The velocity of the wave in of sound depends on bulk modulus and density of the medium derive the formula of velocity

Asked by umeshlochab887 | 12th May, 2021, 06:17: PM

Expert Answer:

Velocity of sound v in a gaseous medium is related to Bulk modulus B  and density ρ .
 
Velocity of sound v is given as
 
begin mathsize 14px style v space equals space square root of B over rho end root end style  .................................(1)
Bulk modulus B is given as
 
begin mathsize 14px style B space equals space fraction numerator s t r e s s over denominator s t r a i n end fraction space equals space fraction numerator increment P over denominator begin display style bevelled fraction numerator negative increment V over denominator V end fraction end style end fraction space equals space minus space V thin space fraction numerator d P over denominator d V end fraction end style  ..............................(2)
When sund waves is propagated in a gaseous medium, compression and rarefaction processes are fast ,
hence we can assume adiabatic condition for change in pressure and volume.
 
For adiabatic process, we have begin mathsize 14px style P V to the power of gamma equals space C end style , where C is constant and γ is ratio of specific heat.
 
Hence we get  begin mathsize 14px style fraction numerator d P over denominator d V end fraction space equals space minus C space gamma space V to the power of negative gamma minus 1 end exponent end style
Hence from eqn.(2) , we get , begin mathsize 14px style B space equals space minus V fraction numerator d P over denominator d V end fraction space equals space C space gamma space V to the power of negative gamma end exponent space equals space gamma space P end style ....................(3)
Density begin mathsize 14px style rho space equals space fraction numerator n space M over denominator V end fraction end style ..........................(4)
where n is number of moles in a volume V and M is molar mass of gas.
 
Using eqn.(3) and eqn.(4) , we rewrite eqn.(1) as
 
begin mathsize 14px style v space equals space square root of fraction numerator gamma space P space V over denominator n space M end fraction end root space equals space square root of fraction numerator gamma space R space T over denominator M end fraction end root end style
In above expression, we have used ideal gas equation , pV = ( n R T )

Answered by Thiyagarajan K | 12th May, 2021, 08:00: PM