Beats — derivation, applications: Visual walkthrough

1.6.20 · D2 · Physics › Oscillations & Waves › Beats — derivation, applications

Step 1 — Ek wave ek point pe: ek akele squiggle ka matlab

KYA: Pehli wave ko hum likhte hain $y_{1} = a cos (2 π f_{1} t)$

Is line ke har symbol ko earn karte hain:

$y_{1}$ — displacement: hawa kitni door rest se push hui, upar ya neeche.
$a$ — amplitude: sabse bada push, wiggle ki crest ki height.
$t$ — time, neeche se tick karta hua.
$f_{1}$ — frequency: har second mein kitne complete up-down wiggles hote hain (unit: hertz, Hz).
$cos$ — ek machine jo ek steadily-growing number ko $+ 1$ aur $- 1$ ke beech smooth back-and-forth mein badal deta hai. Hum $cos$ use karte hain (straight line nahi) kyunki hawa sach mein wapas apni jagah aati hai aur repeat karti hai — cosine sabse simple "returns and repeats" curve hai.
$2 π f_{1} t$ — cosine ko diya jaane wala angle. $2 π$ kyun? Kyunki cosine ek full trip tab complete karta hai jab uska angle $2 π$ se barhta hai. $f_{1} t$ se multiply karne ka matlab: time $t = \frac{1}{f _{1}}$ (ek period) ke baad, angle exactly $2 π$ hai → exactly ek full wiggle. $2 π$ ek units-converter hai "wiggles" se "radians jo cosine chahta hai."

KYUN: Hum ek fixed point ko time ke saath track karte hain (space mein snapshot nahi) kyunki beat woh cheez hai jo tum kaan se sunते ho — ek location, seconds ke saath changing loudness.

PICTURE:

Figure — Beats — derivation, applications

Step 2 — Doosri wave: almost same, lekin drift kar rahi hai

Sab kuch $y_{1}$ jaisa hi hai except $f_{1} \to f_{2}$ , jahan $f_{2}$ $f_{1}$ se bahut thoda different hai. Same amplitude $a$ , same start.

Equal amplitude kyun? Taaki jab do waves eventually ek doosre ke exactly oppose karein, to they can cancel to total silence — sabse clean possible effect. Unequal amplitudes bhi kaam karte hain; wo bas kabhi fully cancel nahi hote (ek leftover hum), jo picture ko muddy kar deta hai.

Same start kyun? Taaki $t = 0$ pe dono crests coincide hon — wo perfectly in step shuru hote hain, aur hum dekh sakte hain unhe drift karte hue. (Agar wo in step shuru na hon? Hum use Step 8 mein handle karte hain — spoiler: beat rate nahi badlata, poora envelope bas sideways slide karta hai.)

PICTURE: $y_{1}$ aur $y_{2}$ ko same board pe draw karo. Bilkul left pe wo saath march karte hain; time ke saath, thoda-sa-faster wala aage nudge karta hai jab tak, baad mein, uski crest doosre ki trough ke upar nahi baith jaati.

Step 3 — Superpose: ripples ko add hone do

KYA: $y = y_{1} + y_{2} = a cos (2 π f_{1} t) + a cos (2 π f_{2} t)$

Ise literally padho: har instant pe, blue curve ki height lo, pink curve ki height add karo, us total pe ek dot lagao. Saare dots trace karo → black resultant.

KYUN: Kyunki woh black curve actual motion hai tumhare kaan ke paas hawa ki — jo cheez tumhara eardrum drive karegi. Beats ke baare mein sab kuch uski shape ke andar hai.

PICTURE: Jahan dono crests upar point karein → tall sum (loud). Jahan crest trough se mile → nearly-zero tak cancel (silent). Black curve clearly bulge aur pinch karti hai.

