SuperpositionJEE Main

Concept

The principle of superposition says overlapping waves add displacement-by-displacement. Two waves of the same frequency combine into a single wave whose amplitude depends on their phase difference: in phase they reinforce (constructive), out of phase they cancel (destructive).

Key formula

Derivation

Represent each wave as a phasor of length $A_1$, $A_2$ with angle $\Delta\phi$ between them. The resultant is the vector sum, whose magnitude follows from the cosine rule: $A_R^2 = A_1^2 + A_2^2 + 2A_1A_2\cos\Delta\phi$.

At $\Delta\phi = 0$ this gives $A_R = A_1 + A_2$ (maximum); at $\Delta\phi = \pi$ it gives $A_R = |A_1 - A_2|$ (minimum).

Intensity goes as amplitude squared, so $I_R = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\Delta\phi$ — the basis of all interference patterns.

Scenarios to explore

• Superposition — Two waves interfering — constructive & destructive.

Real-world applications

• Young's double-slit and thin-film interference.
• Noise-cancelling headphones (destructive interference).
• Antenna arrays and beam forming.

JEE exam tips

• Constructive: $\Delta\phi = 2n\pi$ or path difference $= n\lambda$.
• Destructive: $\Delta\phi = (2n+1)\pi$ or path difference $= (n+\tfrac12)\lambda$.

Common mistakes

• Adding intensities directly instead of amplitudes as phasors.
• Forgetting the cross term $2A_1A_2\cos\Delta\phi$.
• Confusing path difference with phase difference: $\Delta\phi = \tfrac{2\pi}{\lambda}\,\Delta x$.

Exam traps to avoid

• Equal amplitudes give complete cancellation only when exactly out of phase.
• Maximum intensity is $4I$ (not $2I$) for two equal coherent sources of intensity $I$.