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 | /0as 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 | /0as T --> infty
and the same for
/T | (1/T)| sin(vt) sin(vt) dt | /0So 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 | /0for sufficiently large T.
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