# Speed of Sound in Solids Calculator

Created by Rahul Dhari
Reviewed by Steven Wooding
Last updated: Jun 05, 2023

The speed of sound in solids calculator can be used to estimate the speed of sound waves propagating through different solid mediums. Imagine you strike a hammer at an end of a long rod, the generated sound wave will travel through the long rod till the other end and then travel back to you. This calculator will help you with the wave propagation speed or in other words the speed of sound in materials such as solids like a copper or steel rods.

The speed of sound in solids is estimated based on the density and modulus of elasticity of the material. In case you only know the shear modulus of the material you can head over to the Poisson's ratio calculator to find the respective modulus of elasticity. Scroll down to understand the formula behind the calculations.

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

The mechanical wave generated from the vibration of particles in a medium is known as a sound wave. The speed of a sound wave is the distance traveled by the energy of the wave per unit of time. The speed of a sound wave depends on the properties of the medium in which it is traveling. If the sound wave is traveling through a long cylindrical rod of material A having a density, ρ and modulus of elasticity, E, the speed of the sound wave in one-dimension can be written as:

$c = \sqrt{\frac{E}{\rho}}$

Whereas, considering a three-dimensional solid, the speed of sound waves also depends upon the Poisson's ratio, $\nu$ as well as Young's modulus, $E$. You can also express it as a form of other moduli, like shear modulus, $G$, or bulk modulus, $K$. The speed of sound in the longitudinal, $c_\mathrm{l}$ and transverse directions, $c_t$ become:

$\scriptsize c_\mathrm{l} = \sqrt{\frac{E(1-\nu)}{\rho (1 + \nu) (1 - 2\nu)}} \\ \scriptsize c_\mathrm{t} = \sqrt{\frac{G}{\rho}} = \sqrt{\frac{E}{\rho (1 + \nu)}}$

The density and speed of sound in a particular medium are also present in a material’s property known as acoustic impedance ($Z$): learn about it at our acoustic impedance calculator!

## Factors affecting speed of sound

The sound wave propagation speed is affected by different properties of the medium. For example, the stiffness and rigidity of material would have an impact over the speed of sound wave. The higher the modulus of elasticity, the higher the speed of sound. On the other hand, denser materials will result in slower wave speeds. However, the modulus of elasticity has a higher influence over the speed of sound compared to density. Therefore, when comparing the speed of sound across different states it can be observed that:

$\scriptsize V_{\text{solids}} > V_{\text{liquid}} > V_{\text{gas}}$

The speed of sound is also affected by temperature of the medium - check out our speed of sound calculator for more detailed information on this topic!

## Using the speed of sound in solids calculator.

Follow the steps mentioned below to estimate the speed of the sound wave in a solid:

1. Set the units as per the data available. You can also convert the speed into appropriate units using our speed conversion tool.

2. Enter the density of material.

3. Input the modulus of elasticity.

4. Enter the Poisson's ratio of the material.

5. The speed of sound in solids calculator will provide the speed of sound in the material in one and three dimension solids.

Alternatively, if you know the material and its on our list, simply select it from the material drop-down menu to display all its properties.

## Example: Estimate the speed of sound in material.

Let's use the speed of sound in solids calculator in an example problem. Find the speed of sound in a solid long copper rod, given the density of copper is $8,940\ \text{kg/m}^3$ and modulus of elasticity is $117\ \text{GPa}$.

1. Set the calculator units for density and modulus of elasticity as $\text{kg/m}^3$ and $\text{GPa}$ respectively.

2. Enter the density of copper as $8,940\ \text{kg/m}^3$.

3. Input the modulus of elasticity as $117\ \text{GPa}$.

4. Enter the Poisson's ratio of the material as $0.3$.

5. The sound wave speed calculator will return the wave speed in one dimension in m/s as $3,617.6\ \text{m/s}$.

6. The longitudinal and transverse speed of sound would be $4197\ \text{m/s}$ and $2243.6\ \text{m/s}$.

You can calculate the sound absorption coefficient of various solids and understand the phenomenon of the speed of sound in solids better.

Rahul Dhari
Density of material
kg/m³
Modulus of elasticity
Pa
Speed
m/s
Speed of sound in certain solids
Material
Custom ▾
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