QCC Notes
CLASS 11 · PHYSICSJEE MAIN × NEETहिंदी
§ 14.4NCERT Class 11 · Physics · Chapter 14

Acoustic Beats & the Doppler Effect in Sound

QCC Notes · Class 11 Physics NCERT-aligned Updated 26 Sep 2026 4 min read
Phenomenon of Beats:

When two sound waves of slightly different frequencies (f1 and f2) travelling in the same direction superpose, the resultant sound intensity waxes (increases to a maximum) and wanes (drops to a minimum) periodically. This periodic variation in intensity is known as Beats.

1. Mathematical Derivation of Beats

Let two waves of equal amplitude A and slightly different angular frequencies ω1 = 2πf1 and ω2 = 2πf2 superpose at x = 0:

y = y1 + y2 = A sin(ω1t) + A sin(ω2t) = [ 2A cos( (ω1 − ω2)t / 2 ) ] · sin( (ω1 + ω2)t / 2 )

The amplitude of the resultant wave is modulated as:

Amod(t) = 2A cos[ π (f1 − f2) t ]

Since intensity I ∝ Amod2, intensity reaches a maximum when cos2[π(f1 − f2)t] = 1, which occurs twice per modulation cycle. Therefore:

Beat Frequency: fb = | f1 − f2 |
  • Time interval between two successive waxings (or wanings): Tb = 1 / fb = 1 / |f1 − f2|.
  • Human Limit of Perception: Beats are distinctly audible to the human ear only if fb ≤ 10 Hz (due to persistence of hearing, Δt ≈ 0.1 s).

2. Tuning Fork Modifications (Standard JEE / NEET Trap)

  • Loading with Wax: Increases inertia ⇒ Frequency decreases (f ↓).
  • Filing the Prongs: Reduces mass ⇒ Frequency increases (f ↑).

Standard Problem: If fork A (frequency fA) produces 4 beats/s with fork B, then fB = fA ± 4. If loading A with wax decreases the beat frequency, then fA > fB ⇒ fB = fA − 4.

3. The Doppler Effect in Sound

The apparent change in the frequency of sound heard by an observer due to the relative motion between the source of sound, the observer, and the transmitting medium is called the Doppler Effect.

Master Doppler Formula:
f' = f0 [ (v ± vo) / (v ∓ vs) ]

Where:

  • f0: True (original) frequency emitted by source.
  • f': Apparent frequency heard by observer.
  • v: Speed of sound in the stationary medium.
  • vo: Speed of observer; vs: Speed of source.

4. Sign Conventions & Standard Cases

Golden Rule: Direction from Source to Observer (S → O) is taken as positive for sound propagation.

Relative Motion Scenario Formula for Apparent Frequency f' Physical Consequence
Source moves toward stationary Observer f' = f0 [ v / (v − vs) ] Waves compressed; λ' < λ; f' > f0 (Pitch rises)
Source moves away from stationary Observer f' = f0 [ v / (v + vs) ] Waves stretched; λ' > λ; f' < f0 (Pitch falls)
Observer moves toward stationary Source f' = f0 [ (v + vo) / v ] Observer intercepts more waves/s; f' > f0
Observer moves away from stationary Source f' = f0 [ (v − vo) / v ] Observer intercepts fewer waves/s; f' < f0
Both move toward each other f' = f0 [ (v + vo) / (v − vs) ] Maximum apparent frequency
Both move away from each other f' = f0 [ (v − vo) / (v + vs) ] Minimum apparent frequency
Special Situations with Zero Doppler Shift (f' = f0):
  1. When both source and observer are at relative rest (vrel = 0).
  2. When both source and observer move in the same direction with the same speed (vs = vo).
  3. When source moves perpendicular to the line joining source and observer at the point of closest approach.
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