Speed of sound — a = √(γRT) — derivation: Worked examples

3.1.3 · D3 · Physics › Compressible Flow & Aerodynamics › Speed of sound — a = √(γRT) — derivation

Yeh page speed-of-sound formula ke liye "koi surprise nahi" wali drill hai. Hum $a = γ R T$ ko har tarah ke case mein push karte hain jo exam ya real wing pe aa sakta hai: normal air, thandi air, dusri gases, degenerate inputs ( $T \to 0$ pe kya hoga?), classic Newton ki galti, ratios, aur ek word problem. Har example se pehle tum ek Forecast banate ho — pehle answer ka ballpark guess karo, phir khud ko check karo.

Scenario matrix

$a = γ R T$ ke baare mein har problem inhi cells mein se kisi ek mein aata hai. Neeche ke worked examples mein har ek pe ek tag hai jo batata hai ki woh kaun si cell nail karta hai.

#	Case class	Kya badalta hai	Example
A	Baseline — standard sea-level air	kuch nahi (reference case)	Ex 1
B	Cold input — $T$ kam	$T ↓\Rightarrow a ↓$	Ex 2
C	Different gas — naya $γ, R, M$	teeno constants swap karo	Ex 3
D	Ratio / relative — do states compare karo	constants cancel, sirf $T$ ratio bachta hai	Ex 4
E	Degenerate / limiting — $T \to 0$ , $T \to \infty$	boundary behaviour	Ex 5
F	Wrong process trap — isothermal vs adiabatic	$γ$ / Newton error	Ex 6
G	Unit trap — Celsius, ya universal $R$	classic slip pakdo	Ex 7
H	Real-world word problem — Mach + altitude	$a$ ko $M$ mein chain karo	Ex 8
I	Exam twist — ulta solve karo $T$ ke liye	formula invert karo	Ex 9

Worked examples

Worked example Example 5 — Degenerate & limiting inputs

(cell E) $T \to 0 K$ pe $a$ kya hai? Aur $T \to \infty$ kya imply karta hai? Jahan numbers help karein wahan air constants use karo.

Forecast: absolute zero pe molecules chalna band kar dete hain, toh kuch bhi push carry nahi karta — guess $a \to 0$ . Bahut bade $T$ pe $a$ bina bound ke badhta hai (lekin dheere, $T$ ki tarah).

Limit $T \to 0$ : $a = γ R \cdot 0 = 0$ . Yeh step kyun? Root ke neeche product vanish ho jaata hai; zero factor ise khatam kar deta hai. Physically: koi thermal molecular motion nahi → wave ke liye koi carrier nahi.
Limit $T \to \infty$ : $a = γ R T \to \infty$ , lekin yeh $T$ ki tarah badhta hai, toh dheere — $T$ double karne se $a$ sirf $2 \approx 1.41$ se multiply hota hai. Yeh step kyun? Exponent $\frac{1}{2}$ control karta hai ki limit kitni tez blow up hoti hai; yeh sub-linear hai, toh koi runaway nahi.
Sanity numbers: $T = 1152 K$ (= $4 \times 288$ ) pe, $a = 1.4 \cdot 287 \cdot 1152 = 462, 872 \approx 680 m/s$ — exactly $2 \times 340$ , kyunki $4 = 2$ .

Verify: $T = 0$ answer $0$ hai ✓ (koi negative ya complex speed nahi — hamesha physical kyunki kelvin mein $T \geq 0$ ). $4 \times T$ se $2 \times a$ milna $T$ scaling confirm karta hai. ✓

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

Worked example Example 6 — Wrong-process trap: Newton vs Laplace

(cell F) $T = 288 K$ pe isothermal (Newton) speed $a_{iso} = R T$ aur sahi adiabatic (Laplace) speed $a = γ R T$ compute karo. Newton kitne percent galat hai?

Forecast: Newton ne $γ$ drop kar diya, toh uska number $γ = 1.4 \approx 1.18$ factor se chhota hai. Expect karo ~15% too low.

