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

3.1.4 · D2 · Physics › Compressible Flow & Aerodynamics › Mach number M = V - a — subsonic ( - 1), transonic (~1), sup

Shuru karne se pehle, teen words jinpe hum baar baar jaayenge, plain language mein:

parent topic se do speeds yaad karo:

• $V$ = source kitni fast move karta hai,
• $a$ = har ripple kitni fast spread hoti hai (speed of sound),
• aur $M=V/a$ unka ratio hai.

Hum teen characters dekhenge — ek slow dot, ek dot jo exactly sound speed pe move karta hai, aur ek fast dot — aur pictures ko decide karne denge ki kya hota hai.

---

Step 1 — Ek ping, ek growing circle

KYA. Ek single ping ko us waqt freeze karo jab woh janam leti hai, phir time $t$ guzarne do. Woh ek circle ban jaati hai jiska radius steadily grow karta hai.

KYUN. Baad mein jo bhi complicated hoga woh inhi simple circles ka overlap hai. Agar ek circle ko honestly samajh lo, toh cone free mein mil jaata hai.

PICTURE. Figure mein, ping kaa janam black dot pe hua tha. Time $t$ ke baad circle ka edge har direction mein $at$ door tak pahunch jaata hai.

Circle symmetric hai kyunki still air sound ko har taraf equally fast carry karta hai. Yeh yaad rakho: wavefront ko nahi pata source aage kahan gaya. Ek baar ping launch ho gayi, woh apna parent bhool jaati hai.

---

Step 2 — Ek still source: bullseye jaisi rings

KYA. Source ko still rakho aur equal time intervals pe baar baar ping karne do.

KYUN. Yeh $V=0$ baseline hai — "kuch nahi hilta" wala case. Baaki har case isi ka distortion hai, toh pehle ise cleanly dekhna zaroori hai.

PICTURE. Concentric circles, sab same centre share karte hain — ek perfect bullseye. Kahin bhi crowding nahi.

---

Step 3 — Ek slow source ($M<1$): rings aage crowd karti hain par kabhi touch nahi karti

KYA. Ab source ko dheere dheere aage chalao, sound se slow. Successive positions se times $t_0,t_1,t_2,\dots$ pe ping karo.

KYUN. Yeh har airliner ka cruise case hai — subsonic duniya. Hum dekhna chahte hain kyun hawa tab bhi smoothly side hoti hai.

PICTURE. Har nayi circle flight line ke thoda aage se start hoti hai. Centres aage march karte hain, toh circles aage pakk jaati hain aur peeche spread ho jaati hain — lekin important baat yeh hai ki source apni banaayi har circle ke andar rehta hai. Do edges ek single line pe pile nahi hoti.

Kyunki sabse fast khabar hamesha source se aage rehti hai, downstream ki hawa pre-warned hoti hai aur gently side ho jaati hai. Aage ki crowding bas Doppler effect hai (aage zyada pitch) — annoying, violent nahi. Ab bhi koi shock nahi.

---

Step 4 — Exact edge case ($M=1$): saari circles ek flat wall ko kiss karti hain

KYA. Speed badhao jab tak source exactly sound speed pe move kare, $V=a$, yani $M=1$.

KYUN. Yeh "khabar aage rehti hai" aur "source khabar se aage nikal jaata hai" ke beech ka knife-edge hai. Edge cases switch reveal karte hain; inhe skip karna samajh mein ek hole chhod jaata hai.

PICTURE. Ab $Vt=at$ exactly hai. Source apni banaayi har circle ke leading edge pe bilkul baitha hai. Saari front edges ek single vertical wall mein line up ho jaati hain jo source se guzarti hai — dabe pressure ka ek flat plane.

---

Step 5 — Ek fast source ($M>1$): source escape karta hai, aur ek cone appear hota hai

KYA. Sound speed se aage push karo: $V>a$, toh $M>1$. Phir se baar baar ping karo.

KYUN. Yahi parent note ka poora point hai — supersonic flight. Hum chahte hain ki cone apne aap circles se emerge ho.

PICTURE. Ab source har circle se aage bhaagta hai. Har purani circle peeche reh jaati hai, chhoti circles badi ones ke andar nested hain lekin sab dot ke peeche trail kar rahe hain. Unke outer edges ab source ko surround nahi karte — balki woh sab ek common straight line (3D mein, ek cone) pe lean karte hain jo source se nikalti hai. Woh common tangent line hi Mach cone hai; uspe pressure ek shock mein pile up ho jaata hai.

