Problem 4.165

A longitudinal standing wave $\xi=a \cos k x \cdot \cos \omega t$ is maintained in a homogeneous medium of density $\rho$ . Find the expressions for the space density of (a) potential energy $w_{p}(x, t)$ (b) kinetic energy $w_{k}(x, t)$ . Plot the space density distribution of the total energy $w$ in the space between the displacement nodes at the moments $t=0$ and $t=T / 4$ where $T$ is the oscillation period.

\begin{aligned} &\text { 4.165. (a) } w_{p}=1 / 2 \rho a^{2} \omega^{2} \sin ^{2} k x \cdot \cos ^{2} \omega t\\ &\text { (b) } w_{k}=1 / 2 \rho a^{2} \omega^{2} \times\\ &\times \cos ^{2} k x \cdot \sin ^{2} \omega t . \text { See Fig. } 37 \end{aligned}

Elastic Waves. Acoustics.