The Speed of Sound — Question 1

Question 1

Elastic compression and inertia produce the acoustic speed law

A uniform fluid has equilibrium density $\rho_0$ and bulk modulus $B$. In one dimension, small displacement $\xi$ produces pressure perturbation $p=-B\partial_x\xi$. Consider a thin slab of area $A$ and thickness $dx$.

1. Derive the displacement wave equation and sound speed.

2. Evaluate the speed for $B=2.25\times10^9\,\mathrm{Pa}$ and $\rho_0=1000\,\mathrm{kg/m^3}$. Explain why a larger density alone lowers speed.

Question 1 - Solution

Elastic compression and inertia produce the acoustic speed law

(a) Method and calculation

The force difference is $A[p(x)-p(x+dx)]\simeq-Ap_xdx$. Newton's law gives$$\rho_0 A dx\,\xi_{tt}=-A p_xdx=B A dx\,\xi_{xx}.$$$$\xi_{tt}=\frac B{\rho_0}\xi_{xx},\qquad c=\sqrt{\frac B{\rho_0}}.$$The area and slab thickness cancel; this is a local material speed.

(b) Interpretation and further calculation

Substitution yields$$c=\sqrt{\frac{2.25\times10^9}{1000}}=1500\,\mathrm{m/s}.$$At fixed $B$, doubling density gives $c\to c/\sqrt2$: restoring force per deformation is unchanged while inertia increases. Comparing different materials requires both $B$ and $\rho_0$, so density alone cannot rank their speeds.

Physical check

$B/\rho_0$ has units $\mathrm{m^2/s^2}$. Compression must provide a restoring, not destabilizing, acceleration.