CBSE Class 11-science Answered

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 | 12 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 | 12 May, 2021, 08:00: PM

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