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

Since the temperature varies linearly we can write the temperature as a fuction of `x`, which is, the disance from the point `A` towards `B`.
i.e., ` T=T_(1)+(T_(2)-T_(1))/(l)x, [0 lt x lt l]`
hence, `dT=(T_(2)-T_(1))/(l)dx ...(1)`
In order to travel an elemental distance of `dx` which is at a distance of `x` from `A` it will take a time
`dt=(dx)/( alpha sqrt(T)) ....(2)`
From Eqns `(1)` and `(2)`, expressing `dx` in terms of `dT` , we get
`dt=(l)/( alpha sqrt(T))((ldT)/(T_(2)-T_(1)))`
Which on integration gives
`int_(0)^(t)dt=(l)/( alpha(T_(2)-T_(1)))int_(T_(1))^(T_(2))(dT)/(sqrt(T))`
or `t=(2l)/(( T_(2)-T_(1)))(sqrtT_(2)-sqrt(T_(1)))`
Hence the sougth time `t=(2l)/( alpha(sqrt(T_(1))+sqrt(T_(2))))`