The principle of superposition of waves

• Principle of superposition of waves describes how the individual waveforms can be algebraically added to determine the net waveform.
• Waveform tells about the overall motion of the wave.It does not tell about individual particles of the wave.
• Suppose we have 2 waves and
• Example of superposition of waves is Reflection of waves.
• Mathematically: –

Case1:-

• Consider 2waves which are in phase with each other. They have the same amplitude, same angular frequency, and same angular wave number.
• If wave 1 is represented by y₁(x, t) =a sin (kx – ωt).
• Wave 2 is also represented by y₂(x, t) =a sin (kx – ωt).
• By the principle of superposition the resultant wave (2a sin (kx – ωt)) will also be in phase with both the individual waves but the amplitude of the resultant wave will be more.

Case2:-

• Consider when the two waves are completely out of phase.i.e. φ = π
• If wave 1 is represented by y₁(x, t) =a sin (kx – ωt).
• Wave 2 is represented by y₂(x, t) =a sin (kx – ωt+ π).
• =>y₂=asin(π-(-kx+ ωt) =>y₂=-a sin(kx- ωt)
• Therefore by superposition principle y=y₁+y₂=0

Reflection at rigid boundary

• Consider a string which is fixed to the wall at one end. When an incident wave hits a wall,it will exerta force on the wall.
• By Newton’s third law, the wall exerts an equal and opposite force of equal magnitude on the string.
• Since the wall is rigid wall won’t move, therefore no wave is generated at the boundary.This implies the amplitude at the boundary is 0.
• As both the reflected wave and incident wave are completely out of phase at the boundary.Therefore φ=π.
• Therefore, y_i(x, t) = a sin (kx – ωt),
• y_r(x, t) = a sin (kx + ωt + π) = – a sin (kx + ωt)
• By superposition principle y= y_i + y_r =0
• Conclusion: –
  • The reflection at the rigid body will take place with a phase reversal of π or 180.

Standing (Stationary) Waves

• A stationary wave is a wave which is not moving,i.e. it is at rest.
• When two waves with the same frequency,wavelengthand amplitude travelling in opposite directions will interfere they produce a standing wave.

• Explanation:-
  • Consider I^st wave in the figure and suppose we have a rigid wall which does not move. When anincident wave hits the rigid wall it reflects back with a phase difference of π.
  • Consider II^nd wave in the figure, when the reflected wave travels towards the left there is another incident wave which is coming towards right.
  • The incident wave is continuously coming come from left to right and the reflected wave will keep continuing from right to left.
  • At some instant of time there will be two waves one going towards right and one going towards left as a result these two waves will overlap and form a standing wave.
• Mathematically:
• Wave travelling towards left y_l(x,t) =a sin(kx– ωt) and towards right y_r(x,t) =a sin (kx + ωt)
• The principle of superposition gives, for the combined wave
• y (x, t) = y_l(x, t) + y_r(x, t)= a sin (kx – ωt) + a sin (kx + ωt)
• y(x,t)= (2a sin kx) cos ωt (By calculating and simplifying)
• The above equation represents the standing wave expression.
• Amplitude = 2a sin kx.
  • The amplitude is dependent on the position of the particle.
  • The cos ωt represents the time dependent variation or the phase of the standing wave.

Difference between the travelling wave and stationary wave