Shock waves — Mach number, Mach cone — - CRITICAL for rockets -

1.6.22 · Physics › Oscillations & Waves

Mach number KYA hai?

Ratio KYU? Kyunki physically jo matter karta hai woh raw speed nahi balki kya tum apni ripples ko beat kar paate ho. Sound speed temperature/altitude ke saath change hoti hai, isliye wahi 300 m/s upar jaake supersonic hai aur sea level pe subsonic. Ratio $M$ physics ko directly capture karta hai.

Cone KAISE banta hai (scratch se derive karo)

Socho source ek straight line mein constant speed $v_{obj}$ pe move kar raha hai. Time $t = 0$ pe woh point $A$ se ek spherical wavefront emit karta hai. Time $t$ ke baad:

Woh wavefront radius $r = v_{s} t$ tak grow ho gayi hai (sound $v_{s}$ pe bahar ki taraf travel karti hai).
Source apni track pe point $B$ tak distance $d = v_{obj} t$ move kar gaya hai.

Agar $v_{obj} > v_{s}$ , toh source ( $B$ ) ab apni khud ki wavefront sphere ke bahar hai. Har instant ke liye aisa karo, toh saari spheres ka ek common tangent line hoga — envelope. 3D mein woh envelope ek cone hai jiska apex source pe hai.

Half-angle derive karna

Figure — Shock waves — Mach number, Mach cone — - CRITICAL for rockets -

Diagram mein right triangle dekho:

Hypotenuse $A B = v_{obj} t$ (source kitna move hua).
Opposite side (wavefront radius) $= v_{s} t$ .
Cone half-angle $θ$ (the Mach angle) apex $B$ pe baitha hai.

$sin θ = \frac{opposite}{hypotenuse} = \frac{v _{s} t}{v _{obj} t} = \frac{v _{s}}{v _{obj}}$

$t$ cancel ho jaata hai — cone angle time mein constant hai. Aur kyunki $M = v_{obj} / v_{s}$ :

Rockets ke liye yeh CRITICAL KYU hai

Drag spike: $M \approx 1$ cross karte waqt hawa raasta nahi de paati; pressure pile up hota hai → wave drag jump karta hai. Rockets ke pointed noses design kiye jaate hain taaki shock attached aur weak rahe.
Heating: shock cross karne wali hawa compress aur heat hoti hai; hypersonic re-entry pe yahi main thermal load hai (heat shields).
Sonic boom: Mach cone pressure wall hai; jab iska surface tumhare upar se guzarta hai, tum "boom" sunते ho — yeh continuous hai, supersonic jaane ke waqt ek baar ki bang nahi.
Nozzle design: rocket exhaust khud supersonic hai; engineers Mach number use karte hain de Laval nozzles shape karne ke liye.

Worked examples

Common mistake Common errors ko steel-man karo

"Sonic boom us instant hoti hai jab plane sound barrier todta hai." Kyun sahi lagta hai: tum ek baar "barrier todte" ho, isliye yeh single event lagta hai. Fix: boom continuous Mach cone hai jo plane ke peeche poore waqt trail karta hai jab woh supersonic hai. Tum tab sunте ho jab cone tumhare upar se guzarta hai, jo plane ke overhead hone ke kaafi baad hota hai.

" $sin θ = M$ ." Kyun sahi lagta hai: tez object, "bada" effect → multiply karo. Fix: yeh $sin θ = 1/ M$ hai. Tez object ⇒ chota, narrower cone. Limit check karo: $M \to \infty$ se $θ \to 0$ (paper-thin cone), jo physical hai.

"Ek shock $M < 1$ ke liye exist kar sakta hai agar woh 1 ke kareeb ho." Kyun sahi lagta hai: near-transonic flow mein local effects hote hain. Fix: geometric Mach cone ko $sin θ = 1/ M \leq 1$ chahiye, yaani $M \geq 1$ . $M = 1$ se neeche koi real cone angle nahi.

Recall Feynman: 12-saal ke bacche ko samjhao

Socho tum ek still pool mein daud rahe ho, ripples bana rahe ho. Dheere chalo toh ripples tumse aage phail jaati hain. Lekin ripples se tez daud lo, aur tum unhe peeche chhod dete ho — woh ek sharp V-shaped wave mein stack ho jaati hain jo tumhara peecha karti hai, jaise boat ki wake. Ek jet hawa mein wahi karta hai, lekin woh ek 3D ice-cream-cone shape banata hai squished hawa ka. Jab squished hawa ka woh cone tumhare kaanon ke paas se guzarta hai, tum ek loud BOOM sunte ho. Jet jitna tez, cone utna skinny aur pointed.

Mach number kya hai?

Object ki speed ka local speed of sound se ratio,

M = v_{obj} / v_{s}

Mach angle kya hai aur iska formula kya hai?

Shock cone ka half-angle;

sin θ = 1/ M

Mach angle derivation mein time ka factor kyun cancel hota hai?

Kyunki wavefront radius

v_{s} t

aur source displacement

v_{obj} t

dono linearly

t

mein grow karte hain, isliye unka ratio (aur hence cone angle) constant hai.

M \to \infty

hone pe cone ka kya hota hai?

sin θ \to 0

, isliye

θ \to 0

— cone infinitely thin/swept back ho jaata hai.

M < 1

ke liye Mach cone kyun nahi hota?

sin θ = 1/ M > 1

ka koi real solution nahi; source apne khud ke wavefronts ke andar rehta hai, isliye koi shock nahi banta.

Sonic boom physically kya hai?

Mach cone ki pressure wall listener ke upar se guzarती hai, jo sunai deti hai continuously jab tak object supersonic hai — yeh ek-time event nahi hai.

M = 2

pe Mach angle kya hai?

θ = sin^{- 1} (1/2) = 3 0^{\circ}

Rockets ke liye shock waves critical kyun hain?

Woh wave drag cause karti hain (

M = 1

ke paas peak hota hai) aur intense aerodynamic heating, jo nose-cone aur heat-shield design ko drive karti hain.

Connections

Speed of sound in a medium — $v_{s}$ set karta hai, hence har altitude pe $M$ .
Doppler effect — same wavefront geometry; shock $M \geq 1$ limit hai Doppler bunching ka.
Superposition & constructive interference — cone woh jagah hai jahan wavefronts coherently add up hoti hain.
Wave drag and aerodynamic heating — engineering consequence.
De Laval nozzle — rocket engines mein supersonic flow.
Compressible flow / Bernoulli limits — kyun $M = 1$ ke paas incompressible assumptions toot jaati hain.