Sound intensity — decibels (logarithmic scale)

1.6.23 · Physics › Oscillations & Waves

1. Intensity kya hai? (pehle foundation banao)

YE definition kyun? Sound energy carry karti hai. Us energy ko bade area pe spread karo toh har square metre ko thoda share milega (quiet); chhote area pe concentrate karo toh har metre ko zyada milega (loud). Intensity energy flow density measure karti hai.

2. Decibel ko scratch se derive karna

Hume ek aisa number chahiye jo:

har baar intensity ×10 hone pe ek fixed step se badhe,
hearing ke threshold pe 0 se shuru ho.

Step 1 — bel. Pucho "10 ke kitne factors $I$ , $I_{0}$ se upar hai?". Wo number exactly hai $β_{bel} = lo g_{10} (\frac{I}{I _{0}}) .$ Log kyun? Kyunki $lo g (I / I_{0})$ powers of ten count karta hai: agar $I = 1000 I_{0}$ hai toh $lo g_{10} (1000) = 3$ . Har ×10 se 1 add hota hai.

Step 2 — deci. Ek bel ek coarse step hai (pura factor of 10). Hume finer resolution chahiye, isliye hum tenths of a bel = decibels use karte hain. 10 se multiply karo:

ISKO kaise padhein: $β$ ek level hai ( $I_{0}$ se comparison), intensity nahi. Ye dimensionless hai lekin hum "dB" lagaate hain convention yaad rakhne ke liye.

3. Killer property: har ×10 se 10 dB add hota hai

Maano intensity $10 I$ ho jaaye. Toh $β^{'} = 10 lo g_{10} \frac{10 I}{I _{0}} = 10 [lo g_{10} 10 + lo g_{10} \frac{I}{I _{0}}] = 10 (1) + β = β + 10.$

Figure — Sound intensity — decibels (logarithmic scale)

4. Worked examples

5. Common mistakes (steel-manned)

6. Quick reference table

Source	$I$ (W/m²)	$β$ (dB)
Threshold of hearing	$1 0^{- 12}$	0
Whisper	$1 0^{- 10}$	20
Conversation	$1 0^{- 6}$	60
Busy street	$1 0^{- 4}$	80
Jackhammer	$1 0^{- 2}$	100
Pain / damage	$1$	120

Notice karo: har row $I$ mein powers of ten se jump karta hai lekin sirf +20 dB. Yahi compression hai.

Recall Feynman: ek 12-saal ke bachhe ko explain karo

Sound ko rain ki tarah imagine karo jo tumhare haath pe pad rahi hai. Drizzle hai "1 drop", downpour hai "ek million drops". Agar tum ye numbers ek hi chart pe likho, drizzle invisible ho jayegi. Toh drops ki sankhya likhne ki jagah, tum kitne zeros hain wo likhte ho. Decibel basically "zeros count karo, phir ×10" hai. 0 dB = barely-there drizzle, 120 dB = rain ki wall jo hurt karti hai. dB number mein 10 add karne ka matlab hai das guna zyada rain, sirf thoda zyada nahi.

Flashcards

Sound intensity ka SI unit kya hai?

watts per square metre,

W/m^{2}

Sound intensity define karo.

Wave dwara carry ki gayi power per unit area,

I = P / A

Decibel level formula likho.

β = 10 lo g_{10} (I / I_{0})

jahan

I_{0} = 1 0^{- 12} W/m^{2}

Reference intensity

I_{0}

kya hai aur ye kya represent karta hai?

1 0^{- 12} W/m^{2}

; 1 kHz pe human hearing ka threshold.

Agar intensity 10 se multiply ho toh level kitne dB change hoga?

+10 dB.

Agar intensity double ho toh level kitne dB change hoga?

lagbhag +3 dB (

10 lo g_{10} 2

Sound ke liye logarithmic scale kyun use karte hain?

Audible intensity range ~

1 0^{12}

tak failti hai, isliye log ise manageable 0–120 scale mein compress karta hai aur roughly match karta hai ki loudness kaise perceive hoti hai.

60 dB ke do equal sources milke kya level dete hain?

63 dB (intensities add hoti hain → ×2 → +3 dB), 120 dB NAHI.

Point source se distance ke saath intensity kaise vary karti hai?

I = P / (4 π r^{2}) \propto 1/ r^{2}

2 m pe ek sound 80 dB read karta hai; 8 m pe kya hoga (free field)?

r

×4 →

I

÷16 →

- 10 lo g_{10} 16 \approx - 12

dB → 68 dB.

60 dB ko intensity mein convert karo.

I = I_{0} 1 0^{6} = 1 0^{- 6} W/m^{2}

Intensity ke liye

10 lo g_{10}

kyun hai par pressure ke liye

20 lo g_{10}

Kyunki

I \propto p^{2}

, toh

lo g I = 2 lo g p

, prefix double ho jaata hai.

120 dB, 60 dB se kitni guna zyada intense hai?

1 0^{(120 - 60) /10} = 1 0^{6}

= ek million times.

Difference rule

Δ β = 10 lo g_{10} (I_{2} / I_{1})

derive karo.

β_{2} - β_{1}

subtract karo;

lo g I_{0}

terms cancel ho jaate hain,

10 lo g_{10} (I_{2} / I_{1})

bacha rehta hai.

Connections

Sound waves — pressure & displacement (intensity $\propto$ amplitude²)
Inverse-square law for radiation (jahan $1/ r^{2}$ aata hai)
Logarithms and exponentials (scale ka math engine)
Loudness vs intensity — psychoacoustics (perception ≈ log of stimulus, Weber–Fechner)
Wave energy and power ( $I = P / A$ mein $P$ define karta hai)
Doppler effect (companion sound-wave phenomenon)