The shear wave velocity calculator above converts between the stiffness of a solid and the speed at which a shear disturbance travels through it. A shear wave, called an S-wave in seismology and a transverse wave in general physics, moves material sideways relative to the direction the wave is travelling. Its speed depends on only two properties: how strongly the material resists being sheared, and how much mass has to be moved.
Arb Digital publishes free physics calculators that keep the direction of the calculation explicit. Shear wave velocity is measured far more often than it is predicted — a borehole survey gives you a velocity, and the modulus is what you infer from it — so this page runs the relation in both directions rather than assuming everyone starts from a laboratory modulus.
What This Shear Wave Velocity Calculator Does
The hero figure is the shear wave velocity in metres per second. In the first mode it comes straight from the shear modulus and the density. In the second it comes from a measured P-wave velocity and a Poisson's ratio, which is how it is usually obtained from a refraction survey where the shear arrival is hard to pick. In the third mode you supply the velocity and the tool recovers the shear modulus from it, which is the direction geotechnical work almost always runs.
Alongside that, the tool reports the shear modulus, the compressional wave velocity, the ratio between the two velocities, and the vertical travel time through a layer of the thickness you enter. The velocity ratio deserves its own box because it is a direct proxy for Poisson's ratio and one of the most diagnostic quantities in the whole subject: it changes sharply when a rock's pores fill with fluid, while the shear velocity barely moves.
How to Use It
- Pick the mode that matches what you actually have. A modulus from a laboratory test, a P-wave velocity from a survey, or a shear velocity from a borehole are three different starting points and the tool handles each.
- Use the in-place bulk density. Not the dry density and not the grain density. The wave has to accelerate the pore fluid along with the frame.
- Keep Poisson's ratio below 0.5. The relation between the two wave speeds becomes singular at exactly 0.5, and a value at or above it describes something that cannot carry a shear wave.
- Match the strain level. Soil stiffness falls dramatically as strain increases, so a modulus measured in a strong-motion test is not the modulus a seismic wave sees.
- Use the travel time to sanity-check against a record. If the computed time through your layer does not match the arrival you picked, one of the inputs is wrong.
The Formula: How Shear Wave Velocity Is Calculated
The core relation is Vs = √(G ÷ ρ), where G is the shear modulus and ρ the density. It is the same square-root-of-stiffness-over-inertia form that governs every mechanical wave; OpenStax derives the general case and the solid-medium version in 17.2 Speed of Sound, and defines the shear modulus itself in 12.3 Stress, Strain, and Elastic Modulus.
The compressional velocity uses a stiffer effective modulus because the material cannot spread sideways: Vp = Vs √((2 − 2ν) ÷ (1 − 2ν)). Running that backwards gives the second mode, Vs = Vp √((0.5 − ν) ÷ (1 − ν)). And if both velocities are known, Poisson's ratio follows from ν = (Vp² − 2Vs²) ÷ (2(Vp² − Vs²)).
Work the defaults through by hand. A shear modulus of 44 GPa is 4.4 × 1010 Pa. Divided by a density of 2,700 kg/m³ that is 1.6296 × 107 m²/s², and the square root is 4,036.9 m/s. With Poisson's ratio at 0.25 the bracket becomes (2 − 0.5) ÷ (1 − 0.5) = 3, so the P-wave velocity is 4,036.9 × √3 = 6,992 m/s and the velocity ratio is exactly √3, or 1.732. Vertical travel time through 30 m at the shear velocity is 30 ÷ 4,036.9 = 7.43 milliseconds.
Why Shear Waves Cannot Travel Through Liquids
A shear wave is restored by shear stiffness. Displace a small element sideways and the surrounding material pulls it back, because it resists a change of shape. A liquid does not resist a change of shape at all — that is close to the definition of a liquid — so there is no restoring force, no wave, and a shear modulus of zero.
This is not a curiosity. It is the observation that revealed the structure of the Earth's interior. Large earthquakes produce a shadow zone on the far side of the planet where no direct S-wave ever arrives, while P-waves get through with a delay. The only explanation consistent with both is a liquid outer core. A single missing arrival, interpreted correctly, mapped the inside of a planet.
The same physics has an everyday consequence in the Poisson's ratio limit. As a material becomes incompressible, ν approaches 0.5, the bracket in the P-wave relation blows up and the velocity ratio tends to infinity. Saturated soft clays sit close to this limit: their P-wave velocity is essentially the velocity of sound in water, around 1,500 m/s, while their shear velocity may be under 150 m/s. A ratio above ten is normal there and is not an error.
The Velocity Ratio Is More Diagnostic Than Either Velocity
Fill the pore space of a rock with water and its bulk stiffness rises sharply, because water resists compression. Its shear stiffness barely changes, because water resists shear not at all. The P-wave velocity therefore jumps while the shear velocity moves only slightly, from the added mass. The ratio between them is the sensitive quantity.
