Laplace Correction

The velocity of sound is given by \(v=\sqrt{\frac{B}{\rho }}\) and has an experimental value of 332 m/s.

Newtons formula for the speed of sound

Newtons worked on the propagation of sound waves through the air. He assumed that this process of propagation is isothermal. Absorption and release of heat during compression and rarefaction will be balanced, thus, the temperature remains constant throughout the process.

According to Boyle’s law

PV = Constant

Where,

P is pressure

V is the volume of gas.

On differentiating above equation we get-

PdV+VdP=0

⇒ PdV = -VdP

\(\Rightarrow P=-\frac{VdP}{dV}=\frac{dP}{-\left ( \frac{dV}{V} \right )}\)

⇒ P = B

Where, \(B=\frac{dP}{-\left ( \frac{dV}{V} \right )}\) is bulk modulus of air.

The velocity of the sound wave can be written as –

\(v=\sqrt{\frac{B}{\rho }}\)

Thus substituting B =P we get-

\(v=\sqrt{\frac{P}{\rho }}\)

Speed of sound in air

At Normal Temperature and Pressure, the velocity of sound in air is given by –

\(v=\sqrt{\frac{P}{\rho }}\)

Where atmospheric pressure P = 1.1013×105 N/m2

The density of air (𝜌)= 1.293 kg/m3

\(v=\sqrt{\frac{P}{\rho }}=\sqrt{\frac{1.013\times 10^{5}}{1.293}}=280 m/s\)

The value got here does not match with the experimental value. That is 332m/s. Which implies that some correction should be done to Newton’s equation.

Laplace Correction for Newton’s Formula

He corrected the Newtons formula by assuming that, there is no heat exchange takes place as the compression and rarefaction takes place very fast. Thus, the temperature does not remain constant and the propagation of the sound wave in air is an adiabatic process.

For an adiabatic process

PV𝛾 = Constant

Where,

𝛾 is adiabatic index \(\gamma =\frac{C_{p}}{C_{v}}\)

Cp specific heat for constant pressure

Cv specific heat for constant volume.

Differentiating both the sides we get-

\(V^{\gamma }dP+P\gamma V^{\gamma -1}dV=0\)

Dividing both the sides by V𝛾-1

\(VdP+P\gamma VdV=0\) \(P\gamma =-\frac{dP}{\left ( \frac{dV}{V } \right )}=B\)

The velocity of sound is given by

\(v=\sqrt{\frac{B}{\rho }}\)

Substituting B= 𝛾P in above equation we get-

Velocity of sound formula

\(v=\sqrt{\frac{\gamma P}{\rho }}\)

Velocity of sound

Calculate the velocity of sound wave using Laplace correction to Newton’s formula at Normal Temperature and Pressure.

Velocity of the sound formula is given by-

\(v=\sqrt{\frac{\gamma P}{\rho }}\)

Where,

Adiabatic index 𝛾 – 1.4

Where atmospheric pressure P = 1.1013×105 N/m2

The density of air (𝜌)= 1.293 kg/m3

Substituting the values in the equation we get-

\(v=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{1.4\times 1.013\times 10^{5}}{1.293}}=332 m/s\)

Which has a very good match with the experimental value

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