How this calculator works
Wave speed depends on both stiffness and inertia. A long thin rod can expand sideways, while a bulk longitudinal wave in a large solid experiences additional lateral constraint. A transverse wave depends on shear modulus. These formulas assume homogeneous, linear, isotropic elasticity with positive stiffness and -1 < nu < 0.5. Enter dynamic moduli and density appropriate to your specimen and temperature.
Formula
crod = sqrt(E/rho); cT = sqrt(G/rho); G = E/[2(1+nu)]; cL = sqrt[E(1-nu)/(rho(1+nu)(1-2nu))]. nu = (cL²-2cT²)/[2(cL²-cT²)].
Worked example
For E = 32 GPa, rho = 8000 kg/m3 and nu = 0.25, the rod speed is 2000 m/s, shear speed is 1264.911 m/s and bulk longitudinal speed is 2190.890 m/s.
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Frequently asked questions
Why are there two longitudinal speeds?
The rod model allows lateral contraction. The bulk model describes a plane wave in an unbounded solid.
Can a liquid carry this shear wave?
An ordinary equilibrium liquid has no static shear rigidity, so this elastic-solid model does not apply.
Do crystalline materials follow these equations?
Only when an isotropic approximation is adequate. Single crystals can have strongly direction-dependent speeds.