Beats and Slowly Varying Envelopes
When two sinusoids of frequencies f1 and f2 are superposed at one point, their resultant has an amplitude envelope varying at half the angular-frequency difference and successive loudness maxima occur at beat frequency absolute(f1-f2).
Why this shows up in the exam
Tuning-fork identification · Beat-period questions · Combining beats with pipes or Doppler shift
Learn the idea
Nearby frequencies alternately reinforce and cancel, producing beats at their frequency difference. Two almost equal tones slip in and out of phase; the rapid oscillation is heard inside a slowly changing loudness envelope.
🧠 Memory hook: Beats count the gap between frequencies.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- f_beat = |f1 - f2| — number of loudness maxima per second
- cos(2pi f1 t)+cos(2pi f2 t)=2 cos(pi Deltaf t) cos(2pi f_avg t) — carrier-envelope decomposition
How to approach it
- 1Convert observed beats to beats per second
- 2Write the absolute frequency difference
- 3Use loading, tension, or motion information to choose the correct branch
Common slip-ups that cost marks
- •Taking the average frequency as beat frequency
- •Missing the two possible unknown frequencies
- •Using beat theory when frequencies are not close enough to perceive modulation
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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