7.2 Principle of Superposition of Waves

7.2 Principle of Superposition of Waves

If a particle of the medium is simultaneously acted upon by two waves, then the resultant displacement of the particle is the algebraic sum of their individual displacements. This is called principle of superposition.

In other words, the displacements of the individual waves are added together to form a new wave pattern as shown in Fig. 7.4 (a and b), respectively.

(i) If two waves, which overlap each other, have same phase, their resultant displacement will be: $y = y_{1} + y_{2}$ where $y_{1} =$ amplitude of wave 1, $y_{2} =$ amplitude of wave 2, and $y =$ resultant amplitude

Particularly, if $y_{1} = y_{2}$, then resultant displacement will be: $y = 2y_{1}$ or $y = 2y_{2}$

(ii) If two waves, which cross each other, have opposite phase, their resultant displacement will be: $y = y_{1} + (-y_{2}) = y_{1} - y_{2}$

Particularly, if $y_{1} = y_{2}$, then resultant displacement will be $y = 0$.

Thus, if a particle of a medium is simultaneously acted upon by $n$ waves such that its displacement due to each of the individual $n$ waves be $y_{1}$, $y_{2}$, $\ldots$, $y_{n}$, then the resultant displacement of the particle, under the simultaneous action of these $n$ waves is the algebraic sum of all the displacements, i.e.,

This is called principle of superposition of waves. Mathematically, this can also be represented as:

where $y(x,t)$ is the resultant wave, whereas $y_{1}(x,t)$, $y_{2}(x,t)$, $\ldots$, $y_{n}(x,t)$ are the individual waves.

In the context of waves, $y(x,t)$ represents the wave function or wave displacement at a given point $x$ and time $t$. It describes the shape of the wave and its evolution overtime.

The principle of superposition applies to linear waves or small amplitude waves.

Principle of superposition of waves leads to many interesting phenomena:

1. Two waves having same frequency and travelling in the same direction (Interference).
2. Two waves of slightly different frequencies and travelling in the same direction (Beats).
3. Two waves of equal frequency travelling in opposite direction (Stationary waves).