structural:expansion
Three-dimensional solids

In a solid, there is a non-zero stiffness both for volumetric deformations and shear deformations. Hence, it is possible to generate sound waves with different velocities dependenton the deformation mode. Sound waves generating volumetric deformations (compression) and shear deformations (shearing) are called pressure waves (longitudinal waves) and shear waves (transverse waves), respectively. In earthquakes, the corresponding seismic waves are called P-waves (primary waves) and S-waves (secondary waves), respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given by c s o l i d, p = K + 4 3 G ρ = E (1 − ν) ρ (1 + ν) (1 − 2 ν), {\displaystyle c_{\mathrm {solid,p} }={\sqrt {\frac {K+{\frac {4}{3}}G}{\rho }}}={\sqrt {\frac {E(1-\nu)}{\rho (1+\nu)(1-2\nu)}}},} c s o l i d, s = G ρ, {\displaystyle c_{\mathrm {solid,s} }={\sqrt {\frac {G}{\rho }}},} where