The Speed of Sound — Question 5

Question 5

An acoustic measurement determines compressibility rather than density alone

A liquid has measured sound speed $c=1200\,\mathrm{m/s}$ and density $\rho=800\,\mathrm{kg/m^3}$. Its small uniform compression obeys $\Delta V/V=-\Delta p/B$, with acoustic bulk modulus $B=\rho c^2$.

1. Infer $B$ and the fractional compression under $\Delta p=2.0\,\mathrm{MPa}$.

2. If the speed calibration is high by $1\%$ while density is exact, find the exact fractional bias in the inferred modulus.

Question 5 - Solution

An acoustic measurement determines compressibility rather than density alone

(a) Method and calculation

First infer stiffness, then use the small-compression law:$$B=800(1200)^2=1.152\times10^9\,\mathrm{Pa},$$$$\frac{\Delta V}{V}=-\frac{2.0\times10^6}{1.152\times10^9}=-1.736\times10^{-3}.$$The volume decreases by about $0.174\%$ under the stated model.

(b) Interpretation and further calculation

Because the inverse inference squares the speed,$$\frac{B_{\mathrm{meas}}}{B_{\mathrm{true}}}=(1.01)^2=1.0201.$$The modulus is high by $2.01\%$, slightly above the differential estimate $dB/B\simeq2\,dc/c=2\%$. The same constitutive response must apply to the compression process for the inferred acoustic modulus to be usable.

Physical check

The small fractional compression supports linearization; acoustic and slow isothermal moduli need not coincide in a general material.