Speed of sound in solids
Three-dimensional solids In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, it is possible to generate sound waves with different velocities dependent on the deformation mode. Sound waves generating volumetric deformations (compressions) and shear deformations are called longitudinal waves and shear waves, respectively. In earthquakes, the corresponding seismic waves are called P-waves and S-waves, respectively. The sound velocities of these two type waves propagating in a homogeneous 3-dimensional solid are respectively given by: where K and G are the bulk modulus and shear modulus of the elastic materials, respectively, Y is the Young's modulus, and is Poisson's ratio. The last quantity is not an independent one, as . Note that the speed of longitudinal/compression waves depends both on the compression and shear resistance properties of the material, while the speed of shear waves depends on the shear properties only. Typically, compression or P-waves travel faster in materials than do shear waves, and in earthquakes this is the reason that onset of an earthquake is often preceded by a quick upward-downward shock, before arrival of waves that produce a side-to-side motion. For example, for typical steel alloy, K = 170 GPa, G = 80 GPa and = 7700 kg/m3, yielding a longitudinal velocity cl of 6000 m/s. This is in reasonable agreement with cl=5930 m/s measured experimentally for a (possibly different) type of steel. The shear velocity cs is estimated at 3200 m/s using the same numbers. Long rods The speed of sound for longitudinal waves in stiff materials such as metals is sometimes given for "long, thin rods" of the material in question, in which the speed is easier to measure. In rods where their diameter is shorter than a wavelength, the speed of pure longitudinal waves may be simplified and is given by: This is similar to the expression for shear waves, save that Young's modulus replaces the shear modulus. This speed of sound for longitudinal waves in long, thin rods will always be slightly less than the 3-D, longitudinal wave speed in an isotropic materials, and the ratio of the speeds in the two different types of objects depends on Poisson's ratio for the material. The bulk modulus ( or ) of a substance measures the substance's resistance to uniform compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. Its base unit is the pascal.