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# Superposition of Waves and its Interference

### Key Concepts

• Superposition of waves
• Constructive interference
• Destructive interference

### Introduction:

Two sound waves are produced by two different musical instruments when played at the same time. The sound waves of each instrument are unaffected by the other waves that are passing through the same space at the same moment. The sound waves of each instrument are unaffected by the other waves that are passing through the same space at the same moment. Two waves can pass through one another.

Waves don’t carry matter with them they only displace the matter, that is why two waves can occupy the same space at the same time. The combination of two or more overlapping waves is called superposition.

## Superposition of two pulses that are in phase:

• In figure (a), two-wave pulses are traveling toward each other on a stretched rope.  The larger pulse is moving to the right, while the smaller pulse moves towards the left.
• At the moment the two wave pulses meet, at each point along the rope the displacement due to the two pulses are added together and a resultant wave is formed as shown in figure (b).
• When two waves exactly coincide as shown in figure (c), the amplitude of the resultant wave is equal to the sum of the amplitudes of each pulse. This happens according to the superposition principle.
• According to the superposition principle, when two or more waves travel through a medium at the same time, the resultant wave is the sum of the displacement (A1+A2 = A) of the individual wave as shown in the figure.
• We notice that after the two pulses pass through each other, each pulse has the same shape it had before the waves met and each is still traveling in the same direction as shown in figure (d).

### Constructive interference:

When two or more waves travel through the same space at the same time and if the displacements are in phase resultant displacement is the sum of the individual displacement. This is called constructive interference.

## Superposition of two pulses that are out of phase:

• When the pulses on a rope are on opposite sides of the equilibrium position, as they are in figure (a), during the superposition of pulses, the positive and negative displacements are added, as shown in figures (b) and (c), and the resultant wave is the difference between the pulses. This is called destructive interference.
• After superposition, the pulses separate, and their shapes are unchanged as shown in figure (d).

### Destructive interference:

When two or more waves travel through the same space at the same time and if the displacements are out of phase resultant displacement is the difference of the individual displacement. This is called destructive interference.

### Constructive and destructive interference of water waves:

If these waves were water waves coming together, one of the waves would be acting to pull an individual drop of water upward at the same instant and with the same force that another wave would be acting to pull it downward. The result would be no net force on the drop, and there would be no net displacement of the water at that moment.

We have considered the interference produced by two transverse pulse waves. The superposition principle is valid for longitudinal waves as well.

When compression and rarefaction interfere, there is destructive interference.

When two or more compressions interfere or two or rarefactions interfere, there is constructive interference.

### Superposition of non-identical waves:

Pure constructive and pure destructive interference rarely occurs as it requires precisely identical waves. The superposition of most waves produces a combination of constructive and destructive interference and can vary from place to place and time to time. It produces a more complicated-looking wave.

### Summary

• Waves don’t carry matter with them they only displace the matter, that is why two waves can occupy the same space at the same time.
• The combination of two or more overlapping waves is called superposition.
• According to the superposition principle, when two or more waves travel through a medium at the same time, the resultant wave is the sum of the displacement (A1+A2 = A) of the individual wave.
• When two or more waves travel through the same space at the same time and if the displacements are in phase resultant displacement is the sum of the individual displacement. This is called constructive interference.
• When two or more waves travel through the same space at the same time and if the displacements are out of phase resultant displacement is the difference of the individual displacement. This is called destructive interference.

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