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

3.1.3 · Physics › Compressible Flow & Aerodynamics

Speed of sound KYA hai?

Poora game yeh hai ki ideal gas ke liye sahi thermodynamic constraint ke under $\partial p/\partial \rho$ compute karo.

Hum ise KAISE derive karte hain — first principles se

Step 1 — Wave par khade ho jao (control volume trick)

Ek sound wave jo speed $a$ par still gas mein move kar rahi hai, uska analyze karna mushkil hai kyunki sab kuch unsteady hai. Frame kyun switch karein? Ek reference frame mein jo wave ke saath travel kar raha ho, flow steady ho jaata hai — bahut aasaan.

Wave frame mein:

• Gas wave mein speed $a$ par enter karti hai, pressure $p$, density $\rho$ ke saath.
• Gas thodi changed hokar nikалти hai: speed $a-dV$, pressure $p+dp$, density $\rho +d\rho$.

Step 2 — Conservation of mass (continuity)

Mass in = mass out thin wave ke across (area $A$ cancel ho jaata hai): $\rho a=(\rho +d\rho )(a-dV)$

Yeh step kyun? Wave mass pile up nahi karti; jo andar aata hai woh bahar bhi jaana chahiye.

Expand karo aur second-order term $d\rho dV$ drop karo (yeh do tiny cheezein ka product hai → negligible): \rho a = \rho a - \rho\, dV + a\, d\rho \;\Rightarrow\; \boxed{\rho\, dV = a\, d\rho} \tag{1}

Step 3 — Conservation of momentum

Slab par net pressure force = momentum change ki rate (momentum flux). Control volume ke liye: $pA-(p+dp)A=\dot{m}[(a-dV)-a]$

Yeh step kyun? Steady flow ke liye Newton ka 2nd law: force = mass flow rate $\times$ velocity change. Mass flow rate $\dot{m}=\rho aA$.

-dp\,A = \rho a A\,(-dV) \;\Rightarrow\; \boxed{dp = \rho a\, dV} \tag{2}

Step 4 — (1) aur (2) combine karo

(1) se: $dV=ad\rho /\rho$. (2) mein substitute karo: $dp=\rho a\cdot \frac{ad\rho }{\rho }=a^{2}d\rho$ \boxed{a^2 = \frac{dp}{d\rho}} \tag{3}

Yeh kyun important hai: purely mass + momentum se, speed of sound square root of hai ki pressure per unit density change kitna change hota hai. Yahi "stiffness/inertia" wala idea exact form mein hai. Humne abhi tak temperature ya $\gamma$ ke baare mein kuch nahi kaha hai.

Step 5 — Kaun sa process? Adiabatic, isothermal nahi (crucial insight)

Isentropic ideal gas ke liye: $p=C{\rho }^{\gamma }(\text{constant }C)$

Yeh relation kyun? $p{V}^{\gamma }=$ const ko $V\propto 1/\rho$ ke saath combine karne par $p\propto {\rho }^{\gamma }$ milta hai. Yahan $\gamma =c_p/c_v$.

Differentiate karo: $\frac{dp}{d\rho }=C\gamma {\rho }^{\gamma -1}=\gamma \frac{C{\rho }^{\gamma }}{\rho }=\gamma \frac{p}{\rho }$

Toh: a^2 = \frac{dp}{d\rho} = \gamma\,\frac{p}{\rho} \tag{4}

Step 6 — Ideal gas law laao

Ideal gas: $p=\rho RT$ (jahan $R$ specific gas constant hai, $R=R_u/M$). Isliye $p/\rho =RT$: $a^2=\gamma RT\Rightarrow \boxed{a=\sqrt{\gamma RT}}$

$a$ sirf $T$ par kyun depend karta hai (Feynman-check)

$p$ aur $\rho$ dono appear karte hain, lekin unka ratio $p/\rho =RT$ temperature se fix hota hai. Gas ko squeeze karo (raise $p$ aur $\rho$): stiffness badhti hai lekin inertia bhi badhti hai, aur woh cancel ho jaati hain. Sirf random thermal molecular speed — jo $T$ se set hoti hai — bachti hai. Actually $a\sim$ average molecular speed, jo sense banta hai: sound un molecules se aage nahi nikal sakti jo isse carry kar rahi hain.

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• Mach number and flow regimes
• Isentropic relations p ∝ ρ^γ
• Ideal gas law and specific gas constant
• Compressibility and why M > 0.3 matters
• Adiabatic vs isothermal processes
• Stagnation properties and energy equation
• Normal shock waves

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