Principle of Superposition of Waves: The Principle of Superposition of Waves in physics that states that when two or more waves interact, their individual effects combine to create a new wave pattern. This new wave pattern is the result of the superposition of the individual waves, and its amplitude and phase at each point in space are determined by the amplitudes and phases of the individual waves.

The superposition of waves can be either constructive or destructive, depending on the relative phases of the individual waves. Constructive superposition occurs when the waves are in phase, meaning that their peaks and troughs line up, and the resulting wave has an amplitude that is equal to the sum of the amplitudes of the individual waves. Destructive superposition occurs when the waves are out of phase, meaning that their peaks and troughs do not line up, and the resulting wave has an amplitude that is equal to the difference of the amplitudes of the individual waves.

In conclusion, superposition of waves is the principle that states that when two or more waves interact, their individual effects combine to create a new wave pattern. The superposition of waves can be either constructive or destructive, depending on the relative phases of the individual waves.

Principle of Superposition of Waves

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 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, 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)

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.

Types of Superposition of Waves

There are two types of superposition of waves: constructive interference and destructive interference.

• Constructive Interference: Constructive interference occurs when two or more waves are in phase, meaning that their peaks and troughs line up. In this case, the superposition of the waves results in a wave pattern with an amplitude that is equal to the sum of the amplitudes of the individual waves. The resulting wave pattern is larger and has a higher amplitude than either of the individual waves.
• Destructive Interference: Destructive interference occurs when two or more waves are out of phase, meaning that their peaks and troughs do not line up. In this case, the superposition of the waves results in a wave pattern with an amplitude that is equal to the difference of the amplitudes of the individual waves. The resulting wave pattern is smaller and has a lower amplitude than either of the individual waves.

In conclusion, there are two types of superposition of waves: constructive interference and destructive interference. Constructive interference occurs when the waves are in phase, and results in a wave pattern with an amplitude that is equal to the sum of the amplitudes of the individual waves. Destructive interference occurs when the waves are out of phase, and results in a wave pattern with an amplitude that is equal to the difference of the amplitudes of the individual waves.

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By Team Learning Mantras