Q.State Newton's formula for the speed of sound in a gas and the assumption on which it rests. Explain the physical reasoning behind Laplace's correction, and state by roughly what percentage Laplace's formula raises the predicted speed of sound in air over Newton's.

Newton's formula for the speed of sound in a gas, $v_N=\sqrt{P/\rho}$, rests on the assumption that the compressions and rarefactions of the sound wave occur slowly enough for heat to flow freely between neighbouring compressed and rarefied regions of the gas, keeping the whole process at a constant temperature (isothermal), so that the relevant bulk modulus is simply the pressure $P$ itself (from Boyle's law, $PV=\text{constant}$).

However, this predicted speed came out roughly 15-16% lower than the experimentally measured speed of sound in air. Laplace identified the error: sound's compressions and rarefactions actually happen far too rapidly for any significant heat to be exchanged between adjacent regions of the gas within the time available, so the process is in fact adiabatic (no heat exchange), obeying $PV^\gamma=\text{constant}$ instead of Boyle's law, which gives a larger, adiabatic bulk modulus $\gamma P$ (with $\gamma=C_P/C_V$) rather than just $P$.

Laplace's corrected formula, $v_L=\sqrt{\gamma P/\rho}$, is therefore larger than Newton's by exactly the factor $\sqrt{\gamma}$; for air, $\gamma\approx1.4$, so $\sqrt{\gamma}\approx1.183$, meaning Laplace's formula predicts a speed about 18% higher than Newton's -- and this corrected value matches the experimentally observed speed of sound in air closely.

State Newton's assumption (isothermal) and its formula, then explain Laplace's reasoning (adiabatic, since compressions/rarefactions are too fast for heat exchange) and quantify the correction factor as $\sqrt{\gamma}$.

• Reversing which formula is larger (Laplace's corrected value is HIGHER than Newton's, not lower).
• Saying the correction factor is $\gamma$ itself rather than $\sqrt{\gamma}$.