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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*x*1 = *A*1 cos w*t**x*2 = *A*2 cos (w*t* + d)

In writing these equations we have assumed that the initial phase of *x*1 is zero, and that of *x*2 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

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

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 and corresponding to the two motions are also represented. The rotating vector corresponding to the resultant motion is the sum of and ; that is,

As a second case, when d = p we have*x*2 = *A*2 cos (w*t* + p) = –*A*2 cos w*t*

Then the resultant motion is

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

This is easy to understand from examining figure, where the rotating vectors and corresponding to *x*1 and *x*2 are drawn. It can be seen that and 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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