Question

Newton’s formula for speed of sound was modified since
A. Units specifying speed of sound was incorrect
B. Calculated wave speed was much less that experimentally found
C. Calculated wave speed was much higher that experimentally found
D. Formula can be applied to certain wavelength

Answer 1

Hint: Newton’s formula was based on the assumption that the process of sound travel was isothermal. Whereas, the correction that Laplace proposed mentioned that the process was in fact adiabatic. Use this information to determine the velocities of sound before and after correction. Using the expression of bulk modulus in the two processes.

Formula used: The formula for velocity in a gas:
$v=\sqrt{\dfrac{B}{\rho }}$
The formula for Bulk modulus for an isothermal process:
$B=P$
The formula for Bulk modulus for an adiabatic process:
$B=\gamma P$

Complete step-by-step answer:
From purely theoretical consideration, newton gave an empirical relation to calculate the velocity of sound in gas.
$v=\sqrt{\dfrac{B}{\rho }}$… (1)
B is bulk modulus of the gas and $\rho $is the density of medium.
Sound travels through a gas in the form of compressions and rarefactions. Newton assumed that the changes in pressure and volume of a gas, when sound waves are propagated through it, are isothermal. The amount of heat produced during compression is lost to the surroundings and similarly the amount of heat lost during rarefaction is gained from the surroundings, so as to keep the temperature constant.
To find the Bulk modulus B:
P is the initial pressure of the gas, V is the initial volume of the gas.
For a process to be isothermal, the following condition is satisfied:
$PV=\text{constant}$.
And the bulk modulus B is:
$B=P$
Substituting this is equation (1), We get
$v=\sqrt{\dfrac{P}{\rho }}$
This was the formula proposed by Newton.
But Laplace pointed out an error. He challenged one of the assumptions taken by Newton. Which was that the process was isothermal, and corrected it to be adiabatic instead. Giving the following reasons:

(i) Velocity of sound in a gas is quite large. The pulses of compression and rarefaction, therefore, follow one another so rapidly that there is no lime left for any exchange of heat amongst themselves or with the surroundings.

(ii) A gas is a bad conductor of heat. It does not allow the free exchange of heat between compressed layer, rarefied layer and surroundings.
Thus, no exchange of heat is possible, when a sound wave passes through a gas. He modifies the law as:
$v=\sqrt{\dfrac{\gamma P}{\rho }}$,
Because for adiabatic process, the condition ${{\left( PV \right)}^{\gamma }}=\text{constant}$ is satisfied, which gives the bulk modulus B to be:
$B=\gamma P$
Where, $\gamma >1$
Therefore, the correct option is B. As velocity and wavelength are proportional for a constant frequency, and the correction proved the velocity to be more than what was initially taken. Newton's formula calculated the wavelength shorter than it was supposed to be.

Note: Retain that Newton assumed that the changes in pressure and volume of a gas, when sound waves are propagated through it, are isothermal, Laplace considered the process to be adiabatic as the velocity of sound in a gas is quite large.