**Fourier Analysis of Ocean Tides II** **Feature Column Archive**

## 2. Almost orthogonality

We 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 non-periodic one.

**Sample problem.**

Let us start with a simple non-periodic 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 long-term average value of
`cos(vt)`, for any non-zero `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 long-term 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 long-term average value of
`sin`^{2}(vt) is `1/2`, for any non-zero speed `v`.

The graphs of `sin`^{2}(t) (blue) and of the average value of the sine-squared function from `0` to `t` (yellow).

**The solution.**

**To calculate **`B`_{1}: multiply `f(t)` by `sin(t)` and compute twice the long-term 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 long-term avarage value zero. **To calculate **`B`_{2}: multiply `f(t)` by `sin(2`^{1/2}t) and compute twice the long-term average value of the product. This number must be `B`_{2}.

The following graph shows how the multiplications and long-term 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.

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