Speed of Sound in Materials
Longitudinal bar velocity c = √(E/ρ) for common engineering solids.
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The engineering
Stiffness over density under a square root — which is why steel and aluminum, wildly different metals, both ring at about 5,000 m/s: aluminum's modulus is a third of steel's, but so is its density, and the ratio nearly cancels. The specific-stiffness contest that governs sound speed is the same one that governs lightweight structures, and beryllium wins both.
The fine print is geometry: √(E/ρ) is the *thin bar* (rod) velocity. In bulk material, constrained sideways, longitudinal waves run faster — about 5,900 m/s in steel — and that bulk number is what your ultrasonic thickness gauge is calibrated to. Use the wrong velocity and every NDT thickness reading is ~15% wrong; the time-across row is the µs/mm arithmetic those gauges live on.
Where this math comes from
Ernst Chladni — lawyer, musician, and the father of acoustics — measured sound in solids in 1787 by stroking rods and plates and reading the pitch, inferring wave speeds from resonant frequencies and turning bow-dusted 'Chladni figures' into data. Jean-Baptiste Biot made it gloriously direct in 1808: tapping one end of Paris's new 951-meter iron water main and timing the two arrivals — the fast clang through iron, the slow one through air — at the far end.
Newton had priced sound in air in the 1687 Principia and come out 16% low; Laplace's 1816 adiabatic correction fixed the gas case. For solids, the E-and-ρ story matured with elasticity theory, and the twentieth century cashed it in as ultrasonic NDT — Floyd Firestone's 1942 'supersonic reflectoscope' turned echo timing in metal into flaw detection, the ancestor of every weld inspection today.
- 1687Isaac NewtonFirst theoretical sound speed (air) in the Principia.
- 1787Ernst ChladniSound velocities in solids from resonating rods and plates.
- 1808Jean-Baptiste BiotTimes sound racing through an iron water main vs air.
- 1942Floyd FirestoneUltrasonic reflectoscope — sound speed becomes an NDT tool.
See the full timeline of the math behind every calculator →
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