Sound intensity — decibels (logarithmic scale): Foundations

1.6.23 · D1 · Physics › Oscillations & Waves › Sound intensity — decibels (logarithmic scale)

Is page par assume kiya gaya hai ki tumhe kuch nahi pata. the decibel note mein jo bhi letter ya squiggle use hua hai, har ek ko neeche unpack kiya gaya hai, us order mein jisme har ek apne pehle waale par lean kar sake.

1. Energy, power, aur watt (bilkul bottom se)

Picture: socho ek bucket mein paani bhar rahe ho. Total paani energy hai. Tap kitni tezi se chalta hai — litres per second — yeh power hai. Ek ghante mein dheeri trickle aur ek minute mein tez gush same paani deliver kar sakte hain; woh power mein differ karte hain.

Topic ko iska kyun zaroorat hai: ek loud sound total energy ke baare mein nahi hai, yeh is baare mein hai ki har second kitni energy aati hai. Yeh "per second" exactly power hai. Parent note mein jo bhi $P$ dikhta hai woh watts mein measured hai. Dekho Wave energy and power ki wave ki power kahan se aati hai.

Recall Quick check

Agar ek source har 2 seconds mein 6 J sound energy deliver karta hai, toh uski power $P$ kya hai? ::: $P = 6/2 = 3 W$ .

2. Area, aur "per unit area" ( $/ A$ ka idea)

Picture — kyun hum power ko area se divide karte hain: baarish mein apna haath uthao. Tumhare poore haath se takraane wale paani ki rate power jaisi hai. Lekin yeh kitna bhaari lagta hai yeh depend karta hai ki woh rate skin par kitni spread hai — yeh rate per square metre hai. Ek bada umbrella same baarish ko patli spread mein pakadta hai; ek thimble ek concentrated fierce trickle pakadta hai.

Figure — Sound intensity — decibels (logarithmic scale)

Topic ko iska kyun zaroorat hai: intensity power per unit area hai, isliye formula $I = P / A$ kuch bhi matlab rakhne se pehle tumhe area se divide karne mein comfortable hona chahiye.

3. Intensity $I = P / A$ (pehle do ko assemble karna)

Picture: same $3 W$ sound $1 m^{2}$ par deti hai toh $I = 3 W/m^{2}$ ; $6 m^{2}$ par spread ho toh sirf $0.5 W/m^{2}$ milta hai. Same power, different intensity, kyunki area change ho gayi.

Topic ko iska kyun zaroorat hai: $I$ woh quantity hai jo decibel scale measure karti hai. Iske baad jo bhi aata hai woh ek $I$ ka doosre se comparison hai. Energy khud vibrating air se aati hai; dekho Sound waves — pressure & displacement ki actually kya move ho raha hai.

4. Sphere aur inverse-square law ( $4 π r^{2}$ )

Ab hum jaannte hain ki jab tum source se door jaate ho toh $I$ kaise change hoti hai. Yahan sphere ki geometry aati hai.

Picture — kyun sphere at all: ek tiny "point" source sound ko har direction mein equally pump karta hai. Kuch time baad, woh energy ek expanding ball ki skin par hoti hai jo source par centred hoti hai. Source ki saari power $P$ us skin par smear hoti hai.

Toh distance $r$ par intensity hai $I = \frac{P}{4 π r ^{2}} .$

Topic ko iska kyun zaroorat hai: parent note ka worked example (d) ek speaker ko $2 m$ se $8 m$ le jaata hai. Decibels touch karne se pehle intensity ratio find karne ke liye tumhe zaroor $I \propto 1/ r^{2}$ pata hona chahiye.

Recall Quick check

Ek source $3 m$ par $I$ read karta hai. $9 m$ par $I$ ka kya fraction read karega? ::: distance ×3 ⇒ intensity ÷ $3^{2} = 9$ , toh $I /9$ .

5. Powers of ten aur exponents ( $1 0^{- 12}$ , $1 0^{10}$ )

Parent note $1 0^{- 12}$ jaise numbers mein duba hua hai. Yeh hai inka matlab.

Do rules jo tum actually use karte ho:

Multiply karne par exponents add hote hain: $1 0^{a} \times 1 0^{b} = 1 0^{a + b}$ .
Divide karne par subtract hote hain: $\frac{1 0 ^{a}}{1 0 ^{b}} = 1 0^{a - b}$ .

Topic ko iska kyun zaroorat hai: threshold intensity $I_{0} = 1 0^{- 12} W/m^{2}$ hai, aur har intensity ratio powers of ten ka clash hai. Agar tum apne dimaag mein exponents subtract nahi kar sakte, toh decibel formula opaque hi rahega.

6. Logarithms — show ka star ( $lo g_{10}$ )

Powers of ten bade numbers banate hain; logarithms unhe count karke wapas laate hain.