That asymmetry is why the ratio, and Poisson's ratio derived from it, is used as a fluid indicator. A dry sand and the same sand fully saturated have similar shear velocities and very different P-wave velocities. If you are trying to distinguish a change in material from a change in saturation, the ratio separates them in a way neither velocity alone can.
It also means the shear velocity is the better measure of the skeleton itself. Because it ignores the pore fluid's stiffness, it responds to the frame: grain contacts, cementation, void ratio and confining pressure. This is exactly why geotechnical practice standardised on shear velocity rather than P-wave velocity for characterising ground stiffness, as the Pacific Earthquake Engineering Research Center's Guidelines for Estimation of Shear Wave Velocity sets out at length.
Strain Level Changes the Answer by an Order of Magnitude
Soil stiffness is not a constant. It is highest at very small strains and falls steeply as strain grows, which means the shear modulus you use has to match the strain of the problem you are solving. A seismic wave passing through the ground imposes strains around one part in a million. The ground under a foundation at working load sees strains thousands of times larger.
The modulus at seismic strain, usually written as the small-strain or maximum shear modulus, can be several times the modulus that governs settlement. Substituting one for the other is one of the most common errors in the subject, and it produces a velocity that is wrong by a large factor rather than a small one. If your modulus came from a plate load test or a triaxial test at working strain, it is not the right input for a wave calculation.
Confining pressure matters for the same reason. Shear stiffness in a granular material comes from grain contacts, and pressing the grains together harder stiffens them. Shear velocity therefore increases with depth even in a uniform deposit, which is why velocity profiles are reported as profiles rather than single numbers, and why an average over a fixed depth interval is the standard way to summarise a site.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using dry density instead of bulk density — the wave accelerates the pore fluid too, and leaving it out inflates the computed velocity.
- Mixing modulus units — rock moduli are quoted in GPa and soil moduli in MPa, a factor of a thousand that shows up as a factor of about thirty-two in velocity.
- Using a working-strain modulus — wave propagation is a small-strain problem, and the small-strain modulus can be several times larger than the one governing settlement.
- Entering a Poisson's ratio of 0.5 — that describes an incompressible material with no shear wave at all, and the P-wave relation is singular there.
- Treating a high velocity ratio as an error — in saturated soft ground a ratio above ten is genuine, and it is telling you the pore fluid is carrying the compressional wave.
Related Free Tools From Arb Digital
The shear modulus calculator works out G from a laboratory test or from Young's modulus and Poisson's ratio, and the Poisson's ratio calculator covers the elastic constant that links the two wave speeds here. For the compressional side, the speed of sound calculator handles wave speed in gases and liquids, the bulk modulus calculator gives the compressional stiffness, and the acoustic impedance calculator combines velocity and density into the property that governs reflection at an interface. The wave equation calculator ties speed, frequency and wavelength together, and the earthquake magnitude calculator covers the source side of a seismic record. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the speed at which a transverse disturbance travels through a solid, moving material sideways relative to the direction of travel. It equals the square root of the shear modulus divided by the density. In seismology it is the S-wave or secondary wave speed, so called because it always arrives after the faster compressional wave.
Because the effective stiffness restoring a compressional wave is larger. A P-wave is resisted by both the bulk modulus and the shear modulus, since the material is squeezed and cannot spread sideways, while an S-wave is resisted by the shear modulus alone. With the same density in both denominators, the larger stiffness always gives the higher speed.
A shear wave needs a restoring force against a change of shape, and a liquid offers none: its shear modulus is zero. This is how the Earth's liquid outer core was identified. Large earthquakes cast a shadow zone on the far side of the planet where no direct S-wave arrives, while P-waves get through, and only a liquid layer explains both observations.
The bulk density of the material in place, including any pore fluid. The wave has to accelerate everything present, not just the solid frame. Using dry density or grain density leaves mass out of the denominator and produces a velocity that is too high.
Because it responds to pore fluid while the shear velocity does not. Filling pore space with water raises the bulk stiffness sharply and the shear stiffness hardly at all, so the compressional velocity jumps while the shear velocity moves only slightly. The ratio therefore separates a change in saturation from a change in material, which neither velocity can do alone.
No. At exactly 0.5 the material is incompressible, the term linking the two wave speeds becomes singular, and the ratio tends to infinity. Values very close to 0.5 are physically real for saturated soft clays, and they legitimately produce velocity ratios above ten, but the limiting value itself has no finite answer.
In soils, dramatically. Stiffness is highest at very small strains and falls steeply as strain grows. Wave propagation is a small-strain problem, so the small-strain modulus is the correct input and it can be several times the modulus that governs foundation settlement. Substituting one for the other is the most common source of a badly wrong velocity.
This tool is provided for educational and preliminary engineering use. It applies isotropic elastic wave relations to figures you supply, and real ground is layered, anisotropic and strain-dependent in ways these relations do not capture. Site characterisation and any seismic design decision must be based on measured data and reviewed by a qualified geotechnical or geophysical professional.
