S-wave

For the purpose of this explanation, a solid medium is considered isotropic if its strain (deformation) in response to stress is the same in all directions. Let ${\displaystyle {\boldsymbol {u}}=(u_{1},u_{2},u_{3})}$ be the displacement vector of a particle of such a medium from its "resting" position ${\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3})}$ due elastic vibrations, understood to be a function of the rest position ${\displaystyle {\boldsymbol {x}}}$ and time ${\displaystyle t}$. The deformation of the medium at that point can be described by the strain tensor ${\displaystyle {\boldsymbol {e}}}$, the 3×3 matrix whose elements are

${\displaystyle e_{ij}={\frac {1}{2}}(\partial _{i}u_{j}+\partial _{j}u_{i})}$

where ${\displaystyle \partial _{i}}$ denotes partial derivative with respect to position coordinate ${\displaystyle x_{i}}$. The strain tensor is related to the 3×3 stress tensor ${\displaystyle {\boldsymbol {\tau }}}$ by the equation

${\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}e_{kk}+2\mu e_{ij}}$

Here ${\displaystyle \delta _{ij}}$ is the Kronecker delta (1 if ${\displaystyle i=j}$, 0 otherwise) and ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are the Lamé parameters (${\displaystyle \mu }$ being the material's shear modulus). It follows that

${\displaystyle \tau _{ij}=\lambda \delta _{ij}{\bigl (}\sum _{k}\partial _{k}u_{k}{\bigr )}+\mu (\partial _{i}u_{j}+\partial _{j}u_{i})}$

From Newton's law of inertia, one also gets

${\displaystyle \rho \partial _{t}^{2}u_{i}=\sum _{j}\partial _{j}\tau _{ij}}$

where ${\displaystyle \rho }$ is the density (mass per unit volume) of the medium at that point, and ${\displaystyle \partial _{t}}$ denotes partial derivative with respect to time. Combining the last two equations one gets the seismic wave equation in homogeneous media

${\displaystyle \rho \partial _{t}^{2}u_{i}=\lambda \partial _{i}{\bigl (}\sum _{k}\partial _{k}u_{k}{\bigr )}+\mu {\bigl (}\sum _{j}\partial _{i}\partial _{j}u_{j}+\mu \partial _{j}\partial _{j}u_{i}{\bigr )}}$

Using the nabla operator notation of vector calculus, ${\displaystyle \nabla =(\partial _{1},\partial _{2},\partial _{3})}$, with some approximations, this equation can be written as

${\displaystyle \rho \partial _{t}^{2}{\boldsymbol {u}}=\left(\lambda +2\mu \right)\nabla (\nabla \cdot {\boldsymbol {u}})-\mu \nabla \times (\nabla \times {\boldsymbol {u}})}$

Taking the curl of this equation and applying vector identities, one gets

${\displaystyle \partial _{t}^{2}(\nabla \times {\boldsymbol {u}})={\frac {\mu }{\rho }}\nabla ^{2}(\nabla \times {\boldsymbol {u}})}$

This formula is the wave equation applied to the vector quantity ${\displaystyle \nabla \times {\boldsymbol {u}}}$, which is the material's shear strain. Its solutions, the S-waves, are linear combinations of sinusoidal plane waves of various wavelengths and directions of propagation, but all with the same speed ${\displaystyle \beta =\textstyle {\sqrt {\mu /\rho }}}$

Taking the divergence of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity ${\displaystyle \nabla \cdot {\boldsymbol {u}}}$, which is the material's compression strain. The solutions of this equation, the P-waves, travel at the speed ${\displaystyle \alpha =\textstyle {\sqrt {(\lambda +2\mu )/\rho }}}$ that is more than twice the speed ${\displaystyle \beta }$ of S-waves.

The steady-state SH waves are defined by the Helmholtz equation[8]

${\displaystyle (\nabla ^{2}+k^{2}){\boldsymbol {u}}=0}$

where k is the wave number.