Beats — derivation, applications: Question bank — Hinglish deep dive

1.6.20 · D5 · Physics › Oscillations & Waves › Beats — derivation, applications

Yaad dilane ke liye simple words mein, toh yahan koi bhi aisa symbol nahi hai jo tumne pehle na dekha ho:

$f_{1}, f_{2}$ — do sources ki do frequencies (vibrations per second).
$f_{b e a t} = ∣ f_{1} - f_{2} ∣$ — har second mein kitne "loud moments" sunai dete hain.
$f_{a v g} = \frac{f _{1} + f _{2}}{2}$ — woh pitch (note) jo tum actually sunте ho.
Envelope — loudness ka woh slow rise-and-fall jo fast note ke upar baitha hota hai.

True or false — justify

TF1. "Beats ke liye do waves ki bilkul equal amplitude chahiye, warna beats bilkul nahi aate."

False. Unequal amplitudes mein bhi beats hote hain; bas kabhi total silence nahi milti, toh dip ek partial fade hoti hai clean gap ki jagah. Equal amplitude sirf isliye choose ki jaati hai taaki silence complete ho aur algebra tidy rahe.

TF2. "Agar do forks ki frequency identical hai, toh tum zero frequency ke beats sunate ho."

f_{b e a t} = ∣ f_{1} - f_{2} ∣ = 0

ke sense mein True== — "loud–soft–loud" cycle infinite time leti hai, matlab kabhi repeat nahi hoti, toh tum ek steady, unwavering note sunate ho. Zero beats bilkul wahi hai jo tuning target hota hai.

TF3. "Beats ke dauran jo pitch sunai deti hai woh

f_{1}

(higher fork) hoti hai."

False. Audible pitch average

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

hoti hai, jo fast cosine factor se aati hai. Dono individual frequencies itni close hoti hain ki tum unhe alag-alag resolve nahi kar paate.

TF4. "Beats sirf interference hai, aur interference bright aur dark fringes ka ek spatial pattern hai."

Half-true. Beats hain interference (dekho Interference of Waves), lekin yahan yeh interference time mein ek point par hoti hai — loud aur soft moments — fringes jaisa space mein fixed pattern nahi.

TF5. "Kyunki loudness

∣ A (t) ∣

follow karti hai, loudness maxima envelope ki apni oscillation se do baar zyada baar aate hain."

True. Envelope cosine ka upward bump aur downward bump dono loud hote hain, isliye

∣ cos ∣

har envelope cycle mein do baar peak karta hai, rate ko

f_{b e a t} = ∣ f_{1} - f_{2} ∣

tak double kar deta hai.

TF6. "

∣ f_{1} - f_{2} ∣

ko bada karna beats ko hamesha sunna easier bana deta hai."

False. Roughly

10

Hz se zyada hone par swelling itni fast ho jaati hai ki kaan usse rhythm ki tarah follow nahi kar sakta, aur tum beats count karne ki jagah do alag notes (ya ek rough dissonance) perceive karne lagte ho.

TF7. "Ek akela tuning fork apne aap beats produce kar sakta hai."

False. Beats ke liye do superposing frequencies chahiye. Ek pure fork ek steady tone deta hai; tumhe ek doosra source chahiye (doosra fork, ek string, ek reference tone).

TF8. "Formula

f_{b e a t} = ∣ f_{1} - f_{2} ∣

mein absolute-value bars sirf decoration hain."

False. Bars matter karte hain: beat frequency events ki count hai, hamesha non-negative, aur

∣ f_{1} - f_{2} ∣

deliberately yeh throw away karta hai ki kaun sa fork higher hai — isliye sign recover karne ke liye wax/file trick ki zaroorat padti hai.

Spot the error

SE1. "Product

y = 2 a cos (2 π \frac{f _{1} - f _{2}}{2} t) cos (2 π \frac{f _{1} + f _{2}}{2} t)

padhte hue, beat frequency slow cosine ki frequency hai,

\frac{f _{1} - f _{2}}{2}

Error yeh hai ki envelope ki apni frequency par ruk jaana. Loudness

∣ A (t) ∣

par depend karti hai, aur

∣ cos ∣

ke do maxima hote hain har cycle mein, isliye loud moments

2 \times \frac{f _{1} - f _{2}}{2} = ∣ f_{1} - f_{2} ∣

par aate hain.

