Two plane sonic waves of same frequency are travelling in a homogeneous medium. Loudness recorded by a moving detector in the medium varies from `L_(1) = 30 db " to" L_(2) = 43 . 9794 ` dB . Calculate intensity of the two waves.
Give `log_(1) 2 = 0 . 30103 `

Explanation

If due to a sonic wave, intensity at a point is I then loudness at the point is given by
`L = 10 log_(10)()/(I_(0)) dB " where " I_(0) = 10^(-12) Wm^(-2)`
Since minimum loudness recorded is ` L_(1)` , therefore, minimum intensity in the medium is
`I_("min") = I_(0) . "antil " log_(10) ((L_(1))/(10))`
or `I_("min") = 1 xx 10^(-9) Wm^(-2)`
Similarly `I_("max") = I_(0) . "antil" log_(10 ((L_(2))/(10)) = 25 xx 10 ^(-9) Wm^(-2)`
Since two waves are of same frequency, therefore, maximum intensity in the medium will be
`I_("max") = (sqrt(I_(1)) + sqrt(I_(2)))^(2)`
Where `I_(1) and I_(2) ` are intensities of two waves and minimum intensity will be
`I_("min") = (sqrt(I_(1))- sqrtI_(2)))^(2)`
Hence, `(sqrt(I_(1)) + sqrt(I_(2)))^(2) = 25 xx 10^(-9) Wm^(-2) " " . . . (i)`
and `(sqrt(I_(1))-sqrt(I_(2)))^(2) = 1 xx 10^(-9) w m^(-2) " ". . . (ii)`
`I_(2) = 9 xx 10^(-9) W m^(-2)`
`I_(2) = 4 xx 10^(-9) Wm^(-2)`