Traveling Sound Waves — Question 10

Question 10

Three smallness tests reduce to the same acoustic parameter

A progressive plane wave has displacement amplitude $a$, wave number $k$, pressure amplitude $P$, and particle speed amplitude $u_m$. Use $P=Bka$, $u_m=\omega a$, $c=\omega/k$ and $B=\rho_0c^2$. Take $P=140\,\mathrm{Pa}$ and $B=1.4\times10^5\,\mathrm{Pa}$.

1. Show that strain, pressure fraction and particle Mach number coincide in magnitude.

2. Find the maximum allowed pressure amplitude if each parameter must be below $0.01$. Explain why small displacement alone is not an adequate criterion.

Question 10 - Solution

Three smallness tests reduce to the same acoustic parameter

(a) Method and calculation

Eliminate the dimensional amplitude:$$ka=\frac PB,\qquad\frac{u_m}{c}=\frac{\omega a}{\omega/k}=ka.$$$$\epsilon\equiv ka=\frac PB=\frac{u_m}{c}=10^{-3}.$$The given wave satisfies a one-percent small-perturbation threshold.

(b) Interpretation and further calculation

The strict criterion $\epsilon<0.01$ requires$$P<0.01B=1400\,\mathrm{Pa}.$$The same microscopic displacement can produce a much larger strain if wavelength is shorter: $ka=2\pi a/\lambda$. Smallness must be assessed against a relevant length or pressure scale, not from the numerical value of $a$ alone.

Physical check

These equalities refer to a single progressive linear plane wave. They do not replace a nonlinear propagation model once the perturbation becomes large.