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Question

How will standing waves result from the superposition of two waves?


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Solution

Standing wave:

  1. A standing wave is formed by the superposition of two waves traveling in the opposite direction.
  2. The amplitude and the face difference of the waves are the same.
  3. A standing wave is represented by the form, y(x,t)=2Asin(kx)sin(ωt), where A is the amplitude, ω is the angular frequency, k is the wavenumber of the wave.

Formula used:

The resultant wave of two waves is given by the formula yx,t=y1x,t+y2x,t.

Formation of the standing wave:

Suppose two standing waves y1(x,t)=Acos(kx−ωt) and y2(x,t)=−Acos(kx+ωt) are directed opposite each other. Using the formula, the resultant wave is given by

y(x,t)=Acoskx−ωt+-Acoskx+ωt=Acoskx−ωt-coskx+ωt=2Asinkx−ωt+kx+ωt2·sinkx+ωt-kx-ωt2=2Asin2kω2·sin2kx2=2Asinkxsinωt=A'sinωt { cosA-cosB=2sinA+B2sinB-A2}

Where, A'=2Asinkx is the amplitude of the resultant wave.

Hence, the equation y(x,t)=A'sinωt is known as the equation of standing wave and represent a simple harmonic motion of the same frequency of the individual wave.


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