Speed of Sound in Solids Calculator

Compute longitudinal, shear, and extensional sound speeds in any solid from elastic properties, or look up 11 common engineering materials.

🔩 Speed of Sound in Solids Calculator
Young's Modulus (E)
GPa
Poisson's Ratio (ν)0.30
00.50
Density (ρ)
kg/m³
Material
Longitudinal P-wave (v_L)
—
km/h
—
Shear S-wave (v_S)
—
Extensional Bar-wave (v_E)
—
Shear Modulus (G)
—
Bulk Modulus (K)
—
Longitudinal P-wave (v_L)
—
km/h
—
Shear S-wave (v_S)
—
Extensional Bar-wave (v_E)
—
Young's Modulus (E)
—
Poisson's Ratio
—
Density
—

🔩 What is the Speed of Sound in Solids Calculator?

The speed of sound in solids calculator computes how fast compressional (P-wave), shear (S-wave), and extensional (bar-wave) acoustic waves travel through a solid material. Unlike gases, where only one wave type propagates, solids support multiple wave modes due to their shear rigidity. Each wave type propagates at a different speed determined by the material's elastic constants and density.

Common applications include: non-destructive testing (NDT), where ultrasonic pulse-echo measurements determine flaw depth using the known wave speed; seismic exploration, where geophysicists use the P-wave and S-wave velocity ratio to identify rock types; structural health monitoring, where acoustic emission sensors detect crack growth in bridges and pressure vessels; and materials science, where wave speed measurements provide a non-destructive way to determine elastic moduli in small specimens.

A common misconception is that the "speed of sound in steel" is a single value. In fact there are three distinct speeds: the bulk longitudinal P-wave speed (5960 m/s in steel, valid for thick blocks), the shear S-wave speed (3235 m/s), and the extensional wave speed in a thin rod (about 5135 m/s). The correct formula to use depends on the geometry of the part being tested. Only the thin-rod formula applies to slender bars; thick plates and bulk solids require the P-wave formula.

This calculator covers both a formula mode (enter E, Poisson's ratio, and density to compute all three speeds, plus the derived shear modulus G and bulk modulus K) and a preset mode with reference values for 11 materials including steel, aluminum, copper, titanium, concrete, glass, granite, oak, lead, ice, and diamond.

📐 Formula

vL = √(E(1−ν) ÷ (ρ(1+ν)(1−2ν)))
vL = longitudinal (P-wave) speed (m/s) in a bulk solid
vS = √(G / ρ)   shear (S-wave) speed (m/s), G = E / (2(1+ν))
vE = √(E / ρ)   extensional speed (m/s) in a thin rod or bar
E = Young's modulus (Pa); typical steel = 207 GPa
ν = Poisson's ratio (dimensionless, 0 to 0.5); steel = 0.30
ρ = density (kg/m³); steel = 7850 kg/m³
G = shear modulus = E / (2(1+ν)); steel = 79.6 GPa
K = bulk modulus = Eν / ((1+ν)(1−2ν)); steel = 172 GPa
Example (steel): vL = √(207×10&sup9; × 0.70 / (7850 × 1.30 × 0.40)) = 5957 m/s

📖 How to Use This Calculator

Steps

1
Choose Custom Properties or Material Preset - Select Custom Properties to compute wave speeds from Young's modulus, Poisson's ratio, and density, or select Material Preset to look up reference values for 11 common engineering materials.
2
Enter material properties or select a material - For Custom mode: type Young's modulus in GPa, adjust the Poisson's ratio slider (0 to 0.5), and enter density in kg/m3. For Preset mode: choose the material from the dropdown.
3
Read all three wave speeds - The results show the longitudinal P-wave speed, shear S-wave speed, and extensional bar-wave speed in m/s and km/h, plus the derived shear and bulk moduli.

