Isotropic medium

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

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

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

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

Taking the curl of this equation and applying vector identities, one gets $${\displaystyle \partial _{t}^{2}(\nabla \times {\boldsymbol {u}})={\frac {\mu }{\rho }}\nabla ^{2}\left(\nabla \times {\boldsymbol {u}}\right)}$$

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 ${\textstyle \beta ={\sqrt {\mu /\rho }}}$. Assuming that the medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as ${\displaystyle \mu =\rho \beta ^{2}=\rho \omega ^{2}/k^{2}}$^[8] where ω is the angular frequency and k is the wavenumber. Thus, ${\displaystyle \beta =\omega /k}$.

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 faster speed ${\textstyle \alpha ={\sqrt {(\lambda +2\mu )/\rho }}}$.

The steady state SH waves are defined by the Helmholtz equation^[9] $${\displaystyle \left(\nabla ^{2}+k^{2}\right){\boldsymbol {u}}=0}$$ where k is the wave number.

S waves in viscoelastic materials

Similar to in an elastic medium, in a viscoelastic material, the speed of a shear wave is described by a similar relationship ${\displaystyle c(\omega )=\omega /k(\omega )={\sqrt {\mu (\omega )/\rho }}}$, however, here, ${\displaystyle \mu }$ is a complex, frequency-dependent shear modulus and ${\displaystyle c(\omega )}$ is the frequency dependent phase velocity.^[8] One common approach to describing the shear modulus in viscoelastic materials is through the Voigt Model which states: ${\displaystyle \mu (\omega )=\mu _{0}+i\omega \eta }$, where ${\displaystyle \mu _{0}}$ is the stiffness of the material and ${\displaystyle \eta }$ is the viscosity.^[8]

Magnetic resonance elastography

Magnetic resonance elastography (MRE) is a method for studying the properties of biological materials in living organisms by propagating shear waves at desired frequencies throughout the desired organic tissue.^[10] This method uses a vibrator to send the shear waves into the tissue and magnetic resonance imaging to view the response in the tissue.^[11] The measured wave speed and wavelengths are then measured to determine elastic properties such as the shear modulus. MRE has seen use in studies of a variety of human tissues including liver, brain, and bone tissues.^[10]