Modeling Tidal Streams Using a Sine Function

I would like some help on the calculation of tidal streams using tidal diamonds relative to a standard port. Without going into detail, the project concerns the calculation of slack water at various places around the UK coast. Complex tidal streams are not being considered, so the first important simplification is that only those places where the tidal streams approximate a sinusoidal curve are considered.

A typical tidal diamond will display tidal streams relative to HW at a standard port for the period +/- 6hrs. A further simplification (for the purposes of understanding the problem) is that we need only consider the Time offsets relative to High Water at which the tidal stream is zero - thus yielding a single moment in time rather than a period of time at which the water velocity can be deemed 'Slack'

If I take a typical tidal diamond 'D' on chart 121, the source data is

Delta T Speed
(hours)
-6 0.4
-5 -0.5
-4 -1.0
-3 -1.4
-2 -1.3
-1 -1.1
0 -0.6
1 0.4
2 0.9
3 1.4
4 1.4
5 1.1
6 0.6

I can model this data, using a generalized sine function:

f(x) = A*SIN(F*(x-P))+K
A= amplitude
F= frequency, more correctly the Period
P= Phase
K= constant to adjust the overall displacement above or below a mean speed of 0

Using regression I get the following:
f(x)=1.424 * SIN(0.507 * (x - 0.61)) + 0.01

The resulting curve is a good match to the raw data.

The zero-crossing points can be found by inspection (OK, I know the 1st derivative should do this - but I don't know how to do the calculus). They appear to be:

HW - 5.5 (-5:30)
HW + 0.6 (+0:36)
HW + 6.75 (+6:45) <--- this point found by extrapolation

The next stage is to refer these offsets to a particular tide on a particular day. Taking the reference day as 24/05/2005, High Water at Immingham (the standard port for tidal diamonds on chart 121) is as follows:

24/05/2005 07:00 (corrected to BST)
24/05/2005 19:18 (corrected to BST)

Taking the zero cross-over points from above, the following Slack Water times are (ignoring the extrapolated point):

High Water Relative Offset
Immingham -5:30 +0:36
-------------------------------
07:00 01:30 07:36
19:18 13:48 19:54

So far, so good. The values appear reasonable.

The final stage is to produce a graph of speeds centred on midday in the interval +/- 6 hrs. This will effectively shift the curve along the time axis so I can see the zero-crossing points for actual daylight hours. Here is the data:

Time Speed
06:00 -0.60
07:00 -0.05
08:00 0.50
09:00 0.98
10:00 1.31
11:00 1.43
12:00 1.33
13:00 1.03
14:00 0.56
15:00 0.01
16:00 -0.54
17:00 -1.01
18:00 -1.32

The speed is derived using the following function (in Excel):

f(x)=1.424 * SIN(0.507 * 24 * (x - 0.61)) + 0.01

The additional multiplicand (*24) was introduced to convert the times in hours for Excel. The data, when plotted, shows zero-crossing points at 07:00 and 15:00 - neither of which correspond to the known values of 07:36 and 13:48.

I think the problem is the scaling factor "24". Multiplying by 24 will clearly alter the period of the resulting sine curve. Excel treats 06:00 hrs as 0.25, hence the scaling factor of 24 was used to correct this.
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I think there may be 2 problems. The first involves time-shifting the curve to centre it on midday; the 2nd problem is with Excel and how to use it to compute times.

The third issue is the sine function used. There are two forms that I know of:
f(x) = A*SIN(F*(x - P))+K
f(x) = A*SIN(F*x - P)+K
I don't understand the significance of including the phase in the multiplication by period in the first, but not in the second and whether one or the other is the more appropriate here.

The problem is: "How do I create a set of values, centred on 12:00 hrs, using the sine function derived from the supplied raw data?" I have tried to show the steps involved and the assumptions I have made. This has produced a lengthy question, but hopefully it is clear.