The equation for the energy density of a longitudinal wave in a thin rod is

$$w = \rho\cdot \left(\frac{\partial \xi}{\partial t}\right)^2$$

So the energy density seems to be 0 when the deviation ($\xi$) is maximal.

I don't understand how this can be true: if $\xi$ is big, this means that there must be big potential energy, since we have done the work of increasing $\xi$.


1 Answer 1


Although the part of the rod that is most displaced "wants" to get home, it is not because it somehow remembers its original location, but because its neighbors are pulling and pushing it back. Thus we say that the potential energy is most concentrated in those parts that are the most stretched or condensed (they pull/push the most).

The measure of stretchedness or condensedness is given by $\left|\dfrac{\partial \xi}{\partial x}\right|$, which is up to a constant factor equal (thanks to the wave equation) to $\left|\dfrac{\partial \xi}{\partial t}\right|$. So at the points of maximum deviation ($\partial \xi/\partial x=\partial \xi/\partial t=0$) the density is average, and there's no potential energy stored there.


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