$A$ tuning fork vibrating with a frequency of $512$ $Hz$ is kept close to the open end of a tube filled with water. The water level in the tube is gradually lowered. When the water level is $17$ $cm$ below the open end,maximum intensity of sound is heard. If the room temperature is $20^{\circ} C$,calculate:
$(a)$ Speed of sound in air at room temperature.
$(b)$ Speed of sound in air at $0^{\circ} C$.
$(c)$ If the water in the tube is replaced with mercury,will there be any difference in your observations?

(N/A) For maximum intensity of sound wave at the open end of a closed pipe (in the first mode),we have $L = \frac{\lambda}{4}$.
$\therefore \lambda = 4L = 4 \times 0.17 = 0.68 \ m$.
Now,$v = f\lambda = (512)(0.68) = 348.16 \ m/s$.
$(b)$ Speed of sound in air is $v \propto \sqrt{T}$.
$\therefore \frac{v_0}{v_{20}} = \sqrt{\frac{T_0}{T_{20}}}$,where $v_0$ is the speed of sound at $0^{\circ} C$ $(273 \ K)$ and $v_{20}$ is the speed at $20^{\circ} C$ $(293 \ K)$.
$\therefore v_0 = 348.16 \times \sqrt{\frac{273}{293}} \approx 336 \ m/s$.
$(c)$ Mercury is $13.6$ times denser than water. Its surface reflects sound waves much more efficiently than water. Therefore,when water is replaced by mercury,we get a greater intensity of the reflected sound. However,the wavelength and the speed of the sound wave will remain the same.