Step 4 — Woh trick jo sum ko suljhati hai

Hamare paas do cosines ka sum hai. Sum mein throbbing dekhna mushkil hai. Hum ek product chahte hain — ek factor fast wiggle ke liye, ek slow swelling ke liye. Ek trig identity exactly yahi convert karti hai:

Yeh tool kyun, doosra nahi? Hum deliberately sum→product identity choose karte hain (na ki expand karna ya Taylor series) kyunki yeh "do cheezein add" ko "do cheezein multiply" mein badal deta hai, aur multiplication exactly woh language hai ek loud tone (ek factor) ki jo up aur down slowly (doosra factor) dial hoti hai. Woh volume-knob structure is a beat.

Yahan $C = 2 π f_{1} t$ aur $D = 2 π f_{2} t$ . Substitute karo: $y = 2 a slow: half the difference cos (2 π \cdot \frac{f _{1} - f _{2}}{2} \cdot t) fast: the average cos (2 π \cdot \frac{f _{1} + f _{2}}{2} \cdot t)$

Dots note karo: division by $2$ sirf frequency $(f_{1} - f_{2})$ ya $(f_{1} + f_{2})$ pe apply hoti hai — yeh $(\frac{f _{1} - f _{2}}{2})$ hai jo $2 π$ aur $t$ se multiply hota hai, kabhi $t /2$ nahi.

Term by term:

$2 a$ — maximum possible amplitude: jab dono waves saath crest karein, heights $a + a = 2 a$ .
$(\frac{f _{1} - f _{2}}{2})$ — frequencies ka half the difference. Kyunki $f_{1} \approx f_{2}$ , yeh tiny hai → bahut slow cosine.
$(\frac{f _{1} + f _{2}}{2})$ — average frequency. Kyunki $f_{1} \approx f_{2}$ , yeh basically original pitch hai → fast cosine.

PICTURE: Fast average-frequency wiggle aur, uske upar, slow half-difference curve ko overlay karo jo uske peaks ko outline ki tarah hug kare.

Step 5 — Do factors ko naam do: pitch vs. envelope

To poora motion padhta hai "ek steady pitch, jo $A (t)$ se slowly up aur down hoti hai." Yeh bilkul same math hai jaise Amplitude Modulation radio mein.

PICTURE: Envelope ek smooth dashed boundary ki tarah draw ki gayi; fast tone uske andar kaampti hai, fat parts mein ise completely fill karti hai aur pinches pe kuch nahi.

Step 6 — Factor of 2: loudness peaks do baar per envelope cycle kyun aati hain

Yeh crux hai — woh ek step jise zyaadatar log galat samajhte hain.

KYA hum poochhte hain: Sound loud kitni baar hota hai? Woh count-per-second jo hum finally beat frequency, $f_{b e a t}$ kehte hain.

KYUN yeh subtle hai: Loudness parwah nahi karti ki envelope positive hai ya negative. Ek bada upar wala bulge loud hai; ek bada neecha wala bulge equally loud hai — hawa ya to upar ya neeche hard swing kar rahi hai. To loudness envelope ki size follow karti hai, yaani $∣ A (t) ∣$ .

Consequence: Ek plain cosine ek cycle mein below zero dip karta hai. Lekin $∣ cos θ ∣$ un negative dips ko positive humps mein fold karke upar le aata hai — to $cos θ$ ke ek cycle ke liye iske do humps hain.

$c o s envelope cycles at \frac{f _{1} - f _{2}}{2} take ∣ \cdot ∣ ∣ c o s ∣ loudness peaks at 2 \times \frac{f _{1} - f _{2}}{2}$

$f_{b e a t} = 2 \times \frac{f _{1} - f _{2}}{2} = ∣ f_{1} - f_{2} ∣$

Term by term: $2$ aata hai " $∣ cos ∣$ do baar peak karta hai" se; yeh exactly $\frac{1}{2}$ ko cancel karta hai jo envelope frequency ke andar chhupa hua hai, aur clean difference nikalti hai. Hum $∣ f_{1} - f_{2} ∣$ lete hain (absolute value) kyunki "3 beats" aur " $- 3$ beats" same cheez ka matlab hai — loud moments ka ek count negative nahi ho sakta.

