Mach number M = V - a — subsonic ( - 1), transonic (~1), supersonic ( - 1), hypersonic ( - 5): Foundations

3.1.4 · D1 · Physics › Compressible Flow & Aerodynamics › Mach number M = V - a — subsonic ( - 1), transonic (~1), sup

Iससे pehle ki hum formula $M = V / a$ ko padh sakein, hamen isme har symbol ko earn karna hoga — aur har woh symbol jo uske peeche chhupa hua hai. Yeh page toolbox hai. Yahan kuch bhi assume nahi kiya gaya ki tumne parent topic dekha hai; yeh build karta hai jo parent Mach Number note pe rely karta hai.

1. Speed — "kitna tez" ka matlab kya hota hai

Picture: ek arrow jis ki length batati hai ki clock ke ek tick mein tum kitna door jaate ho. Lamba arrow = tez.

Yeh topic iske kyun kaam aata hai: $V$ Mach fraction ke upar hai — woh cheez jise hum compare karte hain messenger ki speed se. "Kitna tez" ka number na ho toh compare karne ke liye kuch bhi nahi.

Figure — Mach number M = V - a — subsonic ( - 1), transonic (~1), supersonic ( - 1), hypersonic ( - 5)

2. Messenger — sound aur uski speed $a$

Picture: pond mein kankad daalo — bahar ki taraf failta hua ring woh message hai jo speed $a$ par travel karta hai. Hawa mein yeh "ring" thodi-si squeezed gas ka ek sphere hoti hai.

Yeh topic iske kyun kaam aata hai: $a$ Mach fraction ke neeche hai. Yeh fluid mein information ki speed limit set karta hai. Shocks ki poori baat " $V$ ne $a$ ko punch mein beat kiya" ke baare mein hai.

3. Temperature $T$ — woh number jo $a$ set karta hai

Picture: buzzing dots ka ek jar. Hot = dots tezi se oodti hain aur zyada takraati hain; cold = dots dhheere chalti hain aur kam takraati hain.

Yeh topic iske kyun kaam aata hai: tez molecules pressure-squeeze ko jaldi pass karte hain, isliye garam gas mein sound tezi se travel karta hai. Kyunki $a$ $T$ par ride karta hai, wahi plane speed alag altitudes par alag Mach numbers deti hai — yahi parent ke Example 2 ka poora point hai.

4. Square root — square ko undo karna

Yeh tool kyun aur koi doosra nahi? Sound wave ki physics (parent mein derive ki gayi) $a^{2} = γ R T$ par aake rukti hai — jo quantity naturally nikaalti hai woh $a$ squared hai. $a$ khud paane ke liye hamen squaring ko undo karna hoga, aur squaring undo karne ka exact tool square root hai. Isliye $a = γ R T$ hai aur kuch nahi.

Yaad rakhne wala consequence: kyunki $a \propto T$ hai, sound speed ko double karne ke liye temperature ko quadruple karna padega. Temperature ka $a$ par gentle grip hai.

5. Gas constants $γ$ aur $R$

Picture: gas ko springs ka ek mattress samjho. $γ$ batata hai har spring kitna stiff hai; $R$ batata hai springs kis cheez se bane hain. $T$ ke saath milkar (woh kitne vibrate ho rahe hain) yeh sound speed fix karte hain.

Yeh topic inke kyun kaam aata hai: yeh gas ki state ( $T$ ) ko ek concrete sound speed $a = γ R T$ mein badalte hain. Inke bina, "hotter = faster sound" ek slogan hi rehta number ki jagah.

6. Ratio — ek speed ko doosri se divide karna

Ratio kyun, difference $V - a$ kyun nahi? Difference units (m/s) carry karta aur alag gases mein alag matlab hota. Ek ratio woh ek hi sawaal ka jawaab deta hai jo physically matter karta hai — "kya tumne messenger ko beat kiya ya nahi?" — ek clean number ke saath. $M = 1$ ek tie hai, chahe gas ya temperature kuch bhi ho.

Number padhna:

$M < 1$ : sound se slower → fluid ko waqt par warning milti hai (subsonic).
$M = 1$ : exact tie (sonic).
$M > 1$ : message se aage nikal gaye → shocks form hote hain (supersonic).

7. Angle $μ$ aur trig tools $sin$ , $arcsin$

Jab $V > a$ hota hai toh wavefronts ek cone mein pile ho jaate hain, aur uski sharpness ek angle se measure hoti hai.

