Shufflebrain

We can let all values of Y above the equator be positive; below the equator, lets let them be negative. We can do the comparable thing with X, except that we'll use the vertical meridian instead of the equator; then values of X on the right side of our circle will be positive and those on the left will be negative. If we plot a graph of sine or cosine values for angle A versus degrees on the circle, we get a wave. The cosine wave starts out at +1 at 0^o (360^o), drops to 0 at 90^o, plunges down to -1 at 180^o and returns to +1 at 360^o, the end of the cycle. Meanwhile, the sine wave starts at 0, swells to +1 at 90^o, drops back to 0 at 180^o, bottoms to -1 at 270^o and returns to a value of 0 at the completion of the 360-degree cycle. We really don't need the triangle anymore (so let's chuck it!): the circumference will suffice.

Notice, though, that the degree scale can get to be a real pain in the neck after a single cycle. Roulette wheel, clock hands, meter dials, orbital planets, components of higher frequency, and the like, rarely quit at a single cycle. But there's a simple trick for shifting to a more useful scale. Remember the formula for finding the circumference of a circle? Remember 2pi r? Recall that pi is approximately 3.14, or 22/7. When we make r equal 1, the value for the circumference is simply 2pi. This would convert the 90^o mark to 1/2 pi, the 180^o mark to 1pi (or just pi)...and so on. When we reach the end of the cycle, we can keep going right on up the pi scale as long as the baker has the dough, so to speak. [8] But we end up with a complete sine or cosine cycle at every 2pi.

With regard to a single sine or cosine wave on the pi scale, what is amplitude? Recall that we said it was maximum displacement from the horizontal plane. Obviously, the amplitude of a sine or cosine wave turns out to be +1. But +1 doesn't tell us where amplitude occurs or even if we have a sine versus cosine wave--or any intermediate wave between a sine and a cosine wave. This is where phase comes in, remember. Phase tells where or when we can find amplitude, or any other point, relative to the reference; i. e., to zero.

Notice that our sine wave reaches +1 at 1/2pi, 2 1/2pi, 4 1/2pi... and so forth. We can actually define the sine wave's phase from this. What about the phase of a cosine wave? Quiz yourself, and I'll put the answer in a footnote.[9]

If someone says, "I have a wave of amplitude +1 with a phase spectrum of 1/2pi, 2 1/2pi, 4 1/2pi," we immediately know that the person is talking about a sine wave. In other words, our ideal system gives us very precise definitions of phase and amplitude. We can also see in the ideal how these two pieces of information, phase and amplitude, actually force us to make what Benjamin Peirce and his son Charles called necessary conclusions! Phase and amplitude spectra completely define our regular waves.

Now let me make a confession. I pulled a philosophical fast one here in order to give us a precise look at phase and amplitude. We know the phase and amplitude of a wave the moment we assert that it is a sine or a cosine wave. Technically, our definition is trivial. To say, "sine wave" is to know automatically where amplitudes occur on our pi scale. But let's invoke Fourier's theorem and apply it to our trivia. If a complicated wave is a series of sine and cosine waves, and those simple waves are their phase and amplitude spectra, then knowing the phase and amplitude spectra for a complicated wave means having a complete definition of it as well. Our trivial definition leads us to a simple explanation of how it is that phase and amplitude completely define even the most complicated waves in existence. It is not easy to explain the inclusiveness of phase and amplitude in the "real" world. But look at how simple the problem becomes in the ideal. First, phase and amplitude define sine and cosine waves. Second, sine and cosine waves define compound waves. It follows quite simply, if perhaps strangely, that phases and amplitudes define compound waves too. But there's a catch.

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We can define the phase and amplitude of sine and cosine waves because we know where to place zero pi--0-- the origin or reference. We know this location because we put the 0 there ourselves. If we are ignorant of where to begin the pi scale, we don't know whether even a regular wave is a sine or a cosine wave, or something in between. An infinite number of points exist between any two loci on the circumference of a circle, and thus on the pi scale. The pure sine wave stakes out one limit, the cosine wave the other, while in between lie and infinite number of possible waves. Without knowing the origin of our wave, we are ignorant of its phase--infinitely ignorant!