Foundations — Speed of sound — a = √(γRT) — derivation

3.1.3 · D1 · Physics › Compressible Flow & Aerodynamics › Speed of sound — a = √(γRT) — derivation

Pehle us formula ko derive karne ke liye, tumhe uske andar ke har letter aur squiggle mein fluent hona chahiye. Ye page har ek ko bilkul kuch nahi se build karta hai, us order mein jo har idea ko pehle waale pe lean karne deta hai. Yahaan kuch bhi assume nahi kiya gaya — agar parent note ne use kiya, toh hum use yahaan build karte hain. (Hum $a$, $\gamma$, $R$ ya $T$ symbols ko kisi bhi calculation mein tab tak use nahi karenge jab tak har ek ka apna section below na ho.)

0 — Show ka star: $a$, speed of sound

Chalte hain us quantity ka naam larte hain jiske baare mein ye poora topic hai, kuch aur karne se pehle.

Tasveer: apne haath thapo. Slightly-squeezed air ka ek patla shell baahir ki taraf daudta hai aur ek pal baad ek dost ke kaan tak pahuncha hai. Us shell ki speed hi $a$ hai. Neeche diya gaya sab kuch sirf is ek number ko compute karne ke liye hai.

1 — "Gas" actually kya hai (har cheez ke peeche ki tasveer)

Ek second ke liye equations bhool jao. Gas tiny balls (molecules) ki ek badi bheed hai jo har direction mein ud rahi hain, ek doosre se aur walls se bounce kar rahi hain. Unke beech mostly empty space hoti hai.

Teen everyday cheezein jo hum is bheed ke baare mein measure kar sakte hain, woh hamare teen main symbols ban jaati hain:

• Ek box mein kitne balls pack hain → density.
• Wo walls pe kitni baar thokti hain → pressure.
• Wo kitni tezi se zip around karti hain → temperature.

Topic mein har cheez in teeno ke beech ka relationship hai. Chalte hain har ek se milte hain.

2 — Density $\rho$ (gas kitni bhaari hai)

Tasveer: figure s01 mein, ek box ke andar balls gino. Usi box mein zyada balls = zyada $\rho$. Sea level par air mein $\rho \approx 1.2kg/{\mathrm{m}}^3$ hoti hai.

Topic ko iske kya zaroorat hai: $\rho$ tug-of-war ka inertia wala half hai. Bhaari gas dhheemi hoti hai — use move karna mushkil hota hai — toh ek push usse zyada slowly travel karti hai. Jab tum baad mein denominator mein $\rho$ dekhoge ($a^2=\gamma p/\rho$), toh wahi inertia sound ko slow kar rahi hoti hai.

3 — Pressure $p$ (gas kitni zor se push karti hai)

Tasveer: har ball jo wall se takraati hai use ek chhota sa kick deti hai. Millions of kicks per second, jod kar aur wall ki area par spread karke, pressure ban jaati hai. Sea-level air $p\approx 101,000\mathrm{Pa}$ par push karti hai.

Topic ko iske kya zaroorat hai: $p$ stiffness wala half hai. Jab tum gas ko squeeze karte ho, uske molecules crowded ho jaate hain aur zyada zor se wapas push karte hain — pressure badhti hai. Woh "push back" hi exactly hai jo ek layer ko doosri layer ko dhakka dene deta hai, sound wave ko pass karte hue. Zyada stiff (squeeze per zyada push-back) → faster sound.

4 — Temperature $T$ (molecules kitni tezi se zip karte hain)

Tasveer: figure s02 mein, cold gas (left) mein chhhote arrows hain — slow, lazy molecules. Hot gas (right) mein lambe arrows hain — fast, frantic molecules. Wahi bheed, zyada energy.

Topic ko iske kya zaroorat hai: poore chapter ka punchline yahi hai ki $a$ sirf $T$ par depend karta hai. Fast molecules "push" ko jaldi pass karte hain — sound literally thermal jiggling par ride karti hai. Hot gas = zippy molecules = faster sound.

5 — Ideal gas law: $p$, $\rho$, $T$ ko ek saath baandhna

Ye teen independent nahi hain — gas ko squeeze karo ya heat karo aur ye saath move karte hain. Unhe link karne wala rule ideal gas law hai.