---

Step 6 — Angle seedha triangle se padhna

KYA. Step 5 ki picture ke andar chhupa right triangle draw karo aur cone ka half-angle $\mu$ padh lo.

KYUN. Ek picture jo formula free mein de de, woh best kind hai. Hum $\mu$ chahte hain $M$ ke terms mein — koi guessing nahi.

PICTURE. Sabse purani ping lo (time $0$ pe janam li). Time $t$ mein:

• source flight line ke saath $Vt$ distance travel kar chuka — yeh triangle ki long side (hypotenuse) hai,
• woh purani circle $at$ radius tak grow ho gayi — yeh triangle ki short side hai, perpendicular cone ke edge ke, source ke start point se cone aur circle ke touch point tak pahunchti hai.

Mach cone edge hypotenuse ka companion hai; flight line aur cone edge ke beech ka angle $\mu$ hai.

Ab picture ko har us case se check karo jo humne draw kiya:

Woh aakhri bullet sabse deep check hai: formula khud jaanta hai ki $M=1$ se neeche koi cone nahi hai. Beautiful.

---

Ek picture summary

Yeh final figure saari chaar duniyaon ko — still, slow, exactly sonic, fast — side by side stack karta hai, taaki tum dekh sako ki wavefronts ek calm bullseye se ek piled-up cone mein kaise jaate hain jab $M$ $1$ se upar chadhta hai.

Recall Feynman retelling — apne words mein bolo

Ek dot imagine karo jo pond mein baar baar pathhar giraaata rehta hai, ek per second, aur saath saath chalta bhi hai. Har pathar se ek ripple banti hai jo same speed se spread hoti hai chahe dot kuch bhi kare — us speed ko $a$ bolo. Dot speed $V$ se chalta hai.

Agar dot ripples se slow chale ($V<a$), woh apni ripples ke andar rehta hai: paani aage hamesha pehle ek ripple dekhta hai, toh woh warned ho jaata hai aur gently side ho jaata hai. Koi pile-up nahi.

Agar dot exactly ripples jaiti speed se chale ($V=a$), woh hamesha apni ripples ke front edge pe baitha rehta hai — woh ek flat wall mein stack ho jaati hain uske aage.

Agar dot ripples se fast chale ($V>a$), woh apni ripples ko peeche chhod deta hai. Purani ripples, sab same rate se grow karti hain jabki unke centres same rate se aage march karte hain, ek straight edge ke saath line up ho jaati hain — ek cone. Aage ka paani koi warning nahi paata: woh ambushed ho jaata hai, aur woh sudden pile-up shock hai.

Us cone ka angle ek right triangle se set hota hai: ripple $at$ grow hui jabki dot $Vt$ chala, toh half-angle ka sine $at/Vt=a/V=1/M$ hai. Yahi poori kahaani hai — circles aur ek triangle.

Recall Predict-first checkpoints

Kaun sa regime hai jisme source apne saare circles ke andar hota hai? ::: Subsonic, $M<1$ (Step 3). $M=1$ pe, wavefronts source ke aage kaun si shape banate hain? ::: Ek single flat wall, half-angle $\mu =90^{\circ }$ (Step 4). Growing circles ek straight tangent kyun share karti hain jab $M>1$ hota hai? ::: Same growth speed $a$ aur same march speed $V$ se nested same-ratio circles ban ti hain, jo hamesha ek common tangent line share karti hain — cone edge (Step 5). Kaun sa trig function $\sin \mu =1/M$ deta hai aur wahi kyun? ::: Sine, kyunki jaani hui circle radius $at$ angle $\mu$ ke opposite hai aur source travel $Vt$ hypotenuse hai (Step 6). $\sin \mu =1/M$ tumhe kya bataata hai jab $M<1$ ho? ::: $1/M>1$, sine ke liye impossible hai, toh koi real cone exist nahi karta — formula khud Mach 1 se neeche shock ko forbid karta hai.

---

Related deepening: is cone ke saath jo shock banti hai uska analysis Normal Shock Waves ($M=1$ flat wall) aur Oblique Shocks & Mach Cone (tilted cone edge) mein hai. $M\approx 0.3$–$0.8$ smooth-flow correction Prandtl–Glauert Correction mein hai, aur viscous effects (ek alag dimensionless ratio) Reynolds Number mein hain.