Speed of Sound in Solids Calculator
Enter a material preset or supply your own Young's modulus, shear modulus, Poisson's ratio, and density to compute all three elastic wave velocities: the longitudinal (P-wave) speed for bulk compression waves, the shear (S-wave) speed for transverse waves, and the Rayleigh surface wave speed. Results update instantly and a step-by-step panel shows every calculation.
How sound travels through solids
Unlike gases, which support only compression (longitudinal) waves, solids can sustain two independent bulk wave types: longitudinal (P-waves) and shear (S-waves). In a P-wave, particles oscillate in the same direction the wave travels, compressing and rarefying the lattice. In an S-wave, particles move perpendicular to propagation, shearing the material. Because solids resist both compression (described by the bulk modulus) and shear (described by the shear modulus), both wave types are possible. A third type, the Rayleigh surface wave, is confined to the free surface of a solid and decays exponentially with depth. Surface waves play a key role in seismology and surface-acoustic-wave (SAW) devices.
The formulas behind this calculator
For a homogeneous, isotropic, linearly elastic solid the wave speeds depend entirely on the density and the elastic moduli. The 3-D longitudinal speed is vL = sqrt[E(1-nu) / (rho(1+nu)(1-2nu))]. The shear speed is vS = sqrt(G/rho), where G is the shear modulus. For a thin rod (diameter much smaller than the wavelength) the appropriate speed is v1D = sqrt(E/rho), which is always slower than vL because lateral inertia no longer plays a role. The Rayleigh speed uses the Viktorov approximation: vR = vS x (0.87 + 1.12*nu) / (1 + nu), valid to better than 0.1% for 0 <= nu <= 0.5. The wavelength at any frequency f is simply lambda = v / f.
Elastic moduli and how they relate
Young's modulus (E) measures stiffness under uniaxial loading: a high E means the material barely stretches when pulled. The shear modulus (G) measures stiffness against twisting or sliding. For an isotropic material these are linked by Poisson's ratio: G = E / [2(1+nu)]. Most metals have nu between 0.25 and 0.35, giving G roughly 0.38E. Rubber-like incompressible materials approach nu = 0.5, which drives the longitudinal denominator to zero and pushes vL toward infinity, reflecting the near-infinite bulk stiffness. Entering a nu very close to 0.5 will correctly yield a very large vL. The bulk modulus K = E / [3(1-2nu)] also appears in seismological P-wave analysis but is not an independent input here because nu already captures the volumetric compressibility once E and G are known.
Applications in non-destructive testing and materials science
Knowing wave speeds is the starting point for almost every ultrasonic technique. In pulse-echo thickness gauging, the wall thickness is d = vL x t / 2, where t is the measured round-trip transit time. In phased-array inspection the beam steering angles are set using Snell's law with these velocities. The vL/vS ratio carries direct physical information: from a measured ratio you can recover Poisson's ratio as nu = (r^2 - 2) / [2(r^2 - 1)] where r = vL/vS. Geophysicists use the same relationship to characterize rock formations from seismic surveys. In process control, continuous wave-speed monitoring can detect microstructural changes such as precipitation hardening or grain-growth in metals because the elastic moduli shift with the microstructure.
Elastic wave speeds in common solids
| Material | Density (kg/m3) | P-wave (m/s) | S-wave (m/s) | Rayleigh (m/s) |
|---|---|---|---|---|
| Steel (structural) | 7850 | 5940 | 3220 | 2960 |
| Aluminum | 2700 | 6320 | 3130 | 2890 |
| Copper | 8940 | 4700 | 2260 | 2090 |
| Brass (70/30) | 8500 | 4430 | 2120 | 1960 |
| Cast iron | 7200 | 4800 | 2600 | 2410 |
| Titanium (Grade 2) | 4500 | 6070 | 3120 | 2880 |
| Nickel | 8900 | 5630 | 2960 | 2740 |
| Gold | 19300 | 3240 | 1200 | 1140 |
| Silver | 10490 | 3600 | 1590 | 1490 |
| Magnesium | 1740 | 5770 | 3050 | 2820 |
| Glass (borosilicate) | 2230 | 5640 | 3280 | 3030 |
| Concrete | 2300 | 4000 | 2370 | 2190 |
Typical values at room temperature. Exact speeds vary with alloy grade, heat treatment, and measurement direction in anisotropic materials.
Frequently asked questions
Why is the longitudinal wave speed always faster than the shear wave speed?
Longitudinal waves benefit from both the bulk stiffness (resistance to volume change) and the shear stiffness, while shear waves involve only shear restoring forces. Mathematically, the ratio vL/vS = sqrt[E(1-nu) / (G(1+nu)(1-2nu))], which is always greater than sqrt(2) for any physically realizable isotropic solid (0 < nu < 0.5). For steel with nu = 0.29, the ratio is about 1.84, meaning P-waves arrive nearly twice as fast as S-waves.
What is the difference between the 3-D longitudinal speed and the thin-rod speed?
The 3-D longitudinal speed (vL) applies when the wave propagates through a bulk medium with unconstrained lateral dimensions much larger than the wavelength. A thin rod is laterally free to contract when compressed (Poisson effect), which effectively reduces the restoring stiffness. The result is the simpler bar-wave speed v1D = sqrt(E/rho), which is always slower than vL. At frequencies where the rod diameter is a significant fraction of the wavelength, dispersion occurs and neither formula is exact.
How do I find the shear modulus if I only know Young's modulus and Poisson's ratio?
For an isotropic material the relationship is G = E / [2(1 + nu)]. For example, steel with E = 200 GPa and nu = 0.29 gives G = 200 / [2 x 1.29] = 77.5 GPa. This calculator accepts G directly so you can override the isotropic approximation for slightly anisotropic materials if you have an independently measured shear modulus.
Why does wave speed matter for ultrasonic inspection?
All time-of-flight measurements in ultrasonic testing convert a transit time into a distance using d = v x t. An error of 1% in the assumed wave speed produces a 1% error in every thickness or depth reading. For safety-critical components such as pressure vessels or aircraft parts, this directly affects whether a flaw is reported as within or outside acceptance criteria. Accurate wave speeds are therefore a fundamental quality-assurance requirement.
Does temperature affect the speed of sound in solids?
Yes. Elastic moduli generally decrease with rising temperature (the lattice softens), while thermal expansion slightly reduces density. For most metals the net effect is a decrease in wave speed of roughly 0.03-0.05% per degree Celsius. High-temperature alloys such as Inconel 718 show a change of about -0.047%/degC. If you are inspecting a component at elevated temperature, apply a temperature correction to the room-temperature speed before converting transit times to thicknesses.
What is a Rayleigh wave and where is it used?
A Rayleigh wave is a surface-guided elastic wave whose energy is concentrated within about one wavelength of the free surface. Particle motion follows a retrograde ellipse (a backwards-rotating ellipse in the plane of propagation). Because the wave hugs the surface it is ideal for detecting surface-breaking cracks and near-surface defects without needing access to the back wall. Rayleigh waves are also the dominant wave type in earthquake surface shaking, and they are used in SAW filters found in mobile phones and GPS receivers.