Problem 4.164

$4.164 .$ The equation of a plane standing wave in a homogeneous elastic medium has the form $\xi=a \cos k x \cdot \cos \omega t .$ Plot: (a) $\xi$ and $\partial \xi / \partial x$ as functions of $x$ at the moments $t=0$ and $t=T / 2$, where $T$ is the oscillation period; (b) the distribution of density $\rho(x)$ of the medium at the moments $t=0$ and $t=T / 2$ in the case of longitudinal oscillations; (c) the velocity distribution of particles of the medium at the moment $t=T / 4 ;$ indicate the directions of velocities at the antinodes, both for longitudinal and transverse oscillations.