Principle of Superposition: Definition, Working Principle, Formula, Superposition Principle, Class 11 Notes

Let us explain principle of superposition mathematical terms.

Step 1: Write down each wave’s equation

Pick two sinusoidal waves traveling in the same direction (same k and ω), but possibly different amplitudes and phases:

$$\begin{array}{c}{y}_1(x,t)=A_1\sin (kx-\omega t+{\phi }_1)\\ y_2(x,t)=A_2\sin (kx-\omega t+{\phi }_2)\end{array}$$ Step 2: Invoke linearity of the wave equation $$\text{The wave equation governing}y(x,t)\text{is linear. That means if each of}{y}_1\text{and}{y}_2\text{individually satisfies the equation, then any sum}$$ $$y(x,t)=y_1(x,t)+y_2(x,t)$$

also satisfies it exactly.

3. Interpret at a fixed point PPP

Imagine standing at a fixed x=P. At time t, the medium “would” be at

$$\begin{array}{c}{y}_1(P,t)\text{if only wave 1 were present,}\\ y_2(P,t)\text{if only wave 2 were present.}\end{array}$$ Since the medium doesn’t “care” which wave caused the displacement—only the net force—it simply reaches the sum:

$$y(P,t)=y_1(P,t)+y_2(P,t)$$

Those who are CBSE Board  students must practice NCERT solutions of the chapter for better performance in the examination.

The following are the different types of superposition of waves:

• Constructive Interference: Whenever 2 waves are travelling in the same direction and are in phase with one another, the amplitude of those waves get added and a resultant wave is obtained. This is known as constructive interference.
• Destructive Interference: Destructive interference happens when two waves arrive out of sync so that a “hill” (crest) of one lines up with a “valley” (trough) of the other. At that moment, their pushes and pulls are opposite, and they cancel each other out which makes the combined wave smaller or even perfectly flat.

Let ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$  be two sources producing progressive waves (disturbance travelling in space given by ${\mathrm{y}}_1$  and ${\mathrm{y}}_2$  )
At point $P$ ,
$y_1=a_1\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t-k{x}_1+{\theta }_1\right)$

$y_2={\mathrm{a}}_2\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t-k{x}_2+{\theta }_2\right)$

$y=y_1+y_2=A\mathrm{s}\mathrm{i}\mathrm{n}(\omega t+\mathrm{\Delta }\varphi )$

Here, the phase difference,

$\mathrm{\Delta }\varphi =\left(\omega t-k{x}_1+{\theta }_1\right)-\left(\omega t-k{X}_2+{\theta }_2\right)$

$=\mathrm{k}\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)+\left({\theta }_1-{\theta }_2\right)=\mathrm{k}\mathrm{\Delta }\mathrm{p}-\mathrm{\Delta }\theta$

where $\mathrm{\Delta }\theta ={\theta }_2-{\theta }_1$

 Here $\mathrm{\Delta }p=\mathrm{\Delta }x$ is the path difference

Figure: 1.3

• Linearity: The medium (string, air, water, etc.) must respond in a straight-line way.
• Small Disturbances: Waves should not be so big that they change the medium’s properties.
• Same Type of Wave: You can only add same types of wave such as two sound waves, two water waves, or two light waves.
• No Energy Loss Features: Things like friction or absorption must be negligible for the simple sum to hold exactly.