Write an expression for the newton's formula for velocity of sound in air. Explain necessary correction made by Laplace.

 Newton's formula for the velocity of sound in air can be expressed as:

Where:

- \( v \) is the velocity of sound,

- \( γ \) (gamma) is the adiabatic index (ratio of specific heats),

- \( P \) is the pressure of the medium (air),

- \( ρ \) (rho) is the density of the medium (air).

Newton's formula provides a good approximation for the velocity of sound in ideal conditions. However, it neglects certain factors that can affect the speed of sound, such as the compressibility and temperature dependence of air.

Laplace introduced a correction to Newton's formula to account for these factors. He modified the formula by considering the adiabatic compression and expansion of air waves. The necessary correction made by Laplace involves incorporating the temperature dependence of the speed of sound and considering the specific heat ratio \( γ \) as a function of temperature.

The corrected formula by Laplace is:

Where:

- \( R \) is the gas constant,

- \( T \) is the temperature of the medium (air),

- \( M \) is the molar mass of the medium (air).

Laplace's correction takes into account the temperature dependency of the speed of sound and provides a more accurate prediction, especially at different temperatures and pressures.