The Speed of Sound — Question 9

Question 9

A thin rod and a bulk solid use different elastic moduli

For the same isotropic material, a slender rod supports long-wavelength axial waves at $c_r=\sqrt{E/\rho}$. An unbounded solid supports longitudinal waves at $c_L=\sqrt{(K+4G/3)/\rho}$. Take $E=70$, $K=70$, $G=26\,\mathrm{GPa}$ and $\rho=2700\,\mathrm{kg/m^3}$; the rounded moduli are supplied for comparison.

1. Compute the two speeds and their ratio.

2. Explain why replacing the rod formula by the bulk formula changes the physical boundary condition, even with identical material.

Question 9 - Solution

A thin rod and a bulk solid use different elastic moduli

(a) Method and calculation

The effective axial stiffnesses are $70\,\mathrm{GPa}$ for the rod and $70+4(26)/3=104.667\,\mathrm{GPa}$ for the bulk wave. Thus$$c_r=5091.8\,\mathrm{m/s},\quad c_L=6226.2\,\mathrm{m/s},\quad c_L/c_r=1.223.$$

(b) Interpretation and further calculation

A long-wavelength rod mode allows lateral contraction at free side surfaces. A plane longitudinal bulk wave has transverse strain constrained by its field geometry. Consequently$$\frac{c_L^2}{c_r^2}=\frac{K+4G/3}{E}=1.4952.$$The speed difference reflects different deformation constraints, not different mass densities. Rod diameter must be small compared with axial wavelength for the simple rod model.

Physical check

Stiffer effective restoring response produces the larger speed. This comparison cannot be decided from $E$ alone.