Rearrange karne par, ye topic ka sabse important combination deta hai: $\frac{p}{\rho }=RT$

Sound ke liye ye kyun important hai: baad mein hum $a^2=\gamma \frac{p}{\rho }$ paate hain. Kyunki $p/\rho =RT$ hai, $p$ aur $\rho$ sirf $RT$ mein collapse ho jaate hain. Yehi wajah hai ki pressure aur density individually drop out ho jaate hain aur sirf temperature bachti hai. Is line ko seekho — ye poori derivation ka hinge hai. Dekho Ideal gas law and specific gas constant.

6 — Specific gas constant $R$ (aur kyun ye $8.314$ nahi hai)

Tasveer: $R$ ko gas ki personality samjho. Light molecules (chhhota $M_{\text{mol}}$) → bada $R$ → same $T$ dene par, wo faster move karte hain aur sound faster carry karte hain. Isliye helium ($M_{\text{mol}}=4$ g/mol, huge $R$) mein sound $1000$ m/s ke paas hoti hai. Dekho Ideal gas law and specific gas constant.

7 — Heat-capacity ratio $\gamma$ ("heat ke liye time nahi" correction)

Ye subtle wala hai — woh symbol jo is topic ko plain gas-law problem se alag banata hai.

Jab tum gas compress karte ho toh do tarike se kar sakte ho:

• Slowly — heat bahar nikal jaati hai, temperature constant rehti hai. Ye isothermal hai.
• Fast — heat ke paas escape karne ka time nahi hota, toh trapped energy bhi gas ko heat karti hai. Ye adiabatic hai.

Ek sound wave gas ko thousands of times per second squeeze karti hai — heat ke leak hone ke liye bahut zyada fast. Toh ye adiabatic hai, aur extra self-heating gas ko slow case se bhi zyada zor se push back karni hai. Woh extra stiffness ek number $\gamma$ mein capture hoti hai. Dekho Adiabatic vs isothermal processes.

Tasveer: $\gamma$ ek "stiffness bonus" hai. Newton ne ise bhool gaya aur $\approx 280$ m/s (isothermal) paya. Laplace ne ise wapas daala aur $340$ m/s paya, reality se match karte hue. $\sqrt{\gamma }$ exactly woh $1.4$ correction hai. Dekho Isentropic relations p ∝ ρ^γ.

8 — Squiggles padhna: $d$, $\partial$, aur derivative

Derivation tiny changes aur slopes ke baare mein baat karti hai, toh tumhe ye notation chahiye.

Tasveer: figure s03 mein curve $p$ against $\rho$ hai. Ek point pick karo, zoom in karo jab tak curve seedha na lage, aur us chhhoti line ki steepness $dp/d\rho$ hai. Zyada steep slope ka matlab hai "ek chhhota squeeze ek badi pressure jump cause karta hai" — ek stiff gas. Isliye $a^2=dp/d\rho$ hai: slope literally hi stiffness hai.

9 — Entropy $s$, "isentropic", aur constant $C$

Ek isentropic ideal gas ke liye, pressure aur density rule $p=C{\rho }^{\gamma }$ se ek saath locked rehte hain.

Topic ko iske kya zaroorat hai: $(\partial p/\partial \rho {)}_s$ mein subscript $s$ wahi hai jo hume isothermal wali ki jagah adiabatic curve $p=C{\rho }^{\gamma }$ par force karta hai. Woh single letter wahi hai jahan $\gamma$ answer mein chhup ke aata hai. Dekho Isentropic relations p ∝ ρ^γ.

10 — Mach number $M$ (kyun hum yeh sab karte hain)

Tasveer: $M=1$ ka matlab hai speed of sound par move karna; $M=0.5$ uska half hai. $M\approx 0.3$ se neeche air aise behave karti hai jaise incompressible ho aur tum yeh sab ignore kar sakte ho. Uske upar, density changes matter karti hain — aur yehi wajah hai ki humein pehle jagah mein $a$ ki zaroorat hai. Dekho Mach number and flow regimes aur Compressibility and why M > 0.3 matters.

Foundations topic ko kaise feed karti hain

Equipment checklist

Right side cover karo aur khud ko test karo — agar koi bhi answer fuzzy lage, derivation se pehle us section ko dobara padho.