Superposition principle

WHAT is the superposition principle?

Key consequences packaged inside this:

• The waves do not scatter off each other; after crossing they emerge unchanged in shape, speed and direction.
• It is what makes interference, beats, and standing waves possible.

WHY is it true? (Derivation from first principles)

The superposition principle is not a separate law — it is a consequence of the wave equation being linear.

A wave on a string obeys: $\frac{\partial^2 y}{\partial t^2} = v^2\,\frac{\partial^2 y}{\partial x^2}$

Step 1 — Suppose $y_1$ and $y_2$ are each solutions. $\frac{\partial^2 y_1}{\partial t^2} = v^2\frac{\partial^2 y_1}{\partial x^2}, \qquad \frac{\partial^2 y_2}{\partial t^2} = v^2\frac{\partial^2 y_2}{\partial x^2}$ Why this step? We start by assuming each wave alone is a valid solution — that's what "being a wave" means.

Step 2 — Try the sum $y = y_1 + y_2$. Plug into the left side: $\frac{\partial^2 (y_1+y_2)}{\partial t^2} = \frac{\partial^2 y_1}{\partial t^2} + \frac{\partial^2 y_2}{\partial t^2}$ Why this step? Differentiation distributes over addition — this is the crucial property that lets the waves "not interfere with each other's physics".

Step 3 — Substitute the known results: $= v^2\frac{\partial^2 y_1}{\partial x^2} + v^2\frac{\partial^2 y_2}{\partial x^2} = v^2\frac{\partial^2 (y_1+y_2)}{\partial x^2}$ Why this step? Factor out $v^2$ and re-combine — we get exactly the wave equation back, but now for $y_1+y_2$.

Conclusion: $y_1+y_2$ also satisfies the wave equation ⇒ the sum is itself a valid wave. The principle holds because the governing equation is linear.

HOW to use it — adding two waves

Take two waves of equal amplitude and frequency travelling the same direction with phase difference $\phi$: $y_1 = A\sin(kx-\omega t), \qquad y_2 = A\sin(kx-\omega t + \phi)$

Add them using $\sin P + \sin Q = 2\sin\!\frac{P+Q}{2}\cos\!\frac{P-Q}{2}$:

$y = 2A\cos\!\Big(\frac{\phi}{2}\Big)\,\sin\!\Big(kx-\omega t+\frac{\phi}{2}\Big)$

Why this matters: The resultant is still a wave of the same frequency, but with amplitude $\boxed{A_{\text{res}} = 2A\cos\!\Big(\frac{\phi}{2}\Big)}$

• $\phi = 0$ → $A_{\text{res}} = 2A$ → constructive (in phase).
• $\phi = \pi$ → $A_{\text{res}} = 0$ → destructive (out of phase).

Worked Examples

Forecast-then-Verify

Common Mistakes (Steel-manned)

Active Recall

Connections

• Interference of waves — direct application of superposition to two coherent sources.
• Beats — superposition of two slightly different frequencies.
• Standing waves — superposition of two identical waves travelling in opposite directions.
• Wave equation — its linearity is the reason superposition holds.
• Phasor method — geometric tool for adding waves with phase differences.
• Simple Harmonic Motion — superposition of SHMs in the same/perpendicular directions.

Concept Map

Hinglish (regional understanding)