Newton's Formula For The Velocity Of Sound In Gas - Practice Questions & MCQ

Speed of sound wave in gas: Newton's formula

The main assumption before deriving the equation is when the sound propagates through a gas, temperature variation in compression and rarefaction is negligible. So, Newton assumed that the exchange of heat with the surrounding, the temperature of the layer will remain the same. Hence this process is isothermal. Thus by using the formula that we have studied in the last concept, we can write that - 

$
v=\sqrt{\frac{B_{\text {isothermal }}}{\rho} \ldots \ldots .(i),(i)}
$

Where $B_{\text {isothermal }}=$ Isothermal Bulk modulus
Now, in the isothermal process, $\mathrm{PV}=$ Constant
Differentiating both sides, we get -

$
\begin{gathered}
P d v=V(-d P) \\
B_{\text {isothermal }}=P=\frac{d P}{\frac{d V}{V}}
\end{gathered}
$

So from the definition of Bulk modulus, we can say that the $\mathbf{P}=\mathrm{B}_{\text {isothermal }}$
$\left(\begin{array}{ll}A s, & \left.B_i=\frac{d P}{\frac{d V}{V}}\right)\end{array}\right.$
So from equation (i), We can write that -

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

This formula is given by Newton, So it is called Newton's formula.

Laplace correction

Laplace Correction gives correction to the speed of sound in the gas. Newton's formula was formulated taking into consideration that sound travels in isothermal conditions, the result so obtained was not matching with the experimental value of the speed of sound.

Thus, Laplace came up with a correction to it that sound travelling through air is a sudden process, it is well known as a Laplace Correction to Newton's Formula.

$
v=\sqrt{\frac{B_{\text {adiabatic }}}{\rho}}
$

Where $B_{\text {adiabatic }}=$ adiabatic bulk modulus
Now, in the adiabatic process, $\mathrm{PV}^V=$ Constant
Differentiating both sides, we get

$
\begin{gathered}
P \gamma V^{\gamma-1} d v=V^\gamma(-d P) \\
B_{\text {adiabatic }}=-\frac{d P}{\frac{d V}{V}}=\gamma P
\end{gathered}
$