9.10 Speed of Sound: Newton's Formula and Laplace's Correction

The speed of sound in a medium is determined by the medium's properties of elasticity and density. A more elastic medium allows for faster wave propagation, while a denser medium slows it down.

The general formula for the speed of a longitudinal wave ($V$) is: $V=\sqrt{\frac{E}{\rho }}$ Where:

• $E$ is the bulk modulus of the medium (a measure of its stiffness).
• $\rho$ is the density of the medium.

For fluids (liquids and gases), the relevant elastic modulus is the Bulk Modulus.

Derived Units→

Newton's Formula (Isothermal Assumption)

Isaac Newton was the first to derive a theoretical formula for the speed of sound in a gas. He made a critical assumption about the nature of the sound wave's propagation.

• Newton's Assumption: Newton proposed that the propagation of sound through a gas is an isothermal process, meaning the temperature of the gas remains constant. He reasoned that the compressions and rarefactions caused by the sound wave occur slowly enough for heat to be exchanged with the surroundings, keeping the temperature uniform.

Derivation

1. For an isothermal process, the relationship between pressure ($P$) and volume ($V$) is described by Boyle's Law: $PV=\text{Constant}$
2. When this relationship is used to derive the Bulk Modulus ($E=\frac{\text{Stress}}{\text{Strain}}=\frac{\Delta P}{\Delta V/V}$), it can be shown that the isothermal bulk modulus is equal to the pressure of the gas: $E_{\text{isothermal}}=P$
3. Substituting this into the general speed of sound equation gives Newton's Formula: $V=\sqrt{\frac{P}{\rho }}$

The Error in Newton's Method

When the values for standard atmospheric pressure ($P\approx 101325{\text{ N/m}}^2$) and the density of air ($\rho \approx 1.293{\text{ kg/m}}^3$) are used, Newton's formula gives a value for the speed of sound of: $V\approx 280\text{ m/s}$ This theoretical value is about 16% lower than the experimentally measured value of approximately 332 m/s. This discrepancy indicated that Newton's initial assumption was incorrect.

Laplace's Correction (Adiabatic Assumption)

Pierre-Simon Laplace corrected Newton's formula by proposing a more accurate assumption about the process.

• Laplace's Assumption: Laplace argued that the compressions and rarefactions in a sound wave occur too rapidly for significant heat exchange to take place between the gas and its surroundings. Therefore, the process is not isothermal but adiabatic (no heat exchange).

Corrected Derivation

1. For an adiabatic process, the relationship between pressure and volume is given by: $P{V}^{\gamma }=\text{Constant}$ Where $\gamma$ (gamma) is the adiabatic index (the ratio of specific heats, $C_p/C_v$). For diatomic gases like air, $\gamma \approx 1.4$.

2. The derivation of the bulk modulus for an adiabatic process shows that it is $\gamma$ times the pressure: $E_{\text{adiabatic}}=\gamma P$

3. Substituting this corrected modulus into the speed of sound equation gives the Newton-Laplace Formula: $V=\sqrt{\frac{\gamma P}{\rho }}$

The Correct Result

Using the value of $\gamma \approx 1.4$ for air, the corrected formula yields: $V=\sqrt{\frac{1.4\times 101325{\text{ N/m}}^2}{1.293{\text{ kg/m}}^3}}\approx 332\text{ m/s}$ This result is in excellent agreement with the measured speed of sound, confirming that sound propagation is indeed an adiabatic process.