How long will it take sound waves to travel a distance `l` between points A and B if the air temperature between them varies linearly from `T_1` and `T_2`? (The velocity of sound in air at temperature `T` is given by `v=alphasqrt(T)`, where `alpha` is a constant)

Explanation

For linearvariation oftemperature, wecanwritetemperature at a distance x from point A is
`T_(x) = T_(1) + (T_(2) - T_(1))/(l) x`
Thus velocity of sound at this point is given as
`v = alphasqrt(T_(1)+((T_(2) - T_(1))/(l)) x)`
or `(dx)/(dt) = v = alphasqrt(T_(1)+((T_(2) - T_(1))/(l)) x)`
or `alpha (dx)/(sqrt(T_(1) + ((T_(2) - T_(1))/(l)))x) = dt`
Integrating the above expression within proper limits, we get
or `underset(0)overset(l)(int)(dx)/(alphasqrt(T_(1)+((T_(2)-T_(1))/(l)))x) = underset(0)overset(t)(int)dt`
`(2I)/(alpha(T_(2)-T_(1)))[sqrt(T_(1) + (T_(2)-T_(1))/(l))x]_(0)^(l) = t`
`t = (2I)/(alpha(T_(2)-T_(1)))[sqrt(T_(2)) - sqrt(T_(1))]`
`t = (2l)/(alpha[sqrt(T_(2)) + sqrt(T_(1))])`
Alternative Solution :
Given that ` v = alphasqrt(T)`
or velocity at point A is `v_(1) = alphasqrt(T_(1))`
velocity at point B is `v_(2) = alpha sqrt(T_(2))`
The average velocity `overline(v) = (v_(1)+v_(2))/(2))`
`overline(v) = (alpha(sqrt(T_(1)) + sqrt(T_(2))))/(2)`