# Wave equation

The wave equation is derived from the equation of motion. The equation of motion for a point mass is Newton's Second Law:

 ${\displaystyle f=ma}$ (1)

For an elastic continuum, the equation of motion is:

 ${\displaystyle \sum _{j=1}^{3}{\frac {\partial \sigma _{ij}}{\partial x_{j}}}=\rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}}$ (2)

where the force ${\displaystyle f}$ on the left of (1) is replaced by the gradient of the stress tensor ${\displaystyle \sigma _{ij}}$, the mass ${\displaystyle m}$ on the right is replaced by the density ${\displaystyle \rho }$, and the acceleration on the right is given explicitly by the second time-derivative of the displacement vector ${\displaystyle u_{i}}$. This is three equations for the three ${\displaystyle (i=1,2,3)}$ components of the unknown displacement ${\displaystyle {\vec {u}}({\vec {x}},t)}$, which is implicit within the stress, on the left.

To make the displacement explicit on the left, we use Hooke's Law:

 ${\displaystyle \sigma _{ij}=\sum _{k,l}^{3}C_{ijkl}\varepsilon _{kl}}$ (3)

where the elastic properties of the medium are given by the stiffness tensor ${\displaystyle C_{ijkl}}$, and the (small) strain ${\displaystyle \varepsilon _{kl}}$is given by:

 ${\displaystyle \varepsilon _{kl}={\frac {1}{2}}[{\frac {\partial u_{k}}{\partial x_{l}}}+{\frac {\partial u_{l}}{\partial x_{k}}}]}$ (4)

Using (3) and (4), equation (2) becomes:

 ${\displaystyle \rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}=\sum _{j=1}^{3}{\frac {\partial }{\partial x_{j}}}\sum _{k,l=1}^{3}C_{ijkl}\varepsilon _{kl}=\sum _{j,k,l=1}^{3}{\frac {\partial }{\partial x_{j}}}C_{ijkl}{\frac {\partial u_{k}}{\partial x_{l}}}}$ (5)

where the symmetries of ${\displaystyle C_{ijkl}}$ simplify the result on the right. This is the equation of motion, with the unknown ${\displaystyle {\vec {u}}({\vec {x}},t)}$ explicit.

If the medium is locally uniform, i.e. if ${\displaystyle C_{ijkl}}$ is locally constant, then (5) becomes

 ${\displaystyle \rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}=\sum _{j,k,l=1}^{3}C_{ijkl}{\frac {\partial ^{2}u_{k}}{\partial x_{j}\partial x_{l}}}}$ (6)

This is the wave equation. When applied to isotropic media, the stiffness tensor ${\displaystyle C_{ijkl}}$ simplifies; this form underlies most exploration seismics. It has solutions of P-waves and shear waves.

What is implied by the assumption of "locally uniformity"? Loosely speaking, it means constant ${\displaystyle C_{ijkl}}$ on the scale of the seismic wavelength. Of course, the subsurface is in fact highly variable in its elasticity, but we conventionally model this as a set of coarse uniform layers. We then solve the wave equation separately in each coarse layer, and match the solutions at the layer boundaries using the boundary conditions of continuity of stress and displacement. However, see the discussion of a more accurate discussion of subsurface heterogeneity on the Equation of Motion page.