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Elastic Waves. Acoustics.

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.

Reveal Answer
 4.165. (a) wp=1/2ρa2ω2sin⁡2kx⋅cos⁡2ωt (b) wk=1/2ρa2ω2××cos⁡2kx⋅sin⁡2ωt. See Fig. 37\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}
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