A plane sound wave propagates along the `x` axis . The damping coefficient of the wave is `gamma=0.0230 m^(-1)`. At the point `x=0` the loudness level is `L=60 dB`. Find `:`
`(a)` the loudness level at a point with coordinate `x=50 m`,
`(b)` the coordinate `x` of the point at which the sound is not heard any more.

Explanation

`(a)` Loudness level in bells `=log((I)/(I_(0)))(I_(0)` is the theshold of audibility . `)`
So, loudness level in decibells, `L=10 log ((I)/( I_(0)))`
Thus loudness level at `x=x_(1)=L_(x_(1))=10 log((I_(x_(2)))/(I_(0)))`
Similarly `L_(x_(1))=10 log ((I_(x_(1)))/( I_(0)))`
Thus `L_(x_(2))-L_(x_(1))=10 log ((I_(x_(2)))/( I_(x_(1))))`
or, `L_(x_(2))=L_(x_(1))+ 10log. (1//2rho a^(2) omega^(2) v e ^(-2 gamma x_(2)))/(1//2 rho a^(2) omega^(2) v e ^(-2 gammax_(1)))=L_(x_(1))+ 10 log e^(-2gamma(x_(2)-x_(1)))`
`L_(x_(2))=L_(x_(1))-20 gamma(x_(2)-x_(1)) log e `
Hence `L^(')=L-20 gamma x log e [` since `(x_(2)-x_(1)) log e`
`=20dB-20 xx 0.23xx50xx0.4343dB`
`=60 dB-10 dB=50dB`
`(b)` The point at which the sound is not heard any more, the loudness level should be zero.
Thus
`0-L-20 gamma x log e ` or `x=(l)/( 20 gamma log e )=(60)/( 20 xx 0.23 xx0.4343)=300m`