Question

How long will it take sound waves to travel the distance between the points A and B, separated by a distance l, if the air temperature between them varies linearly from $T_1$ to $T_2$? The velocity of sound propagation in air is equal to $v=\alpha \sqrt{T}$
Where $\alpha$ is a constant.

• A. $t=\frac{2l\alpha }{(\sqrt{T_2}+\sqrt{T_1})}$
• B. $t=\frac{2\alpha }{l(\sqrt{T_2}+\sqrt{T_1})}$
• C. $t=\frac{2l}{\alpha (T_2+T_1)}$
• D. $t=\frac{2l}{\alpha (\sqrt{T_2}+\sqrt{T_1})}$

Answer 1

The correct option is D $t=\frac{2l}{\alpha (\sqrt{T_2}+\sqrt{T_1})}$

For linear variation of temperature, we can write temperature at a distance x from point A is
$T_x=T_1+\frac{T_2-T_1}{l}x$
Thus velocity of sound at this point is given as
$v=\alpha \sqrt{T_1+\left(\frac{T_2-T_1}{l}\right)x}$
$\frac{dx}{dt}=\alpha \sqrt{T_1+\left(\frac{T_2-T_1}{l}\right)x}$
$\Rightarrow \frac{dx}{\sqrt{T_1+\left(\frac{T_2-T_1}{l}\right)x}}=\alpha dt$
Integrating the above expression within proper limits, we get
${\int }_0^1\frac{dx}{\sqrt{T_1+\left(\frac{T_2-T_1}{l}\right)x}}={\int }_0^t\alpha dt\Rightarrow t=\frac{2l}{\alpha (\sqrt{T_2}+\sqrt{T_1})}$