💡 Example Calculations

Example 1 — Structural Steel (P-wave Ultrasonic Testing)

Steel: E = 207 GPa, nu = 0.30, rho = 7850 kg/m3

1
G = 207 / (2 x 1.30) = 79.6 GPa. K = 207 x 0.30 / (1.30 x 0.40) = 119.4 GPa.
2
v_L = sqrt(207e9 x 0.70 / (7850 x 1.30 x 0.40)) = sqrt(144.9e9/4082) = 5957 m/s.
3
v_S = sqrt(79.6e9/7850) = 3185 m/s. v_E = sqrt(207e9/7850) = 5135 m/s. The P-wave is 1.87 times faster than the S-wave (consistent with nu = 0.30).
P-wave = 5958 m/s | S-wave = 3185 m/s | Bar-wave = 5135 m/s
Try this example →

Example 2 — Diamond (Highest Known Wave Speed)

Diamond: E = 1000 GPa, nu = 0.20, rho = 3510 kg/m3

1
G = 1000 / (2 x 1.20) = 416.7 GPa. K = 1000 x 0.20 / (1.20 x 0.60) = 277.8 GPa.
2
v_L = sqrt(1000e9 x 0.80 / (3510 x 1.20 x 0.60)) = sqrt(800e9/2527) = sqrt(316.6e6) = 17,793 m/s.
3
This is roughly 3 times faster than steel, and about 52 times faster than sound in air at 20°C (343 m/s). Diamond's exceptional stiffness makes it the fastest acoustic transmitter of any natural material.
P-wave = ~17,500 m/s (preset) | ~17,793 m/s (computed)
Try this example →

Example 3 — Lead (Slowest Common Metal)

Lead: E = 16 GPa, nu = 0.44, rho = 11340 kg/m3

1
G = 16 / (2 x 1.44) = 5.56 GPa. High nu = 0.44 means the material is nearly incompressible, dramatically boosting K relative to G.
2
v_L = sqrt(16e9 x 0.56 / (11340 x 1.44 x 0.12)) = sqrt(8.96e9/1959) = 2138 m/s.
3
v_S = sqrt(5.56e9/11340) = 700 m/s. The v_L/v_S ratio = 3.05, much higher than steel's 1.87, reflecting lead's very high Poisson's ratio. This low shear speed makes lead excellent at vibration damping.
P-wave = 2138 m/s | S-wave = 700 m/s
Try this example →

