Speed of Sound in Solids Calculator (longitudinal, bar and shear waves)

Understanding the speed of sound in solids calculator — concept and principle

This calculator estimates how fast sound travels through a solid from its Young's modulus, density and Poisson's ratio. It gives three speeds: the longitudinal (bulk) wave speed used in ultrasonic testing and seismology, the slower thin-rod speed, and the shear wave speed, plus the shear modulus, travel time over a path and acoustic impedance.

The same metal carries longitudinal, rod and shear waves at different speeds.

How it is calculated

vL = √( E(1 − ν) ÷ (ρ(1 + ν)(1 − 2ν)) )

vbar = √(E ÷ ρ)

G = E ÷ (2(1 + ν)), vS = √(G ÷ ρ)

E is Young's modulus in GPa (×10⁹ Pa), ρ the density in kg/m³ and ν Poisson's ratio (between −1 and 0.5).
Travel time = path length ÷ vL; acoustic impedance Z = ρ × vL, shown in MRayl (10⁶ kg/(m²·s)).

The principle

A stiffer material pushes back harder, so the wave moves faster; a heavier material responds more slowly, so the wave moves slower. That is why every formula has the form √(stiffness ÷ density). The stiffness that applies depends on how the material can deform. In a thin rod the sides are free to bulge, so only E acts. In a large block the surrounding material stops sideways bulging, which makes it effectively stiffer and gives the higher longitudinal speed. Shear waves slide layers past each other, so they depend on the shear modulus G and are always slower.

Worked example

Default inputs: E = 70 GPa, ρ = 2,700 kg/m³, ν = 0.33, path 100 mm (values typical of aluminium).

Longitudinal stiffness term: 70×10⁹ × 0.67 ÷ (1.33 × 0.34) = 1.037×10¹¹ Pa; divide by 2,700 and take the root: vL = 6,198 m/s.
Thin-rod speed: √(70×10⁹ ÷ 2,700) = 5,092 m/s.
G = 70 ÷ (2 × 1.33) = 26.32 GPa; vS = √(26.32×10⁹ ÷ 2,700) = 3,122 m/s.
Ratio vL ÷ vS = 1.985.
Travel time over 0.1 m: 0.1 ÷ 6,198 = 16.135 µs; impedance 2,700 × 6,198 = 16.734 MRayl.

Each point is the calculator re-run with only “Density ρ (kg/m³)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Density ρ (kg/m³)	Longitudinal (bulk) wave speed (m/s)
675	12,396
1,350	8,765
2,025	7,157
2,700	6,198
3,375	5,544
4,050	5,061
5,400	4,383

How to read the result

The ratio vL/vS depends only on Poisson's ratio, so it is a quick check on ν. Acoustic impedance matters for ultrasonic inspection: the larger the impedance difference between two materials, the more sound reflects at their boundary. Travel times let you convert a measured echo time into a thickness.

Common mistakes

Using the thin-rod speed for a thick plate or block, or the reverse.

Keep in mind: The formulas assume an isotropic, homogeneous material. Rolled metals, wood, composites and cracked or porous parts have direction-dependent or lower speeds, so calibrate against a reference block for real testing.

Frequently asked questions

Why is sound faster in steel than in air?

Steel is enormously stiffer, and the stiffness gain outweighs its higher density.

Which speed is used for ultrasonic thickness gauges?

The longitudinal speed in the bulk material, usually set by calibrating on a sample of known thickness.

What are P-waves and S-waves?

In seismology, P-waves are longitudinal (pressure) waves and S-waves are shear waves; P-waves arrive first because they are faster.

Sources: W. E. Tefft and S. Spinner, J. Res. NBS 66A(2), 1962 (thin-rod speed and E = 2G(1 + ν)) · University of Wisconsin, EMA 611 Advanced Mechanical Testing, "Experiment 2. Ultrasonics and Transducers"

Checked 9 Nov 2026. W. E. Tefft and S. Spinner, “Cross-Sectional Correction for Computing Young’s Modulus From Longitudinal Resonance Vibrations of Square and Cylindrical Rods”, J. Res. NBS 66A(2), 1962, eq. 1 (thin-rod speed v₀ = √(E/ρ)) and E = 2G(1 + μ); University of Wisconsin, EMA 611 Advanced Mechanical Testing, “Experiment 2. Ultrasonics and Transducers” (longitudinal v = √(C₁₁₁₁/ρ), C₁₁₁₁ = E(1 − ν)/((1 + ν)(1 − 2ν)) for isotropic solids, shear v = √(G/ρ); aluminium E = 70 GPa).

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