Sound Waves — Question 2

Question 2

A displacement gradient identifies a compression

Equilibrium fluid coordinates are $X$. At an instant their displacements are $\xi(X)=a\sin(kX)$, with $a=2.0\,\mu\mathrm m$ and $k=20\,\mathrm{m^{-1}}$. For a small element, mass conservation gives $\rho/\rho_0=1/(1+\partial_X\xi)$.

1. Find the first-order density perturbation and the locations of strongest compression.

2. Compare two small elements centered at $kX=0$ and $kX=\pi$, each initially $1.0\,\mathrm{mm}$ long.

Question 2 - Solution

A displacement gradient identifies a compression

(a) Method and calculation

The separation map is $dx=(1+\xi_X)dX$, where $\xi_X=ak\cos kX$. Linearizing its reciprocal gives$$\frac{\delta\rho}{\rho_0}\simeq-ak\cos kX=-4.0\times10^{-5}\cos kX.$$Maximum compression occurs at $kX=(2m+1)\pi$, where the displacement gradient is most negative.

(b) Interpretation and further calculation

For a sufficiently short element,$$\Delta\ell\simeq\ell_0ak\cos kX.$$Thus the element at phase zero expands by $4.0\times10^{-8}\,\mathrm m$; the one at phase $\pi$ contracts by the same amount. Both center displacements vanish. Compression is determined by neighboring displacement differences, not center displacement.

Physical check

The dimensionless strain $ak=4\times10^{-5}\ll1$ validates linearization. The exact density remains positive because $1+\xi_X>0$.