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7.2.2 Superposition of two SHMs of the same direction and frequency

When a particle is subject to more than one harmonic force, each trying to move the particle in is own direction with SHM, we say that there is an interference of SHMs. Interference effects are easy to observe on the surface of a lake when to stone are thrown into the water. Interference is also important in optics and acoustics. It also is manifested in radio signals.

Consider first the superposition of two simple harmonic motions that produce a displacement of the particle along the same line. Let us begin with the case when both have the same frequency. The displacement of the particle produced by each simple harmonic motion is given by

x1 = A1 cos wt

x2 = A2 cos (wt + d)

In writing these equations we have assumed that the initial phase of x1 is zero, and that of x2 is d. This can always be done by a proper choice of the origin of time. The quantity d is called the phase difference, and is the parameter that really matters.

The resulting displacement of the particle is given by the linear combination

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and is periodic with a period P = 2p/w, since both terms have the same period.

Consider two important special cases. If d = 0 we say that the two motions are in phase. Then the resultant motion is

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Equation shows that the resultant motion is also SHM with the same angular frequency. The motion has an amplitude equal to the sum of the amplitudes of the two motions; that is,

A = A1 + A2

The graphs of the two motions and their resultant have been shown in figure. The two rotating vectors shm and harmonic force corresponding to the two motions are also represented. The rotating vector corresponding to the resultant motion is the sum of shm and harmonic force; that is,

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As a second case, when d = p we have

x2 = A2 cos (wt + p) = –A2 cos wt

Then the resultant motion is

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which shows that the resultant motion is SHM with the same angular frequency and an amplitude equal to the difference of the amplitude of the two motions; that is,

For that reason we say that the motions are in opposition. In figure we indicate, for d = p, the different motion considered and their rotating vectors.

In the general case, where the phase difference is arbitrary, the resultant motion is also SHM with the same angular frequency w and an amplitude given by

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This is easy to understand from examining figure, where the rotating vectors shm and harmonic force corresponding to x1 and x2 are drawn. It can be seen that shm and harmonic force form a fixed angle d. Their resultant OP¢ is given by equation. Note that when in equation we make d = 0 or d = p, we obtain the two cases considered previously.

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