SE2. "

y_{1} = a cos (2 π f_{1} t)

aur

y_{2} = a cos (2 π f_{2} t)

, isliye resultant amplitude hamesha

2 a

hai."

Galat — resultant amplitude time-varying envelope

2 a cos (2 π \frac{f _{1} - f _{2}}{2} t)

hai. Yeh

2 a

sirf loud instants par reach karta hai aur silences par

0

tak drop ho jaata hai.

SE3. "Fork par wax lagane se uski frequency badhti hai kyunki fork ab zyada energy store karta hai."

Error yeh hai ki energy ko frequency se confuse karna. Wax mass

m

add karta hai, aur Simple Harmonic Motion se

f = \frac{1}{2 π} k / m

, zyada mass matlab lower frequency.

SE4. "Ek fork 384 Hz standard ke saath 3 beats/s deta hai; wax lagane se beats faster ho jaate hain, isliye fork 387 Hz tha."

Reasoning apne aap se contradict karta hai. Agar fork 387 Hz hota, toh wax use 384 ki taraf lower karta, isliye beats decrease hote. Beats increase hona matlab hai fork 384 se door gaya, isliye woh shuru mein neeche tha, 381 Hz par.

SE5. "Beats prove karte hain ki do sound waves permanent silence tak cancel ho sakti hain."

Silence sirf instantaneous hoti hai. Kyunki

f_{1} \neq = f_{2}

waves drift karte rehte hain, isliye total cancellation ek moment tak rehti hai aur loudness turant phir se chadhne lagti hai — yahi ise periodic banata hai.

SE6. "Standing waves aur beats same phenomenon hain kyunki dono mein do waves add hoti hain."

Woh superposition algebra share karte hain lekin physically alag hain. Standing Waves mein do equal waves space mein opposite directions mein travel karti hain, fixed nodes deti hain; beats mein ek point par time mein alag frequency ki do waves add hoti hain.

SE7. "Kyunki

f_{b e a t} = ∣ f_{1} - f_{2} ∣

, dono frequencies ko double karne se beat frequency double ho jaati hai."

Beats ko sirf difference control karta hai, scale nahi. Agar

f_{1}, f_{2} \to 2 f_{1}, 2 f_{2}

toh difference

∣2 f_{1} - 2 f_{2} ∣ = 2∣ f_{1} - f_{2} ∣

does double hota hai — lekin isliye nahi ki pitch double hui, isliye ki gap double hua. Dono ko same amount se upar shift karne par beats unchanged rehte hain.

Why questions

WQ1. "Beats actually sunne ke liye

∣ f_{1} - f_{2} ∣

small kyun hona chahiye?"

Envelope

\frac{∣ f _{1} - f _{2} ∣}{2}

par oscillate karta hai; agar yeh large ho toh loudness itni fast change hoti hai ki kaan use rhythm ki tarah track nahi kar sakta, isliye "waah-waah" ki jagah tum do tones ya roughness perceive karte ho.

WQ2. "Pitch average frequency kyun hoti hai, kuch aur kyun nahi?"

Product form ek fast cosine ko

\frac{f _{1} + f _{2}}{2}

par isolate karta hai jo audible oscillation carry karta hai; kaan us carrier ko respond karta hai jabki slow factor sirf uska volume upar-neeche karta hai.

WQ3. "Zero beats perfect tuning match confirm kyun karta hai?"

f_{b e a t} = ∣ f_{1} - f_{2} ∣ = 0

sirf tab hota hai jab

f_{1} = f_{2}

; koi bhi residual mismatch ek slow throb ke roop mein wapas aata, isliye "bilkul koi throb nahi" sabse sharp possible match test hai.

WQ4. "Beats frequency ko fraction of a hertz tak measure kyun kar sakte hain jabki frequencies khud hundreds of hertz hain?"

