S wave

History

In 1830, the mathematician Siméon Denis Poisson presented to the French Academy of Sciences an essay ("memoir") with a theory of the propagation of elastic waves in solids. In his memoir, he states that an earthquake would produce two different waves: one having a certain speed a and the other having a speed \frac{a}{\sqrt 3}. At a sufficient distance from the source, when they can be considered plane waves in the region of interest, the first kind consists of expansions and compressions in the direction perpendicular to the wavefront (that is, parallel to the wave's direction of motion); while the second consists of stretching motions occurring in directions parallel to the front (perpendicular to the direction of motion).

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 \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 k 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 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 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

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.

References

References

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5. (May 2024). "Do viscous fluids support shear waves propagation?".
6. (17 July 1997). "Lecture 16 Seismographs and the earth's interior". University of Illinois at Chicago.
7. (1831). "Mémoire sur la propagation du mouvement dans les milieux élastiques". Mémoires de l'Académie des Sciences de l'Institut de France.
8. (May 2018). "Characterization of Viscoelastic Materials Using Group Shear Wave Speeds". IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control.
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10. (November 2015). "Estimation of material parameters from slow and fast shear waves in an incompressible, transversely isotropic material". Journal of Biomechanics.
11. (10 November 2021). "MR Shear Wave Elastography". University of Utah Health.