Laplace Correction

The variation in pressure and volume of gas caused by the passage of sound waves, according to Laplace, is adiabatic rather than isothermal, as Newton claimed. As a result, the adiabatic bulk modulus is the suitable elasticity modulus. The computation of the speed of sound in a gas has been corrected. When a sound wave propagates through a gas, Newton thought that the pressure–volume changes occur in an isothermal manner. Using the assumption that pressure–volume changes are adiabatic, Laplace was capable of achieving agreement between theory and experiment.

Laplace Correction

• To modify the sound speed in a gas, Laplace correction is used. Laplace devised a theoretical and practical solution to the problem. As a result, the correction to Newton’s Formula is known as a Laplace correction. According to Laplace, sound waves propagate in an adiabatic environment.
• Because air has a low thermal conductivity, compression and rarefaction in the air will occur quickly, leading to heat flowing neither out of nor into the system, indicating an adiabatic situation. For sound waves in an air or gaseous medium, this is referred to as the Laplace Correction.
• The sound speed in the gas is corrected using Laplace correction. Newton estimated the formula for the speed of sound in a gaseous medium, assuming that sound waves propagate in an isothermal medium such as air or gas. This assumption was proved to be false when the speed of sound in the air was measured at 280 m/s, which was incorrect
•  There is a minimal increase in temperature in a compression region and a marginal decrease in temperature in a rarefaction region. According to Laplace, because these pressure changes occur quickly and air is a poor conductor of heat, temperature equalisation between the various zones was improbable. He believed that temperature changes occur under adiabatic conditions, which means that no heat enters or exits the gas from the outside. The heat generated in the compressed layers is completely confined to these layers and does not have time to diffuse across the entire gas body

Laplace Correction Formula

Because the compression and rarefaction motions are exceedingly fast, Laplace revised Newton’s formula by assuming that no heat exchange occurs. As a consequence, the temperature does not stay unchanged, and sound waves are transmitted via an adiabatic process through the air. As a result, in the case of an adiabatic process:

 PV=constant

Here is the ratio of specific heat capacity and is equal to =Cp/Cv

Here Cp is the specific heat at constant pressure and

 Cv is the specific heat at constant volume

By differentiating both we get: 

 γPVγ-1dV+VdP=0

Dividing both sides by Vγ-1 we get:

 dP+ γPV-1dV= 0

 Pγ= -VdPdV=B

As the velocity of sound is given as v= B/ρ

By substituting B=γP We get :

a v= γP/ρ

Speed of Sound in Different Media

Mechanical waves can only travel through matter, therefore sound waves are mechanical waves. As a result, the medium is the stuff through which the waves pass. The configurations of particles and the forces between them differ in solids, liquids, and gases. Solids have the closest particle spacing, while gases have the most space between them. The energy of vibrations can be passed to neighbouring particles more quickly when particles are nearer together.

Speed of sound in Solids

Mechanical waves are sound waves that are defined by the motion of particles in the medium. The density of molecules is greatest in solids and lowest in gases. This means that the particles in solids are closer together than those in liquids or gases. An individual particle in a solid can collide with its neighbouring particle in less time as a result of this. As a result, a disturbance can move far more easily and swiftly through solids than through liquids or gases, and hence the speed of sound is greatest in solids.

Speed of sound in Liquids

Liquid particles are looser than solid particles. Liquids have a lower density of molecules than solids or gases, as a result. As a result, the distance between liquid molecules is higher than that between solid molecules, but still less than that between gas molecules. As a result, the speed of sound in liquids is slower than in solids but faster than in gases.

Conclusion

The speed of sound is defined as the distance travelled by sound waves in a given length of time. Solids, liquids, and gases, on the other hand, cannot travel in a vacuum. Sound waves travel at their fastest in solids. Although the speed of sound in liquids is slower than in solids, it is faster than the speed of sound in gases. The velocity of sound  formula was rectified by Laplace, who assumed that the transmission of sound waves was an adiabatic process and came up with the formula, where is the ratio of specific heat capacity.