Speed of Sound in Solids Calculator — c = √(E/ρ), Longitudinal & Shear Waves

Speed of Sound in Solids Calculator

The speed of sound in solids calculator estimates how fast sound waves propagate through different solid mediums. Imagine you strike a hammer at the end of a long rod — the generated sound wave travels through the rod to the other end and back to you. This calculator gives you that wave propagation speed, in other words the speed of sound in materials such as copper or steel rods.

The speed of sound in solids is estimated from the density and modulus of elasticity of the material. If you only know the shear modulus, you can use the relations below to find the corresponding modulus of elasticity. Scroll down to understand the formulas behind the calculations.

What is a sound wave and how to calculate the speed of a sound wave?

The mechanical wave generated by the vibration of particles in a medium is known as a sound wave. The speed of a sound wave is the distance travelled by the energy of the wave per unit of time, and it depends on the properties of the medium it travels through.

If the sound wave travels through a long cylindrical rod of a material with density ρ and modulus of elasticity E, the one-dimensional speed of the sound wave is:

c = √( E / ρ )

Considering a three-dimensional (bulk) solid, the speed of sound waves also depends on the Poisson's ratio ν as well as the Young's modulus E. The longitudinal (P-wave) speed c_l and the transverse (S-wave / shear) speed c_t become:

c_l = √( E(1 − ν) / [ ρ(1 + ν)(1 − 2ν) ] )
c_t = √( G / ρ ),   where G = E / [ 2(1 + ν) ]

The density and speed of sound in a particular medium are also combined in a material's property known as acoustic impedance (Z = ρ·c).

Factors affecting the speed of sound

The sound wave propagation speed is affected by several properties of the medium:

• Stiffness / rigidity — the higher the modulus of elasticity, the higher the speed of sound.
• Density — denser materials tend to produce slower wave speeds.
• The modulus of elasticity has a stronger influence on the speed of sound than density does.

Therefore, when comparing the speed of sound across different states of matter, sound generally travels fastest in solids, slower in liquids, and slowest in gases:

c_solid > c_liquid > c_gas

Using the speed of sound in solids calculator

1. Choose the wave type: thin rod (1-D) for a long slender bar, or bulk solid (3-D) to also get the longitudinal and transverse (shear) speeds.
2. Select your unit system — Metric (kg/m³, GPa, m/s) or US/Imperial (lb/ft³, Mpsi, ft/s).
3. Pick a material from the list, or choose Custom and enter the density, Young's modulus and Poisson's ratio yourself.
4. Read the resulting speed of sound in m/s, km/h, ft/s and mph.

Example: Estimate the speed of sound in a steel rod

A steel rod has a density of about ρ = 7850 kg/m³ and a Young's modulus of about E = 200 GPa = 200 × 10⁹ Pa. The one-dimensional speed of sound is:

c = √( 200 × 10⁹ / 7850 ) ≈ 5048 m/s

So a hammer strike at one end of a steel rod produces a wave that travels at roughly 5,000 m/s (about 18,000 km/h or 11,000 mph) — far faster than the ≈343 m/s speed of sound in air.

Frequently Asked Questions

Why is sound faster in solids than in air?

Solids are far stiffer (higher modulus of elasticity) than gases, so particle vibrations are transmitted much more quickly, even though solids are denser.

What is the difference between the rod and bulk speeds?

The thin-rod speed c = √(E/ρ) applies to a long slender bar. In a large (bulk) solid, the longitudinal speed c_l is higher because the material is laterally constrained, and a separate transverse (shear) wave travels at c_t = √(G/ρ).

Do I need Poisson's ratio?

Only for the bulk (3-D) mode. The thin-rod speed depends only on the Young's modulus and the density.

What is the speed of sound in copper?

With ρ ≈ 8960 kg/m³ and E ≈ 117 GPa, the thin-rod speed of sound in copper is about 3,610 m/s.