Shock waves — Mach number, Mach cone — - CRITICAL for rockets -: Question bank

1.6.22 · D5 · Physics › Oscillations & Waves › Shock waves — Mach number, Mach cone — - CRITICAL for rocke

Intuition Is page ko kaise use karein

Ye woh thinking traps hain jo Mach-cone ki kahani aapke liye set karti hai — number crunching nahi (woh worked-example pages par hai), balki woh "ruko, kya yeh sach mein sahi hai?" wale moments hain. Answer ko cover karo, apna reason oonchi awaaz mein commit karo, phir reveal karo. Agar aapka reason mujhse alag hai — chahe aapka yes/no match kare — toh aapko lucky chance mila, sahi nahi. Neeche har answer ek reason hai, kabhi akeela verdict nahi.

Yeh deep dive parent Mach number & Mach cone note par lean karta hai; agar koi symbol unfamiliar lage, woh note use zero se build karta hai. Do anchors yaad rakho jo sab kuch handle karte hain: Mach number $M = v_{obj} / v_{s}$ (aapki speed "sound speeds" mein measured) aur Mach angle $θ$ jahan $sin θ = 1/ M$ (trailing cone ka half-angle). Yahan $v_{obj}$ object ki speed hai aur $v_{s}$ local speed of sound — dekho Speed of sound in a medium.

Har trap ke peeche ki picture

Questions attempt karne se pehle, yeh ek diagram memory mein pakka kar lo — neeche ke almost har trap ka matlab is diagram ki misreading hi hai.

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

Sach ya jhooth — justify karo

Har item: decide karo, phir physical reason do. "True/False" akela zero score karta hai.

Sonic boom ek single bang hai jo uss instant hoti hai jab plane

M = 1

cross karta hai.

Jhooth. Boom Mach cone ki pressure wall ka aap par sweep karna hai; yeh poore time plane ke peeche drag hoti rehti hai jab tak woh supersonic rehta hai, isliye koi bhi listener use tab sunta hai jab cone untak pahunche — usually tab jab plane upar se ja chuka hota hai.

Exactly

M = 1

par Mach cone ka half-angle

θ = 9 0^{\circ}

hota hai.

Sach.

sin θ = 1/ M = 1

θ = 9 0^{\circ}

milta hai, matlab "cone" flight path ke perpendicular ek flat plane mein khul gaya hai — wavefronts ek single wall mein pile ho jaati hain bilkul nose par (sound barrier).

Ek faster supersonic object zyada wide (mota) Mach cone produce karta hai.

Jhooth. Faster matlab bada

M

, toh

sin θ = 1/ M

chota, toh

θ

chota — cone thinner aur zyada swept back ho jaata hai. Speed V ko tight kheench leti hai.

Agar object speed of sound se neeche fly kare, toh uske wavefronts phir bhi ek shock cone banate hain, bas ek shallow wala.

Jhooth.

M < 1

ke liye,

sin θ = 1/ M > 1

, jiska koi real angle nahi hai. Source apne hi expanding wavefronts ke andar rehta hai (

B

-inside-the-blue-circle case), toh woh kabhi ek envelope mein pile nahi hoti — na cone, na shock.

Mach angle is baat par depend karta hai ki source kitne time se travel kar raha hai.

Jhooth. Wavefront radius (

v_{s} t

) aur source ka displacement (

v_{obj} t

) dono

t

ke saath linearly badhte hain, toh unka ratio — aur isliye

θ

— fix ho jaata hai jis moment

v_{obj}

aur

v_{s}

fix ho jaate hain. Time cancel ho jaata hai.

Mach number bas object ki speed hai metres per second mein, rescaled.

Jhooth.

M

local sound speed

v_{s}

ke against ek ratio hai, jo khud temperature aur altitude ke saath change hoti hai. Wahi 340 m/s hot sea level par

M < 1

ho sakta hai aur uupar

M > 1

— raw speed nahi batati ki shock forma hoga ya nahi; ratio batata hai.

Wave drag bahut high Mach numbers par sabse badi hoti hai, isliye hypersonic flight mein drag peak hoti hai.

Jhooth (misleading). Correction: wave-drag coefficient sharply spike karta hai bilkul

M \approx 1

ke around (transonic barrier, jahan air raste se hat nahi sakti), phir supersonic range mein ease off aur level down ho jaati hai — yeh zyada maximum tak climb nahi karte high Mach par. Woh transonic spike exactly yahi reason hai ki nose shaping

M \approx 1

ko target karti hai — dekho Wave drag and aerodynamic heating.

