Shufflebrain

Suppose though that instead of a single regular wave we have two waves that are out of phase by a specific amount of pi? We still can't treat phase in absolute terms. But when we have two waves, we can deal with their phase difference--their relative phase value--just as we did in chapter 3 with the hands of the clock. And even though we may not be able to describe them in absolute terms, we will not be vague in specifying any phase differences in our system: our waves or cycles are out of phase by a definite value of pi. If we transfer our waves onto the circle, we can visualize the phase angle. In Fourier analysis, phase takes on the value of an angle.

What about relative phase in compound waves? Let's approach the problem by considering what happens when we merge simple waves to produce daughter waves. In effect, let's analyze the question of interference, but in the ideal. Consider what happens if we add together two regular waves, both in phase and both of the same amplitude. When and where the two waves rise together, they will push each other up proportionally. Likewise, when the waves move down together, they'll drive values further into the minus zone. If the amplitude is +1 in two colliding waves, the daughter wave will end up with an amplitude of +2, and its trough will bottom out at -2. Except for the increased amplitude, the daughter will look like its parents. This is an example of pure constructive interference; it occurs when the two parents have the same phase, or a relative phase difference of 0. The outcome here depends strictly on the two amplitudes, which, incidentally, do not have to be identical, as in the present example.

Next let's consider the consequences of adding two waves that are out of phase by pi, 180 degrees, but have the same amplitude. They'll end up canceling each other at every point, with the same net consequence as when we add +1 to -1. The value of the daughter's amplitude will be 0. In other words, the daughter here won't really be a wave at all but the original horizontal plane. This is an example of pure destructive interference, which occurs when the colliding waves are of equal amplitude but opposite phase.

Now let's take the case of two waves that have equal amplitude but are out of phase by something less than pi; i.e., something less that 180 degrees. In some instances the point of collision will occur where sections of the two waves rise or fall together, thus constructively interfering with each other--like the interference that occurs when we add together numbers of the same sign. In other instances, the interference will be destructive, like adding + and - numbers. The shape of the resulting daughter becomes quite complicated, even though the two parents may have the same amplitude. Yet any specific shape will be uniquely tied to some specific relative phase value.

Now let's make the problem a little more complicated. Suppose we have two regular ways, out of phase by less than pi; but this time imagine that they have different amplitudes. The phase difference will determine where the constructive and destructive interferences occur. Remember, though, that any daughter resulting from the collision of two regular waves will have a unique shape and size; and the resulting shape and size will be completely determined by the phase and amplitude of the two parents. In addition, if we know the phase and amplitude of just one parent, subtracting those values from the daughter will tell us the phase and amplitude of the other parent.

Suppose we coalesce three waves. The result may be quite complicated, but the basic story won't change: the new waves will be completely determined by the phase and amplitude of three, four, five, six or more interacting waves. Or the new compound wave will bear the phase and amplitude spectra that have determined completely by the interacting waves.

How does the last example differ from Fourier synthesis? For the most part, it doesn't. Fourier synthesis reverses the sequence of analysis.

The process is an abstract form of a sequence of interferences that produced the original compound wave. But the compound wave, no matter how complicated it is or how many components contributed to its form, is an algebraic sum of a series of phases and amplitudes.

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