscillation starting at one end, and one if we want to imagine the oscillation starting in the middle.
If we want to imagine the oscillation starting at one end, then
x = A cos (wt)
and if we want to imagine the oscillation starting in the middle, then
x = A sin (wt)
where in both cases, x is the distance of the lantern from the MIDDLE of the oscillation. A is called the amplitude of the oscillation. This is a constant value; it is the furthest the lantern gets from the MIDDLE before it turns round and moves back again. In other words, A is the radius of the circle which the child is running in. t is simply the time, as measured on your stopwatch, since the last oscillation started.
Now in reality we might have a hard time finding out the value of w, and so we use instead a small substitution:
w= 2p/T
where p is the famous pi (=3.14 approx) and T is again the period of the oscillation, the time taken for the lantern to go from any point you like, back to that point again.
And so, if we want to imagine the oscillation as starting at one end, then
x = A cos (2pt/T),
and if we want to imagine the oscillation starting in the middle, then
x = A sin (2pt/T)
Again, with both equations, A= the amplitude of the oscillation (= the radius of the child's running circle), t = the time as measured on your stopwatch since the latest oscillation began, T is the fixed amount of time a complete oscillation takes (which also = the time taken for the child to run a complete circle), and x is the distance of the lantern from the MIDDLE of the oscillation, when the time is t on your stopwatch.
Now, in the case of the tide at the Wellstream, on 12 Oct 1216, we have a situation where the waggoners woke up, to find the estuary at high water. So here we have an oscillationwhich starts at the top, then moves vertically downwards as the tide goes out. so we use
x = A cos (2pt/T)
Translating the terms in this equation into the realistic features of that tide, we see that x is the height of the water above a pre-determined baseline (this baseline is actually 2.56 metres above Crabb Hole on the estuary map) at a time t on your clock, ( your clock started ticking at high water), A is the maximum height of the water, in metres, above the same pre-determined baseline for that particular tide, T is the time from this high water to the next one, and good old p= 3.142 as usual.
The actual values for that tide were as follows:
A was 3.33 metres. High water (clock starts ticking) was at 05.10 GMT. T, the period of the oscillation, was 12 hours 40 minutes.
Now you have all you need!!! Good luck working it all out!! But if you get stuck, then the detailed values for the height of the water as the morning wore on can be seen here.
--- Michael Alan Marshall