A sound wave of frequency 210 Hz travels with a speed of $330 \mathrm{~ms}^{-1}$ along the positive $X$-axis. Each particle of the wave moves a distance of 10 cm between the two extreme points. The equation of the displacement function ( s ) of this wave is ( $x$ in metre, $t$ in second)

A string vibrates in its fundamental mode when a tension $T_1$ is applied to it. If the length of the string is decreased by $25 \%$ and the tension applied is changed to $T_2$, the fundamental frequency of the string increases by $100 \%$, then $\frac{T_2}{T_1}=$

(Linear density of the string is constant)

If the lengths of the open and closed pipes are in the ratio of $2: 3$, then the ratio of the frequencies of the third harmonic of the open pipe and the fifth harmonic of the closed pipe is

The equation of a transverse wave propagating on a stretched string is given by $y=3 \sin (4 x+200 t)$, where $x$ and $y$ are in metre and the time ' $t$ ' is in second. If the tension applied to the string is 500 N , the linear density of the string is