Q.Explain Newtons formula for velocity of sound. What is its limitation?

[!TLDR] Newton's formula v=sqrt(P/rho) assumed isothermal propagation, predicting 279.9 m/s at NTP -- about 16% below the measured ~332 m/s, because sound compressions are actually too fast to stay isothermal (they are adiabatic). [!ANSWER] $v=\sqrt{P/\rho}$; limitation: …

Newton proposed that sound's speed in a medium is $v=\sqrt{E/\rho}$, and specialised this for air by assuming sound propagates ISOTHERMALLY (constant average temperature throughout), so the relevant elastic modulus is the isothermal bulk modulus, equal for a gas to the ambient pressure P -- giving $v=\sqrt{P/\rho}$. Using NTP values ($P=hdg$ with $h=0.76$ m of mercury, mercury density 13600 kg/m³, air density 1.293 kg/m³, $g=9.8\ \text{m/s}^2$), this predicts $v\approx279.9$ m/s. But the experimentally measured speed of sound at 0°C is about 332 m/s -- roughly 16% HIGHER than Newton's prediction, far too large a gap to be experimental error, and Newton himself could not explain it. The limitation: Newton's ISOTHERMAL assumption is physically wrong for sound -- a sound wave's compressions and rarefactions happen far too rapidly (hundreds of times per second for audible frequencies) for the heat generated during compression to actually escape before the next rarefaction, so the true process is ADIABATIC, not isothermal. Laplace fixed this by using the adiabatic modulus $E=\gamma P$ ($\gamma=C_ …

State Newton's formula and its isothermal-pressure specialisation for air, compute (or quote) its NTP prediction, and identify the isothermal assumption as the s …

Explaining the formula but omitting the actual NUMERICAL size of the discrepancy (~16%, 279.9 vs ~332 m/s) or the specific physical reason (compressions too fast for heat exchange) that makes the is …