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 = Cp/Cv is the ratio of two specific heat capacities of the gas.
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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