Sonic-boom cone aur Doppler bunching of wavefronts do unrelated phenomena hain.

Jhooth. Yeh ek hi wavefront geometry hai alag speed ratios par. Doppler effect

M < 1

bunching of fronts ahead of a mover hai; shock cone uska

M \geq 1

limiting case hai jahan woh bunching ek single envelope surface par collapse ho jaati hai.

Error dhundho

Har statement mein ek flaw hai. Use naam do aur correct karo.

sin θ = M

, toh

M = 2

par angle

sin^{- 1} 2

hai."

Relation inverted hai:

sin θ = 1/ M

hai,

M

nahi.

sin^{- 1} 2

ka koi solution hai hi nahi, jisse mistake flag ho jaani chahiye thi. Sahi:

sin θ = 1/2

θ = 3 0^{\circ}

"Wavefront Mach triangle ka hypotenuse hai aur source ka path opposite side hai."

Swap ho gaya. Source zyada distance

v_{obj} t

travel karta hai (hypotenuse, kyunki

v_{obj} > v_{s}

hai), aur wavefront radius

v_{s} t

opposite side hai. Exactly issiliye

sin θ = v_{s} t / (v_{obj} t) = 1/ M < 1

— upar figure se seedha padho.

"Kyunki

M > 1

matlab supersonic hai,

M = 0.99

kaafi close hai ek weak shock cone banane ke liye."

M = 1

se neeche koi partial cone nahi hota.

sin θ = 1/0.99 > 1

ka koi real angle nahi; geometric Mach cone strictly

M \geq 1

require karta hai. Transonic flow mein local supersonic pockets hote hain, lekin koi global trailing cone nahi.

"Jaise

M \to \infty

, cone angle bina bound ke badhta jaata hai."

Ulta hai.

sin θ = 1/ M \to 0

, toh

θ \to 0

— cone infinitely thin aur swept back ho jaata hai, bada nahi. Yeh limit sanity check hai ki

sin θ = 1/ M

sahi hai (na ki

= M

"Wavefronts cone par pile up hote hain kyunki wahan woh destructively interfere karte hain."

Yeh constructive build-up hai. Cone woh tangent envelope hai jahan har emitted sphere touch karti hai, toh unke pressure contributions coherently ek steep wall mein add hote hain — dekho Superposition & constructive interference.

"Rocket exhaust subsonic hai, isliye nozzle shocks ek Mach-cone concern nahi hain."

Properly run De Laval nozzle mein exhaust supersonic hoti hai; engineers Mach number precisely use karte hain us regime ke liye diverging section ko shape karne mein. Nozzle ke andar aur downstream shock structure ek live design issue hai.

Why questions

Mechanism se jawab do, restatement se nahi.

t

Mach-angle derivation mein kyun cancel ho jaata hai, aur yeh physically satisfying kyun hai?

Triangle ki dono legs

t

ke saath scale karti hain (

v_{s} t

aur

v_{obj} t

), toh unka ratio time-independent hai. Physically matlab yeh hai ki cone ek fixed shape hai jo object apne saath drag karta hai — flight continue hone par yeh moti ya patli nahi hoti.

Mach number (ek ratio) sahi variable kyun hai instead of object ki speed ke?

Kyunki shock form hoga ya nahi yeh sirf is baat par depend karta hai ki kya tum apni ripples se aage nikal rahe ho, yaani

v_{obj}

v_{s}

se compare karo. Kyunki

v_{s}

medium ke temperature/density ke saath shift karta hai, sirf ratio

M

shock-or-no-shock physics ko cleanly carry karta hai.

Rockets aur supersonic jets ke pointed noses kyun hote hain?

Ek sharp nose shock ko attached aur weak rakhta hai, compression ko gradually spread karta hai instead of ek strong detached bow shock force karne ke jo wave drag aur heating spike kar de. Blunt bodies ko ek strong stand-off shock milta hai — deliberately re-entry capsules par use kiya jaata hai cooling ke liye, opposite trade-off.

Shock cross karne par air kyun heat up hoti hai, aur hypersonic re-entry par yeh kyun matter karta hai?

Shock air ko almost instantaneously compress karta hai, bulk kinetic energy ko internal energy (temperature) mein convert karta hai. High

M

par yeh compression violent hoti hai, shock heating ko dominant thermal load banati hai — hence heat shields, Wave drag and aerodynamic heating ke mutabiq.

