Isotropic medium s in the Earth versus depth. The negligible S wave velocity in the outer core occurs because it is liquid, while in the solid inner core the S wave velocity is non-zero. 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 \boldsymbol{u} = (u_1,u_2,u_3) be the displacement
vector of a particle of such a medium from its "resting" position \boldsymbol{x}=(x_1,x_2,x_3) due elastic vibrations, understood to be a
function of the rest position \boldsymbol{x} and time t. The deformation of the medium at that point can be described by the
strain tensor \boldsymbol{e}, the 3×3 matrix whose elements are e_{i j} = \tfrac{1}{2} \left( \partial_i u_j + \partial_j u_i \right) where \partial_i denotes partial derivative with respect to position coordinate x_i. The strain tensor is related to the 3×3
stress tensor \boldsymbol{\tau} by the equation \tau_{i j} = \lambda\delta_{i j}\sum_{k} e_{k k} + 2\mu e_{i j} Here \delta_{ij} is the
Kronecker delta (1 if i = j, 0 otherwise) and \lambda and \mu are the
Lamé parameters (\mu being the material's
shear modulus). It follows that \tau_{i j} = \lambda\delta_{i j} \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 \rho \partial_t^2 u_i = \sum_j \partial_j\tau_{i j} where \rho is the
density (mass per unit volume) of the medium at that point, and \partial_t denotes partial derivative with respect to time. Combining the last two equations one gets the
seismic wave equation in homogeneous media \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, \nabla = (\partial_1, \partial_2, \partial_3), with some approximations, this equation can be written as \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 \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 \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 \beta = \sqrt{\mu/\rho}. Assuming that the medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as \mu=\rho \beta^2=\rho \omega^2 / k^2 where
ω is the angular frequency and '''' is the wavenumber. Thus, \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 \nabla \cdot \boldsymbol{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 \left(\nabla^2 + k^2 \right) \boldsymbol{u}=0 where 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 c(\omega) = \omega / k(\omega)=\sqrt{\mu(\omega)/\rho}, however, here, \mu is a complex, frequency-dependent shear modulus and 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: \mu(\omega)=\mu_0+i\omega\eta, where \mu_0 is the stiffness of the material and \eta is the viscosity. == S wave technology ==