PICTURE: Upar envelope $cos$ curve; neeche uska fold-up $∣ cos ∣$ — clearly do loud bumps dikhata hai ek envelope wiggle ke span mein, beat period mark ki gayi.

Step 7 — Edge cases: formula ko uski limits tak push karna

Har honest derivation ko apni extremes mein survive karna chahiye. Teen check karne ke liye:

(a) $f_{1} = f_{2}$ (identical forks). Tab $∣ f_{1} - f_{2} ∣ = 0$ : zero beats. Envelope $A (t) = 2 a cos (0) = 2 a$ ek flat constant hai — bilkul koi swelling nahi, bas ek steady doubled-loudness tone. Yeh tuning target hai: musicians tab tak tune karte hain jab tak throb zero nahi ho jaata.

(b) $f_{1} - f_{2}$ bada hona — jahan "beats" beats rehna band ho jaate hain. Jaise gap barhta hai, envelope faster wiggle karta hai. Transition jo tumhara kaan experience karta hai woh gradual hai, sharp cliff nahi, aur teen regimes se guzarta hai:

Few Hz tak — tum clearly individual swells count kar sakte ho: waah… waah. Yeh true beats hain.
Roughly 10–20 Hz aur upar — swells itni fast aati hain ki individually count nahi ho sakti; kaan alag loud-quiet events sunna band kar deta hai aur iske bajaaye ek harsh, buzzy quality perceive karta hai jise musicians roughness kehte hain.
Kaan ki critical bandwidth se aage (frequency spacing jo tumhara inner ear ab blend nahi kar sakta — roughly 10–15% of the tone's frequency, to mid-range pitches ke liye ~30–100 Hz) — do tones do alag smooth pitches mein separate ho jaate hain aur roughness fade ho jaati hai.

To "beats need $∣ f_{1} - f_{2} ∣ ≲ 10$ Hz" ek countable regime ka rule of thumb hai, hard physical constant nahi — yeh roughly wahan mark karta hai jahan tumhara kaan swells ko ek-ek karke track nahi kar sakta. Exact numbers pitch aur listener pe depend karte hain; envelope ki physics (Step 6) poori tarah unchanged rehti hai — sirf tumhara perception shift hota hai.

(c) Unequal amplitudes $a_{1} \neq = a_{2}$ . Pinch ab zero nahi pahunchti — total silence ki jagah ek leftover quiet hum hai. Same rhythm, shallower dips. Beat frequency unchanged hai; sirf fade ki depth shrink hoti hai.

PICTURE: Teen mini-panels side by side: flat (equal freq), ek fast blur (large gap), aur shallow dips (unequal amplitude).

Step 8 — Edge case: agar waves in step shuru NA hon?

Humne assume kiya tha ki dono waves $t = 0$ pe crests saath shuru karte hain. Reality rarely aise karti hai — do forks ek heartbeat apart strike karo aur ek pehle se doosre se aage hai. To har wave ko apna starting phase do:

$ϕ_{1}, ϕ_{2}$ — initial phase: har wave $t = 0$ pe apni wiggle mein kitni door already hai. Phase $0$ ka matlab "crest pe shuru"; phase $π$ ka matlab "trough pe shuru."

Same sum-to-product identity run karo (ab $C = 2 π f_{1} t + ϕ_{1}$ , $D = 2 π f_{2} t + ϕ_{2}$ ): $y = 2 a cos (2 π \cdot \frac{f _{1} - f _{2}}{2} \cdot t + \frac{ϕ _{1} - ϕ _{2}}{2}) cos (2 π \cdot \frac{f _{1} + f _{2}}{2} \cdot t + \frac{ϕ _{1} + ϕ _{2}}{2})$

Nayi cheezein padho: Slow factor ke andar extra $\frac{ϕ _{1} - ϕ _{2}}{2}$ bas ek constant angle mein add kiya gaya hai — yeh $t$ pe depend nahi karta. Cosine ke angle mein ek constant add karne se poori curve sirf left ya right time mein slide hoti hai; yeh kabhi nahi badlata ki curve kitni fast wiggle karti hai.

