Fourier Analysis of Ocean Tides II Feature Column Archive
2. Almost orthogonalityWe work under the hypothesis that the tidal record at any port, written as a function of time, is a combination of sines and cosines of the form A_{0} + A_{1}cos(v_{1}t) + B_{1}sin(v_{1}t) + A_{2}cos(v_{2}t) + B_{2}sin(v_{2}t) + ... where the ``speeds'' v_{1}, v_{2}, ... are part of a finite collection of possible speeds, determined globally by astronomical considerations. Since we can expect no rational relation between the astronomical constants, the tidal record will not be periodic. But this special structure will allow us to determine the coefficients A_{0}, A_{1}, B_{1}, A_{2}, B_{2}, ... directly from the tidal record, by adapting the Fourier analysis of periodic functions to this nonperiodic one. Sample problem. Let us start with a simple nonperiodic function like f(t) = 2sin(t)  3sin(2^{1/2}t). The graph of f(t) = 2sin(t)  3sin(2^{1/2}t). Because 2^{1/2} is not rational, this function can never repeat itself exactly. Suppose that we know that the function is of the form f(t) = B_{1}sin(t) + B_{2}sin(2^{1/2}t), and are given the curve. How to calculate the coefficients B_{1} and B_{2}? Tools from trigonometry.  any product of sines can be rewritten as a difference of cosines. Specifically,
sin(v_{1}t)sin(v_{2}t) = .5cos([v_{1}v_{2}]t)  .5cos([v_{1}+v_{2}]t).  the longterm average value of cos(vt), for any nonzero v, is zero.
The graphs of cos(t) (blue) and of the average value of the cosine function from 0 to t (yellow).  It follows that the longterm average of sin(v_{1}t)sin(v_{2}t) is zero unless v_{1} = v_{2} or v_{1} = v_{2}. We can mimic the language of ordinary Fourier series and say that under these conditions the two functions sin(v_{1}t) and sin(v_{2}t) are almost orthogonal.
 the longterm average value of sin^{2}(vt) is 1/2, for any nonzero speed v.
The graphs of sin^{2}(t) (blue) and of the average value of the sinesquared function from 0 to t (yellow). The solution. To calculate B_{1}: multiply f(t) by sin(t) and compute twice the longterm average value of the product. This number must be B_{1}. Why? The product is f(t)sin(t) = B_{1}sin^{2}(vt) + B_{2}sin(t)sin(2^{1/2}t). The average value is the sum of the average values of the two terms. The first has average value B_{1}/2 while the second has longterm avarage value zero. To calculate B_{2}: multiply f(t) by sin(2^{1/2}t) and compute twice the longterm average value of the product. This number must be B_{2}. The following graph shows how the multiplications and longterm averaging tease out the coefficients B_{1} = 2 and B_{2} = 3. The graph of f(t) is plotted in blue, the running average of 2f(t)sin(t) is plotted in red, and the running average of 2f(t)sin(2^{1/2}t) is plotted in green.

Welcome to the
Feature Column!
These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Read more . . .
Feature Column at a glance
