Principle of Superposition of Waves

Principle of Superposition of Waves

Consider 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 y₁(x, t) and y₂(x, t). When these two waves overlap, the resultant displacement can be given as y(x,t).

Mathematically, y (x, t) = y₁(x, t) + y₂(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

y₁ = f₁(x–vt),

y₂ = f₂(x–vt)

……….

y_n = f_n (x–vt)

Then, the wave function describing the disturbance in the medium can be described as,

y = f₁(x – vt)+ f₂(x – vt)+ …+ f_n(x – vt)

or, y=∑ _{i=1 to n} = f_i (x−vt)

Let us consider a wave travelling along a stretched string given by, y₁(x, t) = A sin (kx – ωt) and another wave, shifted from the first by a phase φ, given as y₂(x, t) = A sin (kx – ωt + φ)

From the equations, we can see that both the waves have the same angular frequency, the same angular wave number k, and 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).

The above equation can be written as,

y(x, t) = 2A cos (ϕ/2). sin (kx − ωt + ϕ/2)

The resultant wave is a sinusoidal wave, travelling in the positive X direction, where the phase angle is half of the phase difference of the individual waves and the amplitude as [2cos ϕ/2] times the amplitudes of the original waves.

Superposition of Waves – Video Lesson