Beats — derivation, applications — Physics notes

What is a beat?

WHAT we need: $|f_1 - f_2|$ should be small (typically $< 10$ Hz) so the ear can resolve the slow envelope. If the difference is large, you just hear two separate notes.

Derivation from first principles

By the superposition principle the displacements add: $y = y_1 + y_2 = a\cos(2\pi f_1 t) + a\cos(2\pi f_2 t)$

Why this step? Sound waves obey linear superposition — the medium's response just sums.

Use the identity $\cos C + \cos D = 2\cos\!\frac{C-D}{2}\cos\!\frac{C+D}{2}$ :

$y = 2a\,\underbrace{\cos\!\Big(2\pi \tfrac{f_1-f_2}{2}\,t\Big)}_{\text{slow envelope}}\;\underbrace{\cos\!\Big(2\pi \tfrac{f_1+f_2}{2}\,t\Big)}_{\text{fast audible tone}}$

From envelope to beat frequency

That extra factor of 2 is the heart of the derivation — miss it and you halve the answer.

Figure — Beats — derivation, applications

Worked examples

Worked example Example 2 — The sign ambiguity (loading a fork)

A fork of unknown frequency gives 3 beats/s with a 384 Hz standard. The unknown could be 381 or 387. How to decide?

Step: Put a tiny blob of wax on the unknown fork → its frequency decreases.
Why? Wax adds mass, lowering $f = \frac{1}{2\pi}\sqrt{k/m}$ .
Observe: If beats increase (3→4...), unknown was above 384 → it was 387 (moving further away after decreasing? no — moving toward and past). Reason carefully:
- If unknown $=387$ : loading lowers it toward 384 → beats decrease.
- If unknown $=381$ : loading lowers it away from 384 → beats increase.
Answer: Beats decrease ⇒ 387 Hz; beats increase ⇒ 381 Hz. The wax trick resolves the $\pm$ sign that $|f_1-f_2|$ hides.

Recall Feynman: explain to a 12-year-old

Imagine two friends clapping at almost the same speed. At first they clap together — LOUD. Slowly one gets ahead, so one claps in the gaps of the other — it sounds steady/quiet. Then they line up again — LOUD again. That swelling "loud-quiet-loud" is a beat. The more different their clapping speeds, the more often they line up and split apart — that's why a bigger frequency difference gives faster beats. The number of "loud moments" per second equals exactly how many extra claps one friend does per second over the other.

Applications

Tuning musical instruments: Tune a string against a reference until beats vanish ( $f_{beat}=0 \Rightarrow$ exact match).
Determining an unknown frequency: $f_{unknown} = f_{known} \pm f_{beat}$ ; resolve the sign with the wax/file trick (Example 2).
Detecting gas leaks / "beat-frequency oscillators" (heterodyne): mixing a signal with a reference and reading the beat is the basis of radio tuning and Doppler-shift measurements.
Speed/Doppler radar: the beat between transmitted and reflected frequencies gives the target's speed.

Flashcards

What condition on the two frequencies is needed to hear beats?

They must be slightly different (

|f_1-f_2|

small, typically < ~10 Hz) so the ear resolves the slow envelope.

State the beat frequency formula.

f_{beat}=|f_1-f_2|

What frequency do you actually hear as the pitch?

The average,

f_{avg}=\frac{f_1+f_2}{2}

Why is the beat frequency

|f_1-f_2|

and not

\frac{|f_1-f_2|}{2}

Loudness depends on

|A(t)|

(or

A^2

);

|\cos|

has two maxima per envelope cycle, doubling the rate.

Which trig identity is used in the derivation?

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

A fork gives 3 beats/s with a 384 Hz standard; on adding wax beats decrease. Find the unknown.

387 Hz (wax lowers it toward 384, reducing beats).

Why does loading a fork with wax lower its frequency?

Added mass increases

m

f=\frac{1}{2\pi}\sqrt{k/m}

, decreasing

f

Name two practical applications of beats.

Tuning instruments (zero beats = match) and finding an unknown frequency / heterodyne radio & Doppler radar.

Time between successive silences for 6 Hz beats?

1/6\,\text{s}\approx 0.167

s (one beat period).

Connections

Superposition Principle — beats are a direct consequence.
Interference of Waves — beats are interference in time (vs. spatial interference patterns).
Standing Waves — same superposition algebra, but two opposite-direction equal waves.
Doppler Effect — supplies the frequency shift read out as a beat in radar.
Simple Harmonic Motion — wax-loading argument uses $f=\frac1{2\pi}\sqrt{k/m}$ .
Amplitude Modulation — same envelope mathematics in radio.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Beats ka idea bahut simple hai: jab do sound waves ki frequency thodi si alag ho (jaise 256 Hz aur 260 Hz), to dono ko ek saath bajaने par tumhe ek hi note sunai deta hai, lekin uski loudness baar-baar badhti-ghatati hai — "waah... waah... waah". Yeh loudness ka swelling-fading hi beat kehlata hai. Reason: kabhi dono waves ke crest milte hain (LOUD, constructive), kabhi crest-trough milta hai (silence, destructive).

Derivation me hum $y_1=a\cos(2\pi f_1 t)$ aur $y_2=a\cos(2\pi f_2 t)$ add karte hain, aur $\cos C+\cos D$ wali identity lagate hain. Result aata hai: ek fast part jiski frequency average $\frac{f_1+f_2}{2}$ hai (yeh pitch jo tum sunte ho), aur ek slow envelope $2a\cos(2\pi\frac{f_1-f_2}{2}t)$ jo loudness ko control karta hai. Yaad rakho — loudness $|A|$ par depend karti hai, isliye $|\cos|$ ek cycle me do baar peak karta hai, aur beat frequency $=|f_1-f_2|$ ban jaati hai, $\frac{f_1-f_2}{2}$ nahi. Yeh factor-2 wali baat exam me trap hoti hai.

Applications strong hain: instrument tuning me string ko tab tak adjust karo jab tak beats zero na ho jayein — tab frequency exact match hai. Unknown fork ki frequency nikalni ho to $f_{known}\pm f_{beat}$ ; aur $+$ ya $-$ decide karne ke liye fork par thoda wax lagao (mass badhne se frequency ghatti hai) aur dekho beats badhe ya ghate. Radio tuning aur Doppler radar bhi isi "heterodyne" beat principle par chalte hain. Tiny frequency difference ko ear-countable rhythm bana dena — yahi beats ki real power hai.

Beats — derivation, applications

What is a beat?

Derivation from first principles

From envelope to beat frequency

Worked examples

Applications

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Oscillations & Waves

Connections