Traveling Sound Waves — Question 4

Question 4

A proposed pressure profile must satisfy acoustic dispersion

A uniform fluid has $c=300\,\mathrm{m/s}$. Consider $p(x,t)=P\cos(4x-1000t)$ with SI arguments. Free plane sound must satisfy $p_{tt}=c^2p_{xx}$.

1. Test whether this profile is a free sound wave in the given fluid.

2. If wave number remains $4\,\mathrm{m^{-1}}$, find the required frequency. If angular frequency remains $1000\,\mathrm{s^{-1}}$, find the required wavelength.

Question 4 - Solution

A proposed pressure profile must satisfy acoustic dispersion

(a) Method and calculation

Differentiate rather than identifying a drawn waveform as a physical solution:$$p_{tt}=-10^6p,\quad c^2p_{xx}=-(300)^2(4)^2p=-1.44\times10^6p.$$The residual is $p_{tt}-c^2p_{xx}=4.4\times10^5p$, so the proposed nonzero field fails the free-wave equation.

(b) Interpretation and further calculation

At fixed $k$,$$\omega=ck=1200\,\mathrm{s^{-1}},\qquad f=1200/(2\pi)=190.99\,\mathrm{Hz}.$$At fixed $\omega$, $k=1000/300=3.333\,\mathrm{m^{-1}}$ and $\lambda=2\pi/k=1.885\,\mathrm m$. The original phase speed $1000/4=250\,\mathrm{m/s}$ does not match the material speed.

Physical check

The derivative test is equivalent to $\omega^2=c^2k^2$ for a nonzero sinusoid. External forcing would require a different governing equation.