❓ Frequently Asked Questions

How do you calculate the speed of sound in a solid?+
For a bulk solid, the longitudinal P-wave speed is v_L = sqrt(E(1-nu) / (rho(1+nu)(1-2nu))). The shear S-wave speed is v_S = sqrt(G/rho) where G = E/(2(1+nu)). For a thin rod or bar, the extensional speed is v_E = sqrt(E/rho). All three require Young's modulus E in Pa, Poisson's ratio nu, and density rho in kg/m3.
What is the difference between P-wave and S-wave speed?+
P-waves (compressional/longitudinal) involve particle motion parallel to the wave direction and are always faster than S-waves in the same material. S-waves (shear/transverse) involve particle motion perpendicular to wave direction and cannot propagate through liquids. For steel: v_L = 5960 m/s vs v_S = 3235 m/s. The ratio v_L/v_S depends only on Poisson's ratio.
What is the speed of sound in steel?+
Structural steel has three acoustic wave speeds: longitudinal P-wave = 5960 m/s, shear S-wave = 3235 m/s, and extensional bar-wave = 5135 m/s. For ultrasonic testing of steel plates and forgings, the 5960 m/s value is used for perpendicular incidence. Thin steel bars use 5135 m/s.
What material has the highest speed of sound?+
Diamond has the highest P-wave speed of any known natural material at about 17,500 m/s, due to its extreme stiffness (E = 1050 GPa) and relatively low density (3510 kg/m3). Beryllium also ranks very high at about 12,890 m/s. Among common engineering metals, aluminum (6320 m/s) is faster than steel (5960 m/s) because its much lower density compensates for its lower stiffness.
What is the velocity ratio v_L/v_S and why does it matter?+
The ratio v_L/v_S = sqrt((1-nu)/(0.5-nu)) depends only on Poisson's ratio. For steel (nu = 0.30): sqrt(0.70/0.20) = 1.87. For lead (nu = 0.44): sqrt(0.56/0.06) = 3.05. Seismologists and NDT engineers measure this ratio directly from two-sensor time-delay data to identify rock type or detect anisotropy without needing the absolute wave speeds.
Why is the bar-wave speed lower than the bulk longitudinal speed?+
In a bulk infinite medium, lateral expansion is fully constrained, requiring extra stiffness described by the P-wave modulus M = K + 4G/3. In a thin rod, lateral surfaces are free to expand, so only Young's modulus E resists the longitudinal deformation. Since M greater than E for all materials with nu greater than 0, v_L is always greater than v_E = sqrt(E/rho).
How is wave speed used in ultrasonic NDT?+
Ultrasonic probes emit short pulses and measure the round-trip travel time to a reflector. Depth = v x t / 2. For a 10-microsecond echo in steel: depth = 5960 x 0.00001 / 2 = 0.0298 m = 29.8 mm. Any error in the assumed wave speed directly translates to a depth error, so precise knowledge of v for the specific alloy and temperature is essential.
What happens to wave speed near Poisson's ratio of 0.5?+
As nu approaches 0.5 (incompressible limit, like rubber), the bulk modulus K = E*nu/((1+nu)(1-2nu)) approaches infinity because the (1-2nu) term in the denominator approaches zero. This drives v_L toward infinity. However, v_S = sqrt(G/rho) and G = E/(2(1+nu)) remain finite. For nearly incompressible materials like gels and soft tissue, P-wave speeds become very high while S-wave speeds stay low.
Does the speed of sound in concrete change with curing age?+
Yes. Fresh concrete has a P-wave speed of roughly 1500 to 2000 m/s in the first few hours. As it cures and gains strength, speed increases to 3500 to 4500 m/s for high-quality 28-day concrete. This relationship between wave speed and compressive strength is the basis for sonic rebound hammer testing used on construction sites to assess early-age concrete quality without coring.
How does density alone affect wave speed in solids?+
Wave speed scales inversely with the square root of density: v is proportional to 1/sqrt(rho). Doubling density at constant modulus cuts wave speed by factor sqrt(2) = 1.41. This is why dense lead (11,340 kg/m3) has much lower wave speed than aluminum (2700 kg/m3) despite lead having a finite stiffness. High stiffness and low density together maximize acoustic speed.
Can I use this calculator for wood or anisotropic materials?+
The formula mode assumes an isotropic material where properties are the same in all directions. Wood is strongly anisotropic: sound travels about 3 to 4 times faster along the grain than across it. The preset value for oak (3850 m/s) is the along-grain speed. For across-grain measurements, the effective E would be roughly 10 times lower. Use the formula mode with the appropriate directional modulus for your specific geometry.
What is the Bulk Modulus shown in the results?+
The bulk modulus K = E*nu / ((1+nu)(1-2nu)) measures a material's resistance to uniform compression. High K means the material compresses very little under hydrostatic pressure. Diamond (K = 444 GPa) and steel (K = 170 GPa) have very high bulk moduli, while rubber has K near 1 GPa. The P-wave modulus M = K + 4G/3 directly determines the longitudinal wave speed in a bulk solid.
Tags: Speed of Sound in Solids Calculator Longitudinal Wave Speed Calculator Shear Wave Velocity Calculator Elastic Wave Speed Ultrasonic Velocity Calculator Calculator Formula Free Online

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How do you calculate the speed of sound in a solid?

For a bulk (infinite) solid, the longitudinal wave speed is v_L = sqrt(E(1-nu) / (rho(1+nu)(1-2nu))), the shear wave speed is v_S = sqrt(G/rho) where G = E/(2(1+nu)), and the extensional wave speed in a thin rod is v_E = sqrt(E/rho). All three require Young's modulus E, Poisson's ratio nu, and density rho.

