The principle of superposition, the net response for all linear systems at a given time for a given place, caused by two or more stimuli, is the sum of responses which would have been caused by each stimulus individually. So, if the response produced by input A is X and that produced by input B is Y, then the response produced by input A+B is X+Y.

Considering two waves, travelling simultaneously along the same stretched string in opposite directions as shown in the figure above. We can see images of waveforms in the string at each instant of time. It is observed that the net displacement of any element of the string at a given time is the algebraic sum of the displacements due to each wave.

Let us say two waves are travelling alone and the displacements of any element of these two waves can be represented by y_{1}(x, t) and y_{2}(x, t). When these two waves overlap, the resultant displacement can be given as y(x,t).

Mathematically, y (x, t) = y_{1}(x, t) + y_{2}(x, t)

As per the principle of superposition, we can add the overlapped waves algebraically to produce a resultant wave. Let us say the wave functions of the moving waves are

y_{1} = f1(x–vt),

y_{2} = f2(x–vt)

, ………

……….

y_{n} = f_{n} (x–vt)

then the wave function describing the disturbance in the medium can be described as

y = f_{1}(x – vt)+ f_{2}(x – vt)+ …+ f_{n}(x – vt)

or,

\(y=\sum_{i=1}^{n}\; f_{i}(x-vt)\)

Let us consider a wave travelling along a stretched string given by, y_{1}(x, t) = A sin (kx – ωt) and another wave, shifted from the first by a phase φ, given as y_{2}(x, t) = A sin (kx – ωt + φ)

From the equations we can see that both the waves have the same angular frequency, same angular wave number k, hence the same wavelength and the same amplitude A.

Now, applying the superposition principle, the resultant wave is the algebraic sum of the two constituent waves and has displacement y (x, t) = A sin (kx – ωt) + A sin (kx – ωt + φ)

As,

\(\sin A=\sin B=2\sin \frac{1}{2}(A+B)\cos \frac{1}{2}(A-B)\)

The above equation can be written as,

\(y(x,t)=[2\, A\cos \frac{1}{2}\phi ]\sin(kx-wt+\frac{1}{2}\phi )\)

The resultant wave is a sinusoidal wave, traveling in the positive X direction, where the phase angle is half of the phase difference of the individual waves and the amplitude as \([2\cos \frac{1}{2}\phi]\)

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