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An extremely complicated wave may be the product of many component waves. How many? An infinite number, in theory. How, then, does the analyst know when to stop analyzing? The answer suggests another powerful use of Fourier theorem. The analyst synthesizes the components, step by step--puts them back together to make a compound wave. And when the synthesized wave matches the original profile, the analyst knows it's time to quit adding back components. What would you suppose this synthesis is called? Fourier synthesis, what else! Now when Fourier synthesis produces a wave like the original, the analyst knows he or she has the coefficients necessary to calculate the desired Fourier transform; that is, the equation of the compound wave.

Conceptually, Fourier synthesis is a lot like the decoding of a hologram. But before we can talk about this process, we must know more about the hologram itself. And before that, we must dig deeper still into the theoretical essence of waviness.

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The first regular wave in a Fourier series is often called the fundamental frequency or, alternatively, the first harmonic. The subsequent waves, the sine and cosine waves, represent the second, third, fourth, fifth, sixth... harmonics. Computer programs exist that will calculate higher and higher harmonics. In the pre-microchip days, nine was considered the magic number; even today, nine harmonics is enough to approximate compound waves with very large numbers of components. As the analysis proceeds, the discrepancy between the synthesized wave and the original wave usually becomes so small as to be insignificant.

These terms may seem very musical to the reader. Indeed, harmonic analysis is one of the many uses of Fourier's theorem. Take a sound from a musical instrument, for example. The first component represents the fundamental frequency, the main pitch of the sound. Higher harmonics represent overtones. There are odd and even harmonics, and they correspond to sine and cosine waves in the series. I present these terms from harmonic analysis only to illustrate one use of Fourier's theorem. But the theorem has such wide application that it has become a veritable lingua franca among persons who deal with periodic patterns, motions, surfaces, events...and on and on. I see no particular reason why the reader should dwell on terms like "fundamentals" and "odd-and-even harmonics." But for our purposes, it is highly instructive to look into why the component waves of Fourier series bear the adjectives, "sine" and "cosine."[7]

The trigonometrist uses sines and cosines as functions of angles, "function" meaning something whose value depends on something else. A function changes with variations in whatever determines it. Belly fat changes as a function of too many peanut-butter cookies changes location from plate to mouth. Sines and cosines are numerical values that change from 0 to 1 or 1 to 0 as an angle changes from 0 to 90 degrees or from 90 to 0 degrees. The right triangle (with one 90 degree angles) helps us to define the sine and cosine. Sine is the side (Y) opposite an acute angle (A) in a right triangle divided by the diagonal or hypotenuse (r): sin A=Y/r. The cosine is the side (X) adjacent, or next to, an acute angle, divided by the hypotenuse: cos A=X/r.

The famous Pythagorean theorem holds that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the two other sides (r2** = X2** + Y2**, where 2** is square). Suppose we give r the value of 1. Remember that 1 x 1 equals 1. Of course r2** is still 1. Notice that with X2** at its maximum value of 1, Y2** is equal to 0.

Thus when the cosine is at a maximum, the sine is 0, and vice versa. Sine and cosine, therefore, are opposites. If one is odd, the other is even.

Now imagine that we place our right triangle into a circle, putting angle A at dead center and side X right on the equator. Next imagine that we rotate r around the center of the circle and make a new triangle at various steps. Since r is the radius of the circle, and therefore will not vary, any right triangle we draw between the radius and the equator will have an hypotenuse equal to 1, the same value we assigned it before. Of course angle A will change, as will sides X and Y. Now the same angle A can appear in each quadrant of our circle. If angle A does, the result is two result is two non-zero values for both sine and cosine in a 360-degree cycle, which would create ambiguity for us. But watch this cute little trick to avoid the ambiguity.