A particle is performing simple harmonic motion about $x = 0$ with an amplitude '$a$' and periodic time T. The speed of the particle at $x = \dfrac{a}{3}$ will be

Two sources of sound are emitting progressive waves $y_1 = 4\sin 708\,\pi t$ and $y_2 = 3\sin 700\,\pi t$. The sources are placed close to each other. The number of beats heard per second and intensity ratio between waxing and wanning are respectively

Fifth overtone of an open pipe of length $L_0$ is in unison with the fifth overtone of the pipe closed at one end of length $L_c$.
The ratio $L_0$ to $L_c$ is

The lengths of the two pipes open at both ends are L and $(L + L_1)$. If they are sounded together, the beat frequency will be
($v$ = velocity of sound in air)