M = 1

ke paas incompressible Bernoulli reasoning kyun use nahi kar sakte?

Sonic speed ke paas density changes bade aur non-negligible hote hain, isliye incompressible assumption fail ho jaati hai aur pressure–velocity relations ko compression account mein lena padta hai — dekho Compressible flow / Bernoulli limits. Shocks inherently compressible events hain.

Ground par ek listener boom ko tab kyun sunta hai jab jet already upar se ja chuka hota hai?

Cone jet ke peeche trail karta hai, ground se kuch horizontal distance back meit karta hai aircraft ki current position se. Jet ko aage fly karna padta hai aapke upar se pehle uska trailing cone surface aapki spot tak catch kare, characteristic delay deta hai.

θ = 9 0^{\circ}

boundary case kyun hai instead of

θ = 0^{\circ}

ke?

θ = 9 0^{\circ}

M = 1

par hota hai (cone ek plane mein flatten — shock ki onset).

θ \to 0^{\circ}

M \to \infty

extreme hai (infinitely thin cone). Existence boundary

9 0^{\circ}

end hai, jahan

sin θ

pehli baar apni maximum possible value 1 reach karta hai.

Edge cases

Har parameter ko uski limit tak push karo aur batao kya hota hai.

Exactly

M = 1

par Mach cone kya hota hai?

Ek degenerate cone:

θ = 9 0^{\circ}

, toh "cone" nose par baitha motion ke perpendicular ek flat wall hai. Saare wavefronts wahan stack hote hain — yahi transonic sound-barrier pile-up hai.

M \to \infty

(hypersonic limit) par cone ka kya hota hai?

sin θ = 1/ M \to 0

, toh

θ \to 0^{\circ}

: ek infinitely thin, fully swept-back cone flight path se chipka hua. Shock essentially trajectory ke along lie karta hai.

Agar

M < 1

(subsonic) ho — "would-be cone" kaisa dikhta hai?

Koi nahi hota.

sin θ = 1/ M > 1

unphysical hai; source apne spherical wavefronts se enclosed rehta hai, jo simply aage bunch karte hain (ordinary Doppler compression) bina ek envelope surface banaye.

Agar object momentarily rest par ho,

v_{obj} = 0

M = 0

, aur

sin θ = 1/ M

undefined hai (division by zero). Physically wavefronts fixed source par centered concentric spheres hain — perfectly symmetric, koi cone nahi, koi preferred direction nahi.

Agar

v_{s}

path ke along change hota jaaye (thande, paatle air mein climb karna)?

M = v_{obj} / v_{s}

constant

v_{obj}

par bhi change hota hai, toh local Mach angle bhi change hota hai. Ek rocket same speed par neeche subsonic aur uupar supersonic ho sakta hai, kyunki

v_{s}

gir gaya — dekho Speed of sound in a medium.

Jaise flight altitude

h \to 0

, boom ka ground timing kya hota hai?

Cone wall ground se

x = h / tan θ

horizontal lag distance aircraft ke peeche mile karta hai — yahan

x

simply kitna ground par peeche hai jahan cone touch karta hai jab jet height

h

par fly karta hai. Jaise

h \to 0

, woh lag

x \to 0

, toh boom essentially uss moment pahunche jab jet overhead ho — low pass ke liye bahut kam delay.

Recall Ek-line self-test

Cone existence condition? ::: $M \geq 1$ , kyunki $sin θ = 1/ M$ ko $1/ M \leq 1$ chahiye. Speed–angle trade ka direction? ::: Faster ⇒ chota $θ$ ⇒ thinner cone. Kya boom ek event hai ya sweep? ::: Ek sweep — cone wall aap par se guzarti hai jab object supersonic rehta hai.

Connections

Parent: Mach number & Mach cone
Speed of sound in a medium — $v_{s}$ set karta hai, toh har altitude par $M$ set karta hai.
Doppler effect — subsonic wavefront bunching; shock uska $M \geq 1$ limit hai.
Superposition & constructive interference — cone coherent pile-up hai.
Wave drag and aerodynamic heating — in traps ka engineering payoff.
De Laval nozzle — supersonic exhaust design.
Compressible flow / Bernoulli limits — kyun incompressible reasoning $M = 1$ ke paas khatam hoti hai.