Find the damping coefficient v of a sound wave if at distances `r_(1)=10m` and `r_(2)=20 m` from a point isotropic source of sound the sound wave intensity values differ by a factor `eta=4.5`.

Explanation

Intensity of a spherical sound wave emitted from a point source in a homogeneous absorbing medium of wave damping coefficient `gamma` is given by
`I=(1)/(2) rho a^(2) e^(-2gammar)omega^(2) v`
So, Intensity of sound at a distance `r_(1)` from the source
`(I_(1))/(r_(1)^(2))=(1//2rho a^(2) e^(-2gamma r_(1))omega^(2) v)/(r_(1)^(2))`
and intensity of sound at a distance `r_(2)` from the source
`=I_(2)//r_(2)^(2)=(1//2rhoa^(2)e^(-2gammar_(2))omega^(2)v)/(r_(2)^(2))`
But according to the problem `(1)/(eta)(I_(1))/(r_(1)^(2))=(I_(2))/(r_(2)^(2))`
So, `(etar_(1)^(2))/( r_(2)^(2))=e^(2 gamma(r_(2)-r_(1)))`or`1n(etar_(2)^(2))/(r_(1)^(2))=2 gamma(r_(2)-r_(1))`
or,` gamma =(1n(etar_(2)^(2)//r_(1)^(2)))/(2(r_(2)-r_(1)))=6xx10^(-3)m^(-1)`