Beats ek tiny difference ko ek slow, countable rhythm mein convert kar dete hain — 4 seconds mein 2 beats count karne se difference seedha

0.5

Hz pin ho jaata hai, koi fast-timing electronics ki zaroorat nahi.

WQ5. "Radar aur heterodyne radios beats par kyun rely karte hain?"

Ek incoming signal ko reference ke saath mix karne par unke difference ke barabar ek beat produce hoti hai; radar ke liye woh difference Doppler Effect shift hai, isliye beat read karna target ki speed read karna hai — ek small shift measurable bana diya.

WQ6. "Beat envelope mathematically amplitude-modulated radio signal ke saath same kyun hai?"

Dono ek fast carrier hain jo ek slow amplitude factor se multiply hota hai; Amplitude Modulation mein slow factor message hai, beats mein yeh loudness swell hai — identical structure, alag interpretation.

WQ7. "Superposition principle humein

y_{1} + y_{2}

simply add karne kyun deta hai?"

Small disturbances ke liye medium linearly respond karta hai, isliye displacements interact kiye bina sum ho jaate hain — poori derivation Superposition Principle par rest karti hai. Nonlinear media extra frequencies create karte.

Edge cases

EC1. "Jab

f_{1} = f_{2}

exactly ho toh tum kya sunate ho?"

Ek single steady tone jis ki loudness nahi badlti — beat period infinite hai, isliye koi swelling kabhi nahi hoti. Yahi tuner ka goal state hai.

EC2. "Jab

∣ f_{1} - f_{2} ∣

roughly 10 Hz se zyada ho jaata hai toh kya hota hai?"

Throbbing count karne ke liye bahut rapid ho jaati hai aur ek rough, dissonant texture mein blend ho jaati hai; eventually tum beats ki jagah do distinct pitches resolve karte ho.

EC3. "Agar ek wave ki amplitude zero ho, toh kya beats aate hain?"

Nahi. Ek source silent hone par interfere karne ke liye kuch nahi hai, isliye tum sirf surviving steady tone sunate ho — beats ke liye do comparable non-zero amplitudes chahiye.

EC4. "Do forks bilkul out of phase start karte hain (crest against trough) lekin same frequency par — beats aate hain ya nahi?"

Beats nahi, kyunki

f_{1} = f_{2}

. Tumhe fixed phase se set ek constant reduced (ya zero) loudness milti hai; frequency difference ke bina kuch bhi drift nahi karta, isliye kuch swell nahi karta.

EC5. "Wax trick: agar wax lagane par beat frequency unchanged rahe toh?"

Tab tumhari resolution ke andar woh tiny frequency drop ne difference ko measurably shift nahi kiya — ek clear increase ya decrease force karne ke liye zyada wax add karo, kyunki unchanged reading koi sign information nahi deti.

EC6. "Kya beats sirf sound ke liye hote hain, ya kisi bhi wave ke liye?"

Koi bhi wave jo superposition follow kare — light, radio, water — beat kar sakti hai. Hum sound ke liye sunते hain kyunki beat rate us range mein padti hai jo kaan track karta hai; radio ke liye hum ise electronically detect karte hain (heterodyning).

EC7. "Agar do amplitudes bahut unequal hain (maan lo

a

aur

0.05 a

), toh kya phir bhi silence hoti hai?"

Koi asli silence nahi — loudness worst case mein

∣ a - 0.05 a ∣ = 0.95 a

tak dip karti hai, isliye tum gaps ki jagah ek faint pulsing sunate ho. Beat rate phir bhi

∣ f_{1} - f_{2} ∣

hi hoti hai.

Connections

Beats — derivation, applications — parent; yahan har trap wahan ke ek step ko refine karta hai.
Superposition Principle — $y_{1} + y_{2}$ add karne ki permission (WQ7).
Interference of Waves — beats time mein interference hain (TF4).
Standing Waves — same algebra, opposite-direction waves (SE6).
Doppler Effect — woh shift jo radar beat ki tarah read hoti hai (WQ5).
Simple Harmonic Motion — $f = \frac{1}{2 π} k / m$ wax argument (SE3).
Amplitude Modulation — identical carrier-times-envelope structure (WQ6).