Question

How long will it take sound waves to travel the distance l between the points A and B. If the air temperature between them varies linearly from \[{T_1}\] to \[{T_2}\]. The velocity of sound propagation in air is given by \[v = \alpha \sqrt T \] where α is a constant:
A. \[\dfrac{{2l}}{{\alpha \sqrt {{T_1}{T_2}} }}\]
B. \[\alpha l\sqrt {{T_1}/{T_2}} \]
C. \[\alpha l\sqrt {{T_1} + {T_2}} \]
D. \[\dfrac{{2l}}{{\alpha \left( {\sqrt {{T_1} + {T_2}} } \right)}}\]

Answer 1

Hint:The above problem can be resolved using the linear equation of temperature, which depends on the distance between the points. In addition, the mathematical relation for the time taken by the element to cover the distance is used, and on further substituting the values, the resultant equation is obtained. This equation can be integrated with the desired limits to obtain the correct relation.

Complete step by step answer:
As the temperature varies linearly. Then we can write the equation of temperature as the function of x. Here, x is the distance from the point A and Point B. Such that \[0 < x < l\].
Then the equation is,
\[\begin{array}{l}
T = {T_1} + \left( {\dfrac{{{T_2} + {T_1}}}{l}} \right)x\\
dT = \left( {\dfrac{{{T_2} + {T_1}}}{l}} \right)dx..........................\left( 1 \right)
\end{array}\]
The time taken by the element to cover the distance x is given as,
\[dt = \dfrac{{dx}}{{\alpha \sqrt T }}.........................................\left( 2 \right)\]
 From the equation 1 in 2 expressing the dx in terms of dT, we get
\[\begin{array}{l}
dt = \dfrac{{dx}}{{\alpha \sqrt T }}\\
dt = \dfrac{l}{{\alpha \sqrt T }}\left( {\dfrac{{ldT}}{{{T_2} + {T_1}}}} \right)
\end{array}\]
Integrating the above equation as,
\[\begin{array}{l}
dt = \dfrac{l}{{\alpha \sqrt T }}\left( {\dfrac{{dT}}{{{T_2} + {T_1}}}} \right)\\
\int\limits_0^t {dt} = \dfrac{l}{{\alpha \sqrt T }}\int\limits_{{T_1}}^{{T_2}} {\left( {\dfrac{{dT}}{{{T_2} + {T_1}}}} \right)} \\
t = \dfrac{{2l}}{{\alpha \left( {{T_2} + {T_1}} \right)}}\left( {\sqrt {{T_2}} + \sqrt {{T_1}} } \right)\\
t = \dfrac{{2l}}{{\alpha \left( {\sqrt {{T_1}} + \sqrt {{T_2}} } \right)}}
\end{array}\]
Therefore, the time taken by the wave to travel a distance l is \[\dfrac{{2l}}{{\alpha \left( {\sqrt {{T_1}} + \sqrt {{T_2}} } \right)}}\] and option D is correct.

Note: In order to resolve the given condition, one must try to remember the mathematical formula for the temperature in the form of linear dependency with the distance. Moreover, the desired results are to be obtained by applying the additional formula of the time taken to cover the distance by a sound wave. The speed of sound depends on the medium, where the propagation is taking place. Hence the result may vary as per the given conditions.