Sine waves as circular motion

Chapter Sine waves as circular motion

Teach Yourself Electricity and Electronics Third Edition Book
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Teach Yourself Electricity and Electronics Third Edition Book

  • Suppose that you graph the rate of change in the voltage of the wave in Fig. 12-1against time. What will this graph look like? It turns out that it will have a shape that isa sine wave, but it will be displaced to the left of the original wave by one-quarter of acycle. If you plot the relative rate of change against time as shown in Fig. 12-2, you getthe derivative, or rate of change, of the sine wave. This is a cosine wave, having thesame general, characteristic shape as the sine wave. But the phase is different.Sine waves as circular motion21712-2A sine wave representing the rate of change in instantaneous amplitudeof the wave in Fig. 12-1.Sine waves as circular motionA sine wave represents the most efficient possible way that a quantity can alternateback and forth. The reasons for this are rather complex, and a thorough discussion of itwould get into that fuzzy thought territory where science begins to overlap with es-thetics, mathematics, and philosophy. You need not worry about it. You might recall,however, that a sine wave has just one frequency component, and represents a purewave for this reason.Suppose that you were to swing a glowing ball around and around at the end of astring, at a rate of one revolution per second. The ball would describe a circle in space(Fig. 12-3A). Imagine that you swing the ball around so that it is always at the same level;that is, so that it takes a path that lies in a horizontal plane. Imagine that you do this in aperfectly dark gymnasium. Now if a friend stands some distance away, with his or hereyes right in the plane of the ball’s path, what will your friend see? All that will be visibleis the glowing ball, oscillating back and forth (Fig. 12-3B). The ball will seem to move to-ward the right, slow down, then stop and reverse its direction, going back towards theleft. It will move faster and faster, then slower again, reaching its left-most point, at which