Air is a very poor conductor of heat. Hence at a compression, air cannot lose heat due to radiation and conduction. At a rarefaction it cannot gain heat, during the small interval of time. As a result, the temperature throughout the medium does not remain constant.

*Laplace?s correction*

The above discrepancy between
the observed and calculated values was explained by Laplace in 1816. Sound
travels in air as a longitudinal wave. The wave motion is therefore,
accompanied by compressions and rarefactions. At compressions the temperature
of air rises and at rarefactions, due to expansion, the temperature decreases.

Air is a very poor
conductor of heat. Hence at a compression, air cannot lose heat due to
radiation and conduction. At a rarefaction it cannot gain heat, during the
small interval of time. As a result, the temperature throughout the medium does
not remain constant.

Laplace suggested that
sound waves travel in air under adiabatic condition and not under isothermal
condition.

For an adiabatic
change, the relation between pressure and volume is given by

P
V ^{T} = constant

where
γ = C_{p}/Cv is the ratio of two
specific heat capacities of the gas.

Differentiating

P
γ V ^{γ-1} . dV + V^{ γ} dP = 0

P
γ = (V ^{γ-1 }/ V ^{γ-1}) dP/dV

P
γ = -dP / (dV/V) = k

∴ P γ = k (Volume
elasticity) Therefore under adiabatic condition

velocity
of sound v = root( k/p ) = root(γP / p)

This
is Laplace?s corrected formula.

For
air at NTP

γ
= 1.41, ρ = 1.293 kg m^{?3}

This
result agrees with the experimental value of 332 ms^{?1}.

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11th 12th std standard Class Physics sciense Higher secondary school College Notes : Laplace?s correction |

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