Thus, if B and ρ are considered to be the only relevant physical quantities, v = C ( . ) where, as before, C is the undetermined constant from dimensional analysis. The exact derivation shows that C = . Thus, the general formula for longitudinal waves in a medium is: v = ( .
) For a linear medium, like a solid bar, the lateral expansion of the bar is negligible and we may consider it to be only under longitudinal strain. In that case, the relevant modulus of elasticity is Young’s modulus, which has the same dimension as the Bulk modulus. Dimensional analysis for this case is the same as before and yields a relation like Eq. ( .
), with an undetermined C , which the exact derivation shows to be unity. Thus, the speed of longitudinal waves in a solid bar is given by v = Y ( . ) where Y is the Young’s modulus of the material of the bar. Table .
gives the speed of sound in some media. Table . Speed of Sound in some Media Liquids and solids generally have higher speed of sound than gases. [Note for solids, the speed being referred to is the speed of longitudinal waves in the solid].
This happens because they are much more difficult to compress than gases and so have much higher values of bulk modulus. Now, see Eq. ( . ).
Solids and liquids have higher mass densities ( ρ ) than gases. But the corresponding increase in both the modulus (B) of solids and liquids is much higher. This is the reason why the sound waves travel faster in solids and liquids. We can estimate the speed of sound in a gas in the ideal gas approximation.
For an ideal gas, the pressure P , volume V and temperature T are related by (see Chapter ). P V = Nk B T ( . ) where N is the number of molecules in volume V , k B is the Boltzmann constant and T the temperature of the gas (in Kelvin). Therefore, for an isothermal change it follows from Eq.( .