Beats 🥁 — Quick & Fun Notes
When two sound waves have almost the same frequency but not quite, they overlap and create a pleasant “wa-wah” effect in loudness called beats. Your ears hear a tone near the average frequency, yet the volume swells and fades at a new frequency we call the beat frequency. 🎶 :contentReference[oaicite:0]{index=0}
1 ️⃣ Building the Math Step by Step
Start with two waves of the same amplitude a but slightly different angular frequencies \( \omega_1 \) and \( \omega_2 \):
\( s_1 = a\cos(\omega_1 t), \quad s_2 = a\cos(\omega_2 t) \) :contentReference[oaicite:1]{index=1}
Add them (superposition principle) and use the “sum-to-product” identity:
\( s = s_1 + s_2 = 2a\cos\!\bigl(\tfrac{\omega_1-\omega_2}{2}t\bigr)\, \cos\!\bigl(\tfrac{\omega_1+\omega_2}{2}t\bigr) \) :contentReference[oaicite:2]{index=2}
- \( \omega_b = \dfrac{\omega_1-\omega_2}{2} \) controls the envelope (the slow loud–soft pattern).
- \( \omega_a = \dfrac{\omega_1+\omega_2}{2} \) is the rapidly oscillating carrier you mainly hear.
Because \( \omega = 2\pi\nu \), the beat frequency pops out beautifully:
\( \boxed{\; \nu_{\text{beat}} = |\nu_1 – \nu_2| \;} \) 🎯 :contentReference[oaicite:3]{index=3}
2 ️⃣ Why the Volume Swells & Fades
When the “envelope” cosine reaches \( +1 \) or \( -1 \), amplitude = \( 2a \) (loud). When it hits 0, the sound almost disappears (soft). One full loud-to-loud cycle finishes in \( 1/\nu_{\text{beat}} \) seconds. ⌛ :contentReference[oaicite:4]{index=4}
3 ️⃣ Everyday Uses 🎻
- Instrument tuning: Musicians tweak a string until the beats vanish, proving both notes match exactly. 🔧 :contentReference[oaicite:5]{index=5}
- Detecting small frequency shifts: The ear notices a few-hertz difference far better than an absolute pitch change. 🎧
4 ️⃣ Worked Example — Sitar Strings
Two sitar strings produce 5 Hz beats. Increase the tension of one string and the beat frequency drops to 3 Hz. If the other string plays 427 Hz, what was the untuned string’s original pitch? 💡
Tightening raises frequency. Because beats decreased, the untuned string must have started lower than 427 Hz.
\( |427 – \nu_B| = 5 \;\Rightarrow\; \nu_B = 422 \text{ Hz} \)
So the original frequency was 422 Hz. 🎉 :contentReference[oaicite:6]{index=6}
5 ️⃣ Bonus — Standing Waves in a Pipe (One End Closed) 📯
For a pipe that’s open at one end and closed at the other, only “odd” harmonics fit inside. The fundamental (first harmonic) is:
\( \nu_1 = \dfrac{v}{4L} \)
Higher allowed notes are \( 3\nu_1,\; 5\nu_1,\; 7\nu_1,\dots \). If the pipe length is 30 cm and sound speed is 330 m s\(^{-1}\), the fundamental equals 275 Hz. A source at the fourth harmonic (an “even” one) can’t resonate, so the sound suddenly dies when you close an end. 🎺 :contentReference[oaicite:7]{index=7}
High-Yield Ideas for NEET 🏆
- The beat frequency formula \( \nu_{\text{beat}} = |\nu_1 – \nu_2| \).
- The envelope-carrier expression for the superposed wave and how it leads to amplitude modulation.
- Only odd harmonics appear in a pipe with one closed end; fundamental \( \nu_1 = v / (4L) \).
- Raising string tension raises its frequency — handy when interpreting beat-frequency questions.
- The “zero-beat” method is the fastest way to tune instruments accurately.
Keep practicing and let the beats guide you to perfect pitch! 🤗