$sin$ yahan kyun? Time $t$ mein body $V t$ travel karti hai (hypotenuse) jabki uski emit ki gayi wave ka radius $a t$ ho jaata hai (half-angle ke opposite side). Opposite/hypotenuse ka ratio exactly $sin μ = a t / V t = 1/ M$ hai. Geometry hume ek right triangle de rahi hai, isliye woh tool jo right triangles se angles padhta hai — $sin$ — natural choice hai.

Limits seedhe rakho:

$M = 1$ par: $sin μ = 1$ , isliye $μ = 90°$ — path ke perpendicular sound ki ek flat wall.
Jab $M \to \infty$ : $1/ M \to 0$ , isliye $μ \to 0°$ — body se chipka ek needle-thin cone.

Yeh sab Oblique Shocks & Mach Cone mein aur gehraee se samjhaya gaya hai.

8. Differential nudge $d ()$

Parent ki derivation $d p$ , $d ρ$ , $d a$ likhti hai. Yeh typos nahi hain.

Picture: ek pahaadi par khade ho aur ek chhota sa step lo. $d x$ hai kitna east tum stepped; tumne jo thoda sa utha paaya woh $d y$ hai. Do chhote steps multiply ( $d x d y$ ) negligibly tiny hain — ek speck ka speck.

Yeh topic iske kyun kaam aata hai: sound ek weak disturbance hai — sirf extra pressure ki ek whisper. Ise tiny changes ( $d p$ , $d ρ$ ) se model karna hi exactly woh reason hai ki messy conservation laws clean $a^{2} = d p / d ρ$ mein collapse ho jaate hain. Yeh calculus ka woh seed hai jo tum Isentropic Flow Relations mein phir use karoge.

9. Density $ρ$ aur pressure $p$ — squishing pair

Yeh topic inke kyun kaam aata hai: "kya gas squish hoti hai?" ki poori kahani $ρ$ ke change hone ke baare mein hai. $M \approx 0.3$ se neeche density practically nahi hilti aur hum pretend kar sakte hain ki yeh constant hai (incompressible); uske upar, $ρ$ itna change karta hai ki matter karta hai — Compressibility & Bernoulli's Limits aur Prandtl–Glauert Correction dekho.

Prerequisite map

Equipment checklist

Pehle har jawaab predict karo, phir reveal karo.

Symbol

V

kya represent karta hai, aur uski units kya hain?

Flow ya vehicle ki speed, metres per second (m/s) mein.

Speed of sound

a

physically kya hai?

Woh speed jis par fluid mein ek chhota pressure disturbance (the "message") travel karta hai.

a

temperature

T

par kyun depend karta hai?

Garam gas = tez molecules = collisions pressure pulse ko jaldi pass karte hain, isliye sound tezi se travel karta hai.

kya undo karta hai, aur yeh

a = γ R T

mein kyun hai?

Yeh squaring ko undo karta hai; physics

a^{2}

deti hai, isliye hum

a

khud recover karne ke liye root lete hain.

Kya

γ

kisi units mein measure hota hai?

Nahi — yeh ek pure ratio

c_{p} / c_{v} \approx 1.4

hai air ke liye.

M

ek ratio kyun hai na ki difference

V - a

Ratio ek pure number hai, isliye

M = 1

matlab tie hai kisi bhi gas mein kisi bhi temperature par; difference units carry karta aur matlab badal jaata.

M = 0.74

ko words mein padho.

Body local sound speed ke

74%

par move kar rahi hai — subsonic, fluid ko waqt par warning mil jaati hai.

sin μ = \frac{opposite}{hypotenuse}

ka Mach cone ke liye kya matlab hai?

Opposite = wave radius

a t

, hypotenuse = travel ki gayi distance

V t

, isliye

sin μ = 1/ M

arcsin

kya karta hai?

Yeh poochta hai "kaunse angle ka yeh sine hai?" — ratio

1/ M

ko wapas angle

μ

mein badalte hue.

d p

mein little-

d

ka kya matlab hai, aur hum

d ρ d a

kyun drop kar sakte hain?

p

mein ek chhoti change"; do chhoti changes ka product negligibly small hota hai, isliye hum ise discard karte hain.

Kis Mach number se neeche hum air ko incompressible treat karte hain, aur kyun?

Around

M \approx 0.3

, kyunki fractional density change sirf kuch percent hoti hai.