Class 11 > Unit # 12:Acoustics > Newton's Formula for Speed of Sound

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Newton’s Formula & Laplace Correction for the Speed of Sound in Air

Video Lecture: Speed of Sound, Newton's Formula & Laplace Correction

Watch the complete step-by-step video lecture explaining Newton's assumption, the velocity discrepancy, and Laplace's adiabatic correction:

1. Newton’s Formula for the Speed of Sound in Air

The speed of sound in any medium (solid, liquid, or gas) fundamentally depends on the elasticity and density of that medium according to the general relation:

$v = \sqrt{\frac{E}{\rho}}$

Sir Isaac Newton assumed that the propagation of sound waves through air (or other gases) takes place under constant temperature conditions. In other words, he considered the process to be isothermal, meaning Boyle's Law can be applied:

$PV = \text{constant}$

When sound travels, compression regions experience an increase in pressure ($\Delta P$) and a decrease in volume ($\Delta V$), whereas rarefaction regions experience a decrease in pressure and an increase in volume. Let:

• Initial Pressure = $P$, Initial Volume = $V$
• Increase in Pressure = $\Delta P$, Decrease in Volume = $\Delta V$
• Final Pressure = $P + \Delta P$, Final Volume = $V - \Delta V$

Applying Boyle's law to the initial and final states:

$P V = (P + \Delta P)(V - \Delta V)$

$PV = PV - P\Delta V + V\Delta P - \Delta P \Delta V$

Since the changes in pressure ($\Delta P$) and volume ($\Delta V$) are extremely small, their product ($\Delta P \Delta V$) is negligible and can be omitted:

$0 = -P\Delta V + V\Delta P \implies P\Delta V = V\Delta P \implies P = \frac{\Delta P}{\frac{\Delta V}{V}}$

Since Bulk Modulus $B$ (Modulus of Elasticity $E$) is defined as volumetric stress divided by volumetric strain ($E = B = \frac{\Delta P}{\Delta V / V}$), it follows that under isothermal conditions:

$E = P$

Substituting this elasticity into the general wave speed equation yields Newton's Formula:

$v = \sqrt{\frac{P}{\rho}}$

2. Newton’s Defect and Laplace Correction

At Standard Temperature and Pressure (S.T.P.), the theoretical speed of sound calculated using Newton's formula comes out to be approximately $280\text{ m/s}$, whereas the experimentally measured value is around $332\text{ m/s}$. The theoretical value is roughly $15.6\%$ less than the experimental observation—a significant discrepancy that Newton could not resolve.

Laplace's Resolution: The French scientist Pierre-Simon Laplace pointed out that sound waves travel too rapidly through air for heat exchanges to equalize with the surroundings. At compressed regions, particles crowd together, releasing heat and raising the temperature. At rarefied regions, particles spread apart, causing a drop in temperature. Because heat cannot flow in or out quickly enough across these rapid compressions and rarefactions, the process is not isothermal; instead, it is adiabatic.

For an adiabatic process, the gas obeys Poisson's law:

$P V^\gamma = \text{constant} \quad \text{--- (i)}$

Where $\gamma = \frac{C_p}{C_v}$ is the ratio of molar heat capacities.

Differentiating equation (i) for a change in pressure from $P$ to $P + \Delta P$ and volume from $V$ to $V - \Delta V$:

$(P + \Delta P)(V - \Delta V)^\gamma = P V^\gamma$

Expanding and applying Binomial theorem while neglecting higher-order small terms, Laplace derived the adiabatic elasticity as $E = \gamma P$. Substituting this into the speed formula gives Laplace's corrected expression:

$v = \sqrt{\frac{\gamma P}{\rho}}$

For diatomic air, $\gamma = 1.42$. Multiplying Newton's value by $\sqrt{1.42}$ raises the theoretical speed to match the experimental value of $332\text{ m/s}$ at $0^\circ\text{C}$, bringing theory and experiment into excellent agreement.

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