Newton’s Formula and Factors Affecting for Velocity of Sound in a Gas

Newton’s Formula for Velocity of Sound in a Gas

When a sound wave travels through the medium, there alternate rarefactions and compression and rarefactions are produced.

Newton assumed that in compression heat is produced by vibrating particles which transfer into surrounding. The rarefaction heat is lost by particles so heat is taken from surrounding. As a whole heat lost is equal to heat gain i.e. sound wave travels on air by isothermal process.

In isothermal process,

\begin{align*} PV &= \text {constant} \\ \text {Differentiating equation} \: (i)\: \text {we get}, \\ PdV + VdP &= 0 \\ \text {or,} \: PdV & = -VdP \\ \text {or,} \: P &= - V\frac {dP}{dV} \\ \text {or,} \: P &= \frac {dP}{(-dV/V)} = B\\ \therefore B = \text {Bulk modulus of gas air}\\ \therefore B &= \frac {dP}{(-dV/V)} \\ \text {or,} \: P &= B \\\text {where, P} = \text {atmospheric pressure} \\ \text {So, the velocity of sound in air is written by} \\ V &= \sqrt {\frac {E}{\rho }} = \sqrt {\frac {B}{\rho }} \\ \text {or,} \: \sqrt {\frac {P}{\rho }} \: [\therefore P = B]\\ \text {At NTP, Atmospheric pressure (p)} &= 760\: \text {mm of Hg} \\ &= 760\: \text {mm of Hg} \\ &= 1.013\times 10^5 N/m^2 \\ \text {density of air at NTP,} = \rho = 1.293 \: kg/m^3 \\ \text {Velocity of sound in air at NTP becomes,} \\ V &= \sqrt {\frac {1.013 \times 10^5}{1.293}} \\ \therefore V &= 280 \: m/s \\ \end{align*}

Laplace Correction for Velocity of Sound

Newton’s formula for velocity is wrong as it gives value 280 m/s which is wrong because velocity of sound calculated by modern experiment is 332 m/s.

After 100 years Laplace corrected the Newton’s assumption for velocity of sound. Newton assumed that the compression and rarefaction is slow process so sound waves propagate through an isothermal process in gas. According to Laplace, the processes of compression and rarefaction occur so rapidly that neither heat is transferred to the surrounding during rarefaction. Thus, the temperature in different region does not remain constant. So, the sound waves in a gas propagate through an adiabatic process. The equation of an adiabatic process is

\begin{align*} PV^{\gamma} &= \text {constant} \dots (iv) \end{align*}

Where γ is the ratio of molar heat capacity of the air at constant pressure to that at constant volume (C_p/C_v =γ). Differentiating equations (iv) on both sides, we get

\begin{align*} V^{\gamma} dP + P(\gamma V^{\gamma - 1}dV) &= 0 \\ \text {Dividing this equation by } V^{\gamma - 1} \: \text {we get} \\ \text {or,} \: VdP + \gamma P\: dV = 0 \\ \text {or,} \: \gamma P &= - \frac {VdP}{dV} \\ \text {or,} \: \gamma P &= - \frac {dP}{dV/V} \\ \text {or,} \: \gamma P &= B \dots (v) \\ \text {On substituting this value of B in equation of the wave,} \\ v &= \sqrt {\frac {B}{\rho}}, \text {we get} \\ v &= \sqrt {\frac {\gamma P}{\rho}} \dots (vi) \\ \text {For air}, \gamma = 1.4 \text {and at NTP, velocity of sound in air is given by} \\ v &= \sqrt {\frac {\gamma P}{\rho}} \\ &= \sqrt {\frac {1.4 \times 1.013 \times 10^5}{1.293}} \\ &= 331.2 ms^{-1} \end{align*}

Thus result closely agrees with the experimental value. Thus, Laplace’s formula gives the correct velocity of the sound in air.

Factors Affecting the Velocity of Sound in a Gas

1. Effect of temperature
   The ideal gas equation is given by
   \begin{align*} PV &= n RT \\ \text {or,} PV &= \frac {m}{M}RT \: \left ( m = \text {actual mass of gas molecules} \: M = \text {molar mass} \right ) \\ \text {or,} \: \frac {PV}{m} &= \frac {RT}{M} \\ \therefore \frac {P}{\rho } &= \frac {RT}{M} \\ \text {According to Laplace correction} \\ v &= \sqrt {\frac {\gamma P}{\rho }} \\ &= \sqrt {\frac {\gamma RT}{M }} \\ \text {For any gas medium}\\ \gamma , R, M \text {are constant} \\ \therefore v \propto \sqrt {T} \\ \end{align*}
   For two different gas medium V₁ and V₂ be the velocities of sound at temperature T₁ and T₂ then,
   $$ \frac {v_1}{v_2} = \sqrt {\frac {T_1}{T_2}} $$
   Hence, velocity of sound is directly proportional to square root of temperature of gas. If the temperature increases then velocity of sound increases.
2. Effect of pressure
   $$\text {Since,}\: v = \sqrt {\frac {\gamma P}{\rho }} = \sqrt {\frac {\gamma RT}{M}} $$
   at constant temperature, the velocity of sound is independent to the pressure.
3. Effect of density
   \begin{align*} \text {Since,}\: v = \sqrt {\frac {\gamma P}{\rho }} \\ \text {at constant pressure,} \: v \propto \frac {1}{\sqrt {\rho }} \\ \end{align*}
   i.e. velocity of sound in inversely proportional to density of a medium.
4. Effect of humidity
   The density of humid air is low than that of dry air. Since, the velocity of sound is inversely proportional to the square root of density. So, a velocity of sound is more humid air than in dry air.
5. Effect of direction of wind
   The velocity of sound increases with the same direction of wind but velocity of sound decreases in opposite direction of the wind.
   $$ v = v_s \pm v_w $$
6. Effect of amplitude, frequency and wavelength
   The velocity of sound is independent to the amplitude, frequency and wavelength of around but if amplitude becomes very high then the velocity of sound may change.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa,. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.