## (This version uses integration)

Once the working hypothesis is established, that the astronomical tidal function for any given port is a sum of a certain number of constituents whose frequencies are known a priori then the amplitudes and phases of the constituents may be determined by Fourier analysis.

To put the sum in more standard form, a constituent
Hcos(vt + phi) will be rewritten as Acosvt + Bsinvt
(with A = Hcos(phi) and B = -Hsin(phi) as usual).

The fundamental trigonometric identities that make Fourier analysis work imply that for different speeds v and w

```         /T
|
(1/T)|  cos(vt) cos(wt) dt  ---> 0
|
/0
```
as T --> infty

and similarly for the products cos(vt) sin(wt), cos(vt) sin(vt), and sin(vt) sin(wt), whereas

```         /T
|
(1/T)|  cos(vt) cos(vt) dt  ---> 1/2
|
/0
```
as T --> infty

and the same for

```         /T
|
(1/T)|  sin(vt) sin(vt) dt
|
/0
```
So if R(t) is the record from a tide gauge at the port in question, the cosine and sine amplitudes A and B for a particular speed v may be determined by

```         /T
|
A = (2/T)| R(t) cos(vt) dt
|
/0

/T
|
B = (2/T)| R(t) sin(vt) dt
|
/0
```
for sufficiently large T.

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