Newton: $a_{iso} = 287 \times 288 = 82, 656 \approx 287.5 m/s$ . Yeh step kyun? Isothermal $p = ρR T$ se $d p / d ρ = R T$ milta hai, koi $γ$ nahi — yeh exactly woh tempting error hai jo parent note mein hai.
Laplace (sahi): $a = 1.4 \times a_{iso} = 1.183 \times 287.5 \approx 340 m/s$ . Yeh step kyun? Real wave adiabatic hai, $a^{2}$ ko $γ$ se multiply karti hai, matlab $a$ ko $γ$ se.
Error: $\frac{340 - 287.5}{340} = 0.154 = 15.4%$ too low.

Verify: do speeds exactly $1.4 = 1.183$ se differ karti hain ✓, aur $287.5$ m/s Newton ka famous ~15% miss hai. ✓

Worked example Example 7 — Unit traps: Celsius & universal

$R$ (cell G) Ek student likhta hai $a = 1.4 \times 8.314 \times 15$ "air at $1 5^{\circ}$ C" ke liye. Kya do galtiyan hain, aur sahi value kya hai?

Forecast: $R_{u} = 8.314$ use karna $287$ ki jagah number ko ~ $287/8.314 \approx 5.9 \times$ chhota bana deta hai, aur $288 K$ ki jagah $15$ bhi bahut galat hai. Unka answer nonsense hoga (tens of m/s).

Unki (galat) value: $1.4 \times 8.314 \times 15 = 174.6 \approx 13.2 m/s$ . Yeh step kyun? Dikhao ki double error kitna far off land karti hai — real sound speed ke paas kahin nahi.
Temperature fix karo: $1 5^{\circ} C = 15 + 273 = 288 K$ . Yeh step kyun? Formula ko absolute temperature chahiye; Celsius ka zero false hai.
Constant fix karo: $R = 287$ use karo (specific), $R_{u} = 8.314$ (universal) nahi. Yeh step kyun? $a$ per-kilogram physics hai; $R_{u}$ per-mole hai. Pehle molar mass se divide karo: $8314/29 \approx 287$ .
Sahi value: $1.4 \times 287 \times 288 \approx 340 m/s$ .

Verify: galat answer ( $13.2$ ) absurd hai (ek bicycle tez hai) ✓; corrected wala $340$ hai ✓.

Worked example Example 8 — Real-world word problem: Mach at altitude

(cell H) Ek jet 11 km pe true airspeed $V = 250 m/s$ pe cruise karta hai jahan $T = 217 K$ hai. Kya yeh compressible regime mein hai? Dekho Compressibility and why M > 0.3 matters.

Forecast: altitude pe sound slow hai (~295 Ex 2 se), toh $M = 250/295$ comfortably 0.3 se upar hai — haan, compressible; shayad high-subsonic ~0.85.

Local sound speed: $a = 1.4 \times 287 \times 217 \approx 295 m/s$ (Ex 2 se). Yeh step kyun? Mach local $a$ use karta hai, jo local temperature se set hota hai — sea-level value nahi.
Mach number: $M = \frac{V}{a} = \frac{250}{295} \approx 0.847$ . Yeh step kyun? $M = V / a$ plane ki speed ko compare karta hai ki pressure signals kitni tez travel karte hain; yeh decide karta hai ki density changes matter karti hain ya nahi.
Thresholds se compare karo: $0.847 > 0.3$ → compressible, aur $< 1$ → abhi bhi subsonic.

Verify: $M \approx 0.85$ ✓ — ek realistic airliner cruise Mach. Note karo ki wahi $250$ m/s sea level pe ( $a = 340$ ) $M = 0.74$ deta hai: altitude pe same speed "zyada transonic" hai kyunki $a$ drop ho gaya. ✓

Quick recall

Recall Answers cover karo — kya tumne har cell hit ki?

Agar sirf $T$ gire toh $a$ kaise change hota hai? ::: Yeh girta hai, kyunki $a \propto T$ (cell B/D).
$T_{2}$ aur $T_{1}$ pe sound speeds ka ratio? ::: $T_{2} / T_{1}$ — constants cancel ho jaate hain (cell D).
$T \to 0 K$ pe $a$ kya hai? ::: $0$ — wave carry karne ke liye koi thermal motion nahi (cell E).
Newton ki isothermal error kitni badi hai? ::: ~15% too low; usne $γ$ factor drop kar diya (cell F).
Do classic unit slips? ::: Kelvin ki jagah Celsius; specific $R$ ki jagah universal $R_{u}$ (cell G).
Measured $a$ se $T$ nikalne ke liye? ::: $T = a^{2} / (γ R)$ (cell I).