What is the difference between P-wave and S-wave speed?

P-waves (primary, longitudinal) are compressional waves where particle motion is parallel to wave travel direction. S-waves (secondary, shear) have particle motion perpendicular to wave direction. P-waves are faster: for steel, v_L = 5960 m/s vs v_S = 3235 m/s. S-waves cannot travel through liquids since liquids have no shear stiffness.

What is the speed of sound in steel?

The longitudinal P-wave speed in structural steel is about 5960 m/s (21,456 km/h). The shear S-wave speed is about 3235 m/s. The extensional bar-wave speed (for thin rods) is about 5135 m/s. These values assume E = 207 GPa, nu = 0.30, rho = 7850 kg/m3.

Why is sound faster in solids than in gases?

Speed of sound equals sqrt(elastic modulus / density). Solids have enormous elastic moduli (steel E = 207,000 MPa vs air E_bulk = 0.142 MPa), and this stiffness advantage greatly outweighs the higher density, giving solids wave speeds 10 to 20 times faster than air.

What is Poisson's ratio and how does it affect wave speed?

Poisson's ratio nu is the negative ratio of lateral strain to axial strain during uniaxial loading. It ranges from 0 (no lateral deformation) to 0.5 (incompressible). Higher nu increases the P-wave speed relative to the bar-wave speed, because bulk waves constrain all lateral motion. For steel nu = 0.30 gives v_L/v_E = 5960/5135 = 1.16.

What material has the highest speed of sound?

Diamond has the highest P-wave speed of any known bulk material at about 17,500 m/s, due to its extreme stiffness (E = 1050 GPa) and low density (3510 kg/m3). For comparison, steel has v_L = 5960 m/s and aluminum has v_L = 6320 m/s.

How does ultrasonic testing use wave speed?

Ultrasonic non-destructive testing (NDT) sends a short pulse into a material and measures the time for the echo to return from defects or the far wall. Defect depth = (v x time) / 2. Knowing the material's wave speed precisely is essential for accurate depth measurement. Both P-waves and S-waves are used depending on the defect orientation.

What is the speed of sound in concrete?

The P-wave speed in concrete ranges from about 2500 to 4500 m/s depending on mix design, water-cement ratio, and cure age. High-quality dense concrete averages about 3100 m/s. This is used in sonic echo testing to assess pile integrity and locate voids or delaminations in concrete structures.

Does temperature affect the speed of sound in solids?

Yes, though less dramatically than in gases. As temperature increases, both the elastic modulus and density decrease slightly. For steel, the P-wave speed decreases by about 0.5 m/s per degree Celsius (roughly 0.01% per degree). At very high temperatures near melting point the drop becomes significant.

What is the velocity ratio v_L/v_S and why does it matter?

The velocity ratio v_L/v_S = sqrt((1-nu)/(0.5-nu)) depends only on Poisson's ratio. For nu = 0.3 (steel): sqrt(0.7/0.2) = 1.87. For nu = 0.25: sqrt(0.75/0.25) = sqrt(3) = 1.73. This ratio is used in seismology and NDT to identify materials by measuring both wave types and computing their speed ratio.

📌 Quick Tips

💡Longitudinal (P-wave) speed is always higher than shear (S-wave) speed in the same material. For most metals, v_L is about 1.7 to 2 times v_S.
💡The extensional (thin-rod) speed v_E = sqrt(E/rho) is lower than v_L because it allows lateral deformation, whereas bulk waves are constrained by all surrounding material.
💡Higher Young's modulus means stiffer material, which transmits sound faster. Diamond (E = 1050 GPa) holds the record at about 17,500 m/s for P-waves.
💡Poisson's ratio of 0.5 represents an incompressible material (like rubber). As nu approaches 0.5, the bulk wave speed approaches infinity while the shear wave speed stays finite.
💡Non-destructive testing (NDT) uses both P-waves and S-waves to detect cracks: P-waves for overall depth, S-waves for cracks perpendicular to the beam.