Picture — ek ruler jahan har tick ×10 hai:

Ek ordinary ruler par, steps add hote hain (1, 2, 3). Ek log ruler par, steps multiply hote hain (1, 10, 100, 1000) phir bhi equally spaced positions 0, 1, 2, 3 par aate hain. Log us position ko read off karta hai. Isliye ek trillion-fold range ek tidy 0-to-12 spread mein collapse ho jaati hai.

Common mistake Log sirf positive, dimensionless numbers "khata" hai

Kyun care karein: $lo g_{10} (x)$ sirf $x > 0$ ke liye defined hai — $10$ ki koi bhi power $0$ ya negative number nahi deti. Aur $x$ ek pure number hona chahiye (koi units nahi), kyunki " $10$ to the power of $3 W/m^{2}$ " nonsense hai. Is topic mein fix: hum hamesha log ko ratio $I / I_{0}$ khilaate hain, jo positive hai (intensities kabhi negative nahi hoti) aur dimensionless hai (units cancel ho jaate hain). Toh log hamesha happy hai. Edge case — negative dB: agar $I$ reference se quieter hai, toh $I / I_{0} < 1$ hai, aur 1 se chhote number ka log negative hota hai (e.g. $lo g_{10} (0.1) = - 1$ ). Toh $β$ negative ho sakta hai — $0 dB$ threshold se fainter sound, say, $- 5 dB$ read karta hai. Negative dB bilkul valid hai, iska bas matlab hai "reference se below".

Recall Quick check

$lo g_{10} (1 0^{6})$ kya hai? ::: $6$ — kyunki $10$ ko power $6$ par $1 0^{6}$ milta hai.

7. Ratios, reference $I_{0}$ , aur "dimensionless"

Picture: $I_{0} = 1 0^{- 12} W/m^{2}$ woh faintest sound hai jo ek healthy kaan detect kar sakta hai — loudness ki "zero line". Har sound ko us baseline ke multiple ke roop mein quote kiya jaata hai, toh 0 dB ka matlab hai "sabse quietest audible sound ke barabar", "koi sound nahi" nahi.

Topic ko iska kyun zaroorat hai: decibel formula $β = 10 lo g_{10} (\frac{I}{I _{0}})$ log mein ratio $I / I_{0}$ feed karta hai. Reference skip karo toh tum units wali cheez ka log le rahe hoge — meaningless.

8. Decibel unit aur Greek letters ( $β$ , $Δ$ )

Topic ko iska kyun zaroorat hai: difference rule $Δ β = 10 lo g_{10} (I_{2} / I_{1})$ do situations compare karne ke liye $Δ$ use karta hai (do distances, do violins ki sankhya), aur har answer dB mein stamped hota hai. $Δ$ ko "gap between" aur dB ko "level ka unit" samajhna zaroori hai.

Prerequisite map

Ise top-down padho: joules watts banate hain, watts aur area intensity banate hain; alag se, powers of ten logarithm banate hain; dono streams decibel par milte hain.

Equipment checklist

Khud test karo — tumhe har ek ka jawaab parent note padhne se pehle dena aana chahiye.

Power ko kaunse letter se denote karte hain, aur uska unit kya hai?

P

; the watt, jisme

1 W = 1 J/s

"Per unit area" tumhe kya karne ko kehta hai, words mein?

Quantity ko us area se divide karo jis par woh spread hai, har square metre ke liye ek amount deta hai.

Intensity ki definition aur uska unit likho.

I = P / A

W/m^{2}

mein measured.

Radius

r

ke sphere ka surface area kya hota hai?

A = 4 π r^{2}

Agar point source se distance double ho jaye, toh intensity ka kya hota hai?

Yeh quarter ho jaati hai (

I \propto 1/ r^{2}

Exponent rules use karke

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

evaluate karo.

1 0^{- 2 - (- 12)} = 1 0^{10}

lo g_{10} (x)

kaunsa sawaal answer karta hai?

"10 ko kis power se

x

milta hai?"

Woh log rule batao jo decibels ko kaam karta hai.

lo g_{10} (a \times b) = lo g_{10} (a) + lo g_{10} (b)

— multiplication addition ban jaata hai.

Log ka argument

I / I_{0}

positive aur dimensionless kyun hona chahiye?

Log zero ya negatives ke liye undefined hai, aur "10 to the power of a number-with-units" meaningless hai, isliye units cancel hone chahiye.

β

ki negative value (negative dB) ka kya matlab hai?

Sound reference

I_{0}

se fainter hai, toh

I / I_{0} < 1

aur uska log negative hai.

Level

β

ka unit kya hai, aur yeh humein kya yaad dilaata hai?

Decibel (dB); yeh flag karta hai ki

β

ek level hai

I_{0}

se compared, raw intensity nahi.

Δ β

mein symbol

Δ

ka kya matlab hai?

"Change in" ya difference

β_{2} - β_{1}