The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time. The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements. This is true of waves which are finite in length (wave pulses) or which are continuous sine waves.
Superposition of two opposite direction wave pulses
The animation at left shows two Gaussian wave pulses are travelling in the same medium but in opposite directions. The two waves pass through each other without being disturbed, and the net displacement is the sum of the two individual displacements.
It should also be mentioned that this medium is nondispersive (all frequencies travel at the same speed) since the Gaussian wave pulses do not change their shape as they propagate. If the medium was dispersive, then the waves would change their shape.
Hence, the superposition of waves can lead to the following three effects:
- Whenever two waves having the same frequency travel with the same speed along the same direction in a specific medium, then they superpose and create an effect known as the interference of waves.
- In a situation where two waves having similar frequencies move with the same speed along opposite directions in a specific medium, then they superpose to produce stationary waves.
- Finally, when two waves having slightly varying frequencies travel with the same speed along the same direction in a specific medium, they superpose to produce beats.
Here you can see two waves ( in red and green) superposing to form the resultant wave (in purple). The diagram clearly shows only three points where it shows how the displacement of the resultant wave is calculated.
For example, when time = 0, the displacement of the red wave was -0.9 (signs must be taken into account as displacement is a vector quantity) and the displacement of the green wave was -2.1. As a result, the displacement of the resultant wave will be the vector sum of those, (-0.9)+(-2.1) = -3. This is simply the principle of superposition.
The principle of superposition states that, when two or more waves of the same type cross at some point, the resultant displacement at that point is equal to the sum of the displacements due to each individual wave.
The displacements are added vectorially. However, usually the vibrations or oscillations are in a single plane, so the displacements can be added algebraically (just including the +/−+/− sign).
The principle depends on the medium behaving linearly when the waves pass through; i.e. when the parts of the medium have twice the displacement then it has twice the restoring force. (For very large amplitudes this breaks down and harmonics are obtained).
The idea and the language transfers across to electromagnetic waves in a vacuum, even though there is no mechanical displacement of the medium.
When the waves pass beyond that point of intersection, they separate out again and the waves are completely unchanged (unless the medium has been overstretched).
When superposition occurs, for slow mechanical waves the amplitude can be observed. For high frequency waves, such as sound or EM waves, the intensity is usually measured, which is a measure of the energy of the wave in the region: the energy is proportional to the square of the amplitude, Intensity∝𝐸∝𝐴2Intensity∝E∝A2.
The resulting intensity at the point of intersection can be greater or smaller than the intensity due to each of the superposing waves. Superposition is important for explaining phenomena such as interference, diffraction and standing waves.