# Dictionary:Isotropic media

A medium is isotropic if all directions of wave propagation are equivalent, and that case of isotropy has the most simple symmetry. In this case, the material has only two independent elastic moduli, called Lame's and shear modulus, ${\displaystyle \lambda }$ and ${\displaystyle G}$, which are related to the stiffness tensor ${\displaystyle C_{ijkl}}$ by: $$C_{ijkl}=[\lambda\delta_{ij}\delta_{kl}+G(\delta_{ij}\delta_{kl}+\delta_{ij}\delta_{kl})]$$ where ${\displaystyle \delta }$ is the Kronecker delta function, ${\displaystyle \delta _{ij}\equiv 1}$ for ${\displaystyle i=j}$ , and ${\displaystyle \delta _{ij}\equiv 0}$ for ${\displaystyle i\neq j}$, with i,j,k,l=1,2,3. The stiffness tensor ${\displaystyle C_{ijkl}}$ of isotropic media is explicitly written as the stiffness matrix (in Voigt notation ) ${\displaystyle C_{IJ}}$: $$C_{IJ}^{iso}= \begin{bmatrix} C_{33} & C_{12} & C_{12} & 0 & 0 & 0\\ C_{12} & C_{33} & C_{12} & 0 & 0 &0\\ C_{12} & C_{12} & C_{33} & 0 & 0 & 0\\ 0 &0 &0 &C_{44} &0 &0\\ 0 &0 &0 &0 &C_{44} &0\\ 0 &0 &0 &0 &0 &C_{44}\\ \end{bmatrix} \equiv \begin{bmatrix} \lambda+2G & \lambda & \lambda & 0 & 0 & 0\\ \lambda & \lambda+2G & \lambda & 0 & 0 &0\\ \lambda & \lambda & \lambda+2G & 0 & 0 & 0\\ 0 &0 &0 &G &0 &0\\ 0 &0 &0 &0 &G &0\\ 0 &0 &0 &0 &0 &G\\ \end{bmatrix}$$ where λ+2G = M = K +4G/3 = ρVP2 is the longitudinal modulus, and G = ρVS2, with ρ the density, and K the bulk modulus.