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

3.1.4 · Physics › Compressible Flow & Aerodynamics

Mach number KYA hai?

Speed of sound khud koi fixed number nahi hai; ek ideal gas ke liye yeh sirf temperature par depend karta hai: $a = γ R T$ jahan $γ = c_{p} / c_{v}$ (specific heats ka ratio, air ke liye $\approx 1.4$ ), $R$ specific gas constant hai (air ke liye $287 J/(kg\cdotK)$ ), aur $T$ kelvin mein absolute temperature hai.

Speed of sound KAISE derive karte hain? (first principles se)

Formula

$a = γ R T$ derive karna Derive kyun karein? Taaki $M$ ek black box na rahe — $a$ ek choti disturbance ki physics se aata hai.

Step 1 — Wave par baith jao. Ek weak pressure pulse ko consider karo jo still gas (pressure $p$ , density $ρ$ ) mein speed $a$ se move kar rahi hai. Wave ke reference frame mein jump karo: ab gas wave mein $a$ speed se andar flow kati hai aur $a - d a$ speed se bahar, properties $p + d p$ , $ρ + d ρ$ ke saath. Yeh step kyun? Ek steady frame simple conservation laws use karne deta hai instead of time-dependent waalon ke.

Step 2 — Mass ka conservation wave ke across (area constant): $ρ a = (ρ + d ρ) (a - d a)$ Expand karo aur second-order term $d ρ d a$ drop karo: $ρ d a = a d ρ \Rightarrow d a = \frac{a}{ρ} d ρ$ Yeh step kyun? Mass create nahi ho sakta; yeh velocity change ko density change se link karta hai.

Step 3 — Momentum ka conservation (force = rate of momentum change), phir se steady, thin wave: $p - (p + d p) = ρ a [(a - d a) - a]$ $- d p = - ρ a d a \Rightarrow d p = ρ a d a$ Yeh step kyun? Net pressure force gas ko accelerate karta hai; pressure change ko velocity change se jodata hai.

Step 4 — Combine karo. $d a = \frac{a}{ρ} d ρ$ ko $d p = ρ a d a$ mein substitute karo: $d p = ρ a \cdot \frac{a}{ρ} d ρ = a^{2} d ρ \Rightarrow a^{2} = \frac{d p}{d ρ}$ Yeh step kyun? Yeh fundamental result hai: sound speed² = gas kitna stiff hai (density ke response mein pressure).

Step 5 — Sound adiabatic + reversible (isentropic) hoti hai. Sound wave itni tezi se oscillate karti hai ki heat leak nahi ho pati, isliye $p / ρ^{γ} = const$ , jo deta hai $\frac{d p}{d ρ} = γ \frac{p}{ρ}$ . Ideal gas $p = ρR T$ ke saath: $a^{2} = γ \frac{p}{ρ} = γ R T \Rightarrow a = γ R T$ Yeh step kyun? Newton ne originally isothermal use kiya tha ( $a = p / ρ$ ) aur ~18% kam value mili — Laplace ne yeh realize karke fix kiya ki process adiabatic hai, isothermal nahi.

Chaar flow regimes

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

Compressibility $M \sim 0.3$ ke paas kyun "switch on" hoti hai

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Quick self-test (answers chhupao, pehle predict karo)

Q: $M$ kaunsi do speeds se banta hai? → $V$ (flow) ko $a$ (local sound speed) se divide karo.
Q: $a$ , $T$ par kyun depend karta hai? → Zyada garam gas, molecules ki tezi collisions pressure pulse transmit karti hain.
Q: Physically kya hota hai jab $M > 1$ ? → Body apne pressure signals ko outrun kar leti hai → shock waves / Mach cone.
Q: Mach angle formula? → $sin μ = 1/ M$ .
Q: Transonic ( $M \approx 1$ ) mushkil kyun hota hai? → Coexisting subsonic aur supersonic pockets ke saath body par local shocks.
Q: "Compressible" ke liye threshold? → ~ $M = 0.3$ ( $d ρ / ρ \sim \frac{1}{2} M^{2} \approx 5%$ ).

Recall Feynman: 12-saal ke bacche ko explain karo

Imagine karo tum daud rahe ho aur crowd ko "watch out!" chilla rahe ho. Jab tum apni awaaz se slower daudo, log time par sunte hain aur side ho jaate hain — smooth (subsonic). Jab tum apni awaaz ke bilkul barabar daudo, warning mushkil se tumse pehle pahunchti hai — chaotic (transonic). Jab tum apni awaaz se tezi daudo, tum logon se takraa jaate ho jo kabhi sunte hi nahi the — woh "takkar" ek shock wave hai (supersonic). Mach number bas measure karta hai tumhari running speed tumhari shouting speed se kaise compare hoti hai. Aur thande din par sound slower chalti hai, isliye same running speed "zyada dangerous" ho jaati hai — zyada Mach.

Flashcards

Mach number kya hai?

Local flow speed aur local speed of sound ka ratio,

M = V / a

(dimensionless).

Ideal gas mein speed of sound?

a = γ R T

— sirf absolute temperature par depend karta hai.

Sound speed

γ R T

kyun hai aur

p / ρ

kyun nahi?

Sound waves adiabatic hoti hain (heat transfer ke liye time nahi), isliye

d p / d ρ = γ p / ρ

, jo

γ

factor add karta hai (Laplace ka correction).

a^{2} = d p / d ρ

derive karo.

Thin wave ke across mass + momentum dete hain

d a = (a / ρ) d ρ

aur

d p = ρ a d a

; combine karne par milta hai

d p = a^{2} d ρ

Subsonic regime ki range aur feature?

M < 1

(≈

< 0.8

): smooth flow, koi shocks nahi, fluid "pre-warned" hai.

Transonic regime aur yeh mushkil kyun hai?

M \approx 0.8

–

1.2

: mixed subsonic aur supersonic pockets ke saath body par local shock waves.

Supersonic regime ka feature?

M > 1

(~5 tak): flow apne pressure signals ko outrun karta hai → shock waves aur Mach cone.

Hypersonic regime?

M > 5

: severe heating, thin shock layers, real-gas effects (dissociation, ionization).

Mach angle formula aur derivation?

sin μ = 1/ M

, envelope cone ki geometry se: wave radius

a t

over travel distance

V t

M = 2

par Mach angle?

μ = arcsin (0.5) = 30°

Mach 1 altitude ke saath alag kyun hota hai?

a = γ R T

; thandi high-altitude air se

a

kam hoti hai, isliye same true airspeed se zyada

M

milta hai.

M \approx 0.3

compressibility threshold kyun hai?

d ρ / ρ \sim \frac{1}{2} M^{2} \approx 5%

wahan; iske neeche density change negligible hai (incompressible).

M = 1

par Mach angle kya hota hai?

90°

— wavefronts ek flat normal wall of sound form karte hain.

Connections

Speed of Sound in Gases — $a = γ R T$ ki derivation $M$ ko underpin karti hai.
Isentropic Flow Relations — $M$ use karta hai $T_{0} / T$ , $p_{0} / p$ , $ρ_{0} / ρ$ relate karne ke liye.
Normal Shock Waves — kya hota hai jab $M > 1$ flow abruptly decelerate hoti hai.
Oblique Shocks & Mach Cone — $sin μ = 1/ M$ se geometry.
Compressibility & Bernoulli's Limits — kyun $M < 0.3$ ⇒ incompressible.
Prandtl–Glauert Correction — lift par transonic compressibility corrections.
Reynolds Number — doosra key dimensionless number (viscous, compressible nahi, effects).