Beats — derivation, applications — Hinglish notes

1.6.20 · Physics › Oscillations & Waves

Beat kya hota hai?

HUMEIN KYA CHAHIYE: $∣ f_{1} - f_{2} ∣$ chhota hona chahiye (typically $< 10$ Hz) taaki ear slow envelope resolve kar sake. Agar difference zyada ho, to aap bas do alag notes sunenge.

First principles se Derivation

Superposition principle se displacements add hote hain: $y = y_{1} + y_{2} = a cos (2 π f_{1} t) + a cos (2 π f_{2} t)$

Yeh step kyun? Sound waves linear superposition follow karti hain — medium ka response sirf sum karta hai.

Identity use karo $cos C + cos D = 2 cos \frac{C - D}{2} cos \frac{C + D}{2}$ :

$y = 2 a slow envelope cos (2 π \frac{f _{1} - f _{2}}{2} t) fast audible tone cos (2 π \frac{f _{1} + f _{2}}{2} t)$

Envelope se beat frequency tak

Woh extra factor of 2 derivation ki jaan hai — isko miss karo to answer adha ho jata hai.

Figure — Beats — derivation, applications

Worked examples

Worked example Example 2 — Sign ambiguity (fork ko loading karna)

Ek unknown frequency ka fork 3 beats/s deta hai 384 Hz standard ke saath. Unknown 381 ya 387 ho sakta hai. Kaise decide karein?

Step: Unknown fork par wax ka chhota sa blob lagao → uski frequency decrease hoti hai.
Kyun? Wax mass add karta hai, $f = \frac{1}{2 π} k / m$ ko lower karta hai.
Observe: Agar beats increase hoin (3→4...), unknown 384 se upar tha → woh 387 tha (decrease hone ke baad further away move kiya? nahi — toward aur past move kiya). Dhyan se reason karo:
- Agar unknown $= 387$ hai: loading use 384 ki taraf lower karta hai → beats decrease hoti hain.
- Agar unknown $= 381$ hai: loading use 384 se aur door lower karta hai → beats increase hoti hain.
Answer: Beats decrease hoin ⇒ 387 Hz; beats increase hoin ⇒ 381 Hz. Wax trick woh $\pm$ sign resolve karti hai jo $∣ f_{1} - f_{2} ∣$ chhupa deta hai.

Recall Feynman: 12 saal ke bachche ko samjhao

Socho do dost almost same speed se taali baja rahe hain. Pehle woh saath taali bajate hain — LOUD. Dheere dheere ek aage nikal jata hai, isliye ek doosre ke gaps mein taali bajata hai — steady/quiet lagta hai. Phir woh line up ho jaate hain — LOUD phir se. Woh swelling "loud-quiet-loud" hi beat hai. Unki clapping speeds mein jitna zyada fark, utni baar woh line up aur split apart hote hain — isliye bada frequency difference faster beats deta hai. "Loud moments" ki sankhya per second exactly itni hai jitni extra taaliyaan ek dost doosre se zyada bajata hai per second.

Applications

Musical instruments tune karna: Ek string ko reference ke against tune karo jab tak beats khatam na ho jaayein ( $f_{b e a t} = 0 \Rightarrow$ exact match).
Unknown frequency determine karna: $f_{u nk n o w n} = f_{k n o w n} \pm f_{b e a t}$ ; sign wax/file trick se resolve karo (Example 2).
Gas leaks detect karna / "beat-frequency oscillators" (heterodyne): ek signal ko reference ke saath mix karna aur beat padhna radio tuning aur Doppler-shift measurements ka basis hai.
Speed/Doppler radar: transmitted aur reflected frequencies ke beech ka beat target ki speed deta hai.

Flashcards

Beats sunne ke liye do frequencies par kya condition chahiye?

Woh slightly different honi chahiye (

∣ f_{1} - f_{2} ∣

chhota, typically < ~10 Hz) taaki ear slow envelope resolve kar sake.

Beat frequency formula batao.

f_{b e a t} = ∣ f_{1} - f_{2} ∣

Aap pitch ke roop mein actually kaunsi frequency sunते ho?

Average,

f_{a v g} = \frac{f _{1} + f _{2}}{2}

Beat frequency

∣ f_{1} - f_{2} ∣

kyun hai na ki

\frac{∣ f _{1} - f _{2} ∣}{2}

Loudness

∣ A (t) ∣

(ya

A^{2}

) par depend karti hai;

∣ cos ∣

ke ek envelope cycle mein do maxima hote hain, rate double ho jaati hai.

Derivation mein kaunsi trig identity use hoti hai?

cos C + cos D = 2 cos \frac{C - D}{2} cos \frac{C + D}{2}

Ek fork 384 Hz standard ke saath 3 beats/s deta hai; wax lagane par beats decrease hoti hain. Unknown find karo.

387 Hz (wax use 384 ki taraf lower karta hai, beats reduce hoti hain).

Wax se fork load karne par frequency kyun decrease hoti hai?

Added mass

m

f = \frac{1}{2 π} k / m

mein increase karta hai,

f

decrease karta hai.

Beats ke do practical applications batao.

Instruments tune karna (zero beats = match) aur unknown frequency find karna / heterodyne radio aur Doppler radar.

6 Hz beats mein successive silences ke beech ka time?

1/6 s \approx 0.167

s (one beat period).

Connections

Superposition Principle — beats iska direct consequence hain.
Interference of Waves — beats time mein interference hain (vs. spatial interference patterns).
Standing Waves — same superposition algebra, lekin do opposite-direction equal waves.
Doppler Effect — woh frequency shift deta hai jo radar mein beat ki tarah read hota hai.
Simple Harmonic Motion — wax-loading argument $f = \frac{1}{2 π} k / m$ use karta hai.
Amplitude Modulation — same envelope mathematics radio mein.

Beats — derivation, applications