Elastic Field Equation

The derivation of coupled wave equations in terms of elastic strains differs only slightly from the description in terms of elastic displacements. The constitutive equation in terms of strains is  

$${\bf\sigma_1} ~=~ \displaystyle \mathop{\mbox{${\bf c_{11}}$... ...}_{\mbox{$\sim$}} ~{\partial \over{\partial t}} {\bf\epsilon_3}$$ (20)

Substituting Equation 20 into 5 gives the following wave equation in terms of elastic strains (omitting the source term):  

$$\displaystyle \mathop{\mbox{${\bf \nabla^{T}}$}}_{\mbox{$\si... ...on_3})~= ~\rho~{\partial^2 \over{\partial t^2}}{\bf \epsilon_1}$$ (21)