S wave

Theory

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 ${\displaystyle \mathit{u}=(u_1,u_2,u_3)}$ be the displacement vector of a particle of such a medium from its "resting" position ${\displaystyle \mathit{x}=(x_1,x_2,x_3)}$ due elastic vibrations, understood to be a function of the rest position ${\displaystyle \mathit{x}}$ and time ${\displaystyle t}$. The deformation of the medium at that point can be described by the strain tensor ${\displaystyle \mathit{e}}$, the 3×3 matrix whose elements are $${\displaystyle e_{ij}=\frac{1}{2}\left({\mathrm{\partial }}_i{u}_j+{\mathrm{\partial }}_j{u}_i\right)}$$

where ${\displaystyle {\mathrm{\partial }}_i}$ denotes partial derivative with respect to position coordinate ${\displaystyle x_i}$. The strain tensor is related to the 3×3 stress tensor ${\displaystyle \mathit{\tau }}$ 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 $${\displaystyle {\tau }_{ij}=\lambda {\delta }_{ij}\sum _k{\mathrm{\partial }}_k{u}_k+\mu \left({\mathrm{\partial }}_i{u}_j+{\mathrm{\partial }}_j{u}_i\right)}$$

From Newton's law of inertia, one also gets $${\displaystyle \rho {\mathrm{\partial }}_t^2{u}_i=\sum _j{\mathrm{\partial }}_j{\tau }_{ij}}$$ where ${\displaystyle \rho }$ is the density (mass per unit volume) of the medium at that point, and ${\displaystyle {\mathrm{\partial }}_t}$ denotes partial derivative with respect to time. Combining the last two equations one gets the seismic wave equation in homogeneous media $${\displaystyle \rho {\mathrm{\partial }}_t^2{u}_i=\lambda {\mathrm{\partial }}_i\sum _k{\mathrm{\partial }}_k{u}_k+\mu \sum _j({\mathrm{\partial }}_i{\mathrm{\partial }}_j{u}_j+{\mathrm{\partial }}_j{\mathrm{\partial }}_j{u}_i)}$$

Using the nabla operator notation of vector calculus, ${\displaystyle \mathrm{\nabla }=({\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)}$, with some approximations, this equation can be written as $${\displaystyle \rho {\mathrm{\partial }}_t^2\mathit{u}=\left(\lambda +2\mu \right)\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathit{u}\right)-\mu \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathit{u}\right)}$$

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

This formula is the wave equation applied to the vector quantity ${\displaystyle \mathrm{\nabla }\times \mathit{u}}$, 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 $\beta =\sqrt{\mu /\rho }$. Assuming that the medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as ${\displaystyle \mu =\rho {\beta }^2=\rho {\omega }^2/k^2}$ 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 ${\displaystyle \mathrm{\nabla }\cdot \mathit{u}}$, which is the material's compression strain. The solutions of this equation, the P waves, travel at the faster speed $\alpha =\sqrt{(\lambda +2\mu )/\rho }$.

The steady state SH waves are defined by the Helmholtz equation $${\displaystyle \left({\mathrm{\nabla }}^2+k^2\right)\mathit{u}=0}$$ 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. 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.

S wave technology

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. This method uses a vibrator to send the shear waves into the tissue and magnetic resonance imaging to view the response in the tissue. 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.