KYUN yeh matter karta hai: Beat frequency set hoti hai ek kitni fast envelope oscillate karta hai, aur woh speed abhi bhi $\frac{f _{1} - f _{2}}{2}$ → fold → $∣ f_{1} - f_{2} ∣$ hai. Unchanged. Sirf visible effect yeh hai ki pehla loud moment thode different instant pe hota hai. Clapping picture mein: agar do dost thoda out of sync shuru karein, to "loud-quiet-loud" pattern identical hai — bas thoda midway through shuru hota hai.

PICTURE: Same beat pattern Step 5 se, do baar draw ki gayi — ek baar $ϕ_{1} - ϕ_{2} = 0$ ke saath aur ek baar nonzero offset ke saath — envelope ko bodily sideways shifted dikhata hai lekin loud moments ke beech identical spacing ke saath.

Ek-picture summary

Sab ek saath: do near-twin ripples (blue, pink) → unka black sum → dashed envelope apne swell trace karta hua → fold-up $∣ envelope ∣$ loud moments mark karta hua, ek beat-period $1/∣ f_{1} - f_{2} ∣$ apart spaced.

Recall Feynman retelling — poora walkthrough plain words mein

Do air-pushes tumhare kaan tak pahunchte hain, almost same speed pe wiggling. Hawa ka ek point watch karo: akele har wave ek smooth cosine wiggle hai (Step 1–2). Hawa sirf ek jagah ho sakti hai, to uska real motion do wiggles added hai (Step 3) — aur jahan crests agree hain woh tall hai, jahan crest trough se milti hai woh flatten ho jaata hai.

Woh "do wiggles ka sum" clumsy hai, to hum ek trig identity use karte hain ise product ke roop mein rewrite karne ke liye (Step 4): ek fast wiggle average speed pe — woh pitch hai jo tum suno — ek slow wiggle half the difference pe times — yeh ek volume knob hai jo up aur down turn ho raha hai (Step 5).

Yahan woh trick hai jis pe sabka pair phisal jaata hai (Step 6): tumhara kaan ek downward bulge utna hi loud sunta hai jitna ek upward, to yeh slow curve ke negatives ko positives mein upar fold karta hai — har slow cycle ke liye do loud moments deta hai. Woh doubling "half" ko cancel karta hai, aur clean answer nikalti hai: loud moments per second $= ∣ f_{1} - f_{2} ∣$ .

Corners check karo (Steps 7–8): identical forks → koi throb nahi (isi pe tune karo!); jaise forks zyaada different hote hain swells speed up hoti hain jab tak count nahi ho sakti — pehle buzzy roughness, phir do alag notes; mismatched loudness → dips bilkul silence tak nahi pahunchti; aur agar forks out of step shuru hon, to poora throb sirf time mein sideways slide karta hai — same rhythm. Yeh poori kahani hai, aur yeh same math hai jo radios air se ek station nikalne ke liye use karte hain.

Connections

Superposition Principle — Step 3 superposition hai, aur kuch nahi.
Interference of Waves — beats interference hai time mein laid out.
Standing Waves — same sum-to-product algebra, lekin oppositely-travelling waves ke liye.
Amplitude Modulation — Step 5 ka envelope radio carrier ke modulation se identical hai.
Doppler Effect — woh frequency difference supply karta hai jo radar beat ke roop mein read out karta hai.
Simple Harmonic Motion — har single wave (Step 1) ek point pe SHM hai.