Traveling Sound Waves — Question 9

Question 9

Density and velocity obey a local conservation law

Linear acoustic continuity is $\delta\rho_t+\rho_0u_x=0$ and the pressure response is $p=c^2\delta\rho$. For $p=P\cos(kx-\omega t)$, use $\omega=ck$ and no background flow.

1. Derive $u$ directly from continuity and recover the progressive-wave impedance relation.

2. For $P=4.0\,\mathrm{Pa}$, $c=400\,\mathrm{m/s}$ and $\rho_0=1.0\,\mathrm{kg/m^3}$, calculate density amplitude and maximum fractional density change.

Question 9 - Solution

Density and velocity obey a local conservation law

(a) Method and calculation

The density field is $\delta\rho=(P/c^2)\cos\Phi$. Hence$$\delta\rho_t=\frac{\omega P}{c^2}\sin\Phi,\qquad u_x=-\frac{\omega P}{\rho_0c^2}\sin\Phi.$$Integrating in $x$ with no uniform-flow component gives$$u=\frac{\omega P}{\rho_0c^2k}\cos\Phi=\frac P{\rho_0c}\cos\Phi.$$

(b) Interpretation and further calculation

The numerical density amplitude is$$\delta\rho_m=\frac4{400^2}=2.50\times10^{-5}\,\mathrm{kg/m^3},\qquad\frac{\delta\rho_m}{\rho_0}=2.50\times10^{-5}.$$The velocity amplitude is $4/(1\times400)=0.010\,\mathrm{m/s}$. Continuity and momentum therefore lead to the same impedance relation.

Physical check

The fractional density perturbation equals $u_m/c$ for a progressive linear wave. Both are small here.