The Tidal Analyzer, (Kelvin, opposite p. 304).

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
*H*cos(*vt* + *p*)
will be rewritten using a standard trigonometric identity
as *A*cos*vt* + *B*sin*vt*
(with *A* = *H*cos*p* and *B* = -*H*sin*p*).

There are three facts about sine and cosine finctions that make Fourier analysis work.

**First fact**. In the long run, the average value of
any function of the form sin(*vt*) or cos(*vt*) must be zero. This
is clear from looking at the graphs of these functions: each
positive contribution to the average is exactly cancelled by a
negative one.

**Second fact**. For *different*
speeds *v* and *w* the average value of the product

cos(

sin(

**Third fact**. The average value of the
products

sin(

goes to exactly 1/2 if the averages are taken over longer and longer time intervals. First of all, in each case the two factors are always in phase, in fact equal, so their product is always either the square of a positive number or the square of a negative number, or zero, but in any case never negative, so there can be no cancellation. Why is the average exactly 1/2? Since the graphs of the sine function and the cosine function are so similar, we can expect that in the long run sine-squared and cosine-squared would have

sine-squared + cosine-squared = 1

which holds everywhere must hold for the averages as well. Since the two averages are equal, and add up to one, they must each equal 1/2. Click here for a more standard formulation, where the averages are defined as integrals.

Suppose, to take a simple example, that the tidal curve from an ideal port was exactly

*A*(*t*) = *A*_{0} + *A*_{1} cos(*vt*) +
*B*_{1} sin(*vt*) + *A*_{2} cos(*wt*) +
*B*_{2} sin(*wt*),

where we know the ``speeds'' *v* and *w*, and we want to
calculate the coefficients *A*_{0}, *A*_{1}, *A*_{2},
*B*_{0}, *B*_{1}, *B*_{2}.

** A_{0}**. Since by ``Fact one'' each of the sine and cosine terms
averages out to zero in the long run, the long-term average
of the tidal height function must be exactly

** A_{1}**. We multiply

So multiplying *A*(*t*) by cos(*vt*)
and averaging over a long time
period produces 1/2 the coefficient *A*_{1}.

** B_{1}**. The same phenomenon happens when we multiply

** A_{2}, B_{2}** are obtained in the same way: multiply
by cos(

What we just did for five terms works just as well for fifty:
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

The Harmonic Analyzer
(Green & Lloyd, p. 46).

is in the Kelvin Museum in Glascow.

Back to Main Tide page.

Back to Tony's Home Page.

Math Dept, SUNY

Stony Brook NY

October 12 1998