Briefly explain the concept of super position principle.

Explanation

When a jerk is given to a stretched string which is tied at one end, a wave pulse is produced and the pulse travels along the string. Suppose two persons holding the stretched string on either side a jerk simultaneously, then these two wave pulses move towards each other, meet at some point and move away from each other with their original identity. Their behaviour is very different only at the crossing/meeting points, this behaviour depends on whether the two pulses have the same or different shape as shown in Figures.

When the pulses have the same, at the crossing, the total displacement is the algebraic sum of thier individual displacements and hence its net amplitude is higher than the amplitudes of the individual pulses. Whereas, if the two pulses have same amplitude but shapes are `180^(@)` out of phase at the crossing point, the net amplitude vanishes at that point and the pulses will recover their densities after crossing. Only waves can posses such a peculiar property and it is called superposition of waves. This means that the principles of superposition explains the net behaviour of the waves when they overlap. Generating to any number of waves i.e., if two or more waves in a medium move simultanesouly, when they overlap, their total displacement is the vector sum of the individual displacements.
To express mathematically, consider two functions which characterize the displacement of the waves, for example,
`y_(1) = A_(1) sin(kx - omega t)`
and `y_(2) =A_(2)cos (kx - omega t)`
Since, both `y_(1)` and `y_(2)` satisfy the wave equation (solutions of wave equation) then their algebraic sum
`y=y_(1) + y_(2)`
also satisfies the wave equation. This means, the displacements are additive. Suppose we multiply `y_(1)` and `y_(2)` with some constant then their amplitude is scaled by that constant Further, if `C_(1)` and `C_(2)` are used to multiply the displacements `y_(1)` and `y_(2)`, respectively, then, their net displacment y is
`y=C_(1)y_(1) + C_(2)y_(2)`
This can be generalized to any number of waves. In this case of n such waves in more than one dimension the displacements are written using vector notation.
Here, the net displacement `bar(y)` is
`bar(y) = sum_(i=1)^(n)C_(1)bar(y_(1))`
The principle of superposition can explain the following:
(a) Space (or spatial) Interference (also known as Interference)
(b) Time (or Temporal) Interference (also known as Beats)
( c) Concept of stationary waves.
Waves that obey principle of superposition are called linear waves (amplitudfe is much smaller than their wavelengths). In general, if the amplitude of the wave is not small that they are called non-linear waves.