# S-wave velocity

The velocity of secondary waves, aka shear waves.

Conversion formulas — edit
The elastic properties of homogeneous isotropic linear elastic materials are uniquely determined by any two moduli. Given any two, the others can thus be calculated. Key reference: Mavko, G, T Mukerji and J Dvorkin (2003), The Rock Physics Handbook, Cambridge University Press.

$\dagger \ X = \sqrt{9\lambda^2 + 2E\lambda + E^2}$

$(V_\mathrm{P},\,V_\mathrm{S})$ $(\mu,\,\lambda)$ $(E,\,\lambda)\,\dagger$ $(E,\,\mu)$ $(K,\,\lambda)$ $(K,\,\mu)$ $(K,\,E)$ $(\nu,\,\lambda)$ $(\nu,\,\mu)$ $(\nu,\,E)$ $(\nu,\,K)$
P-wave velocity
$V_\mathrm{P}=\,$
$V_\mathrm{P}$ $\sqrt{\tfrac{\lambda+2\mu}{\rho}}$ $\sqrt{\tfrac{E - \lambda + X}{2\rho}}$ $\sqrt{\tfrac{\mu\,(E-4\mu)}{\rho\,(E-3\mu)}}$ $\sqrt{\tfrac{3K - 2\lambda}{\rho}}$ $\sqrt{\tfrac{K+\frac43\mu}{\rho}}$ $\sqrt{\tfrac{3K \left(3K + E\right)}{\rho \left(9K - E\right)}}$ $\sqrt{\tfrac{\lambda (1 - \nu)}{\rho\nu}}$ $\sqrt{\tfrac{2 \mu (1-\nu)}{\rho (1 - 2\nu)}}$ $\sqrt{\tfrac{E(1 - \nu)}{\rho(1 + \nu)(1 - 2\nu)}}$ $\sqrt{\tfrac{3K(1 - \nu)}{\rho(1 + \nu)}}$
S-wave velocity
$V_\mathrm{S}=\,$
$V_\mathrm{S}$ $\sqrt{\tfrac{\mu}{\rho}}$ $\sqrt{\tfrac{E - 3\lambda + X}{4\rho}}$ $\sqrt{\tfrac{\mu}{\rho}}$ $\sqrt{\tfrac{3 (K - \lambda)}{2 \rho}}$ $\sqrt{\tfrac{\mu}{\rho}}$ $\sqrt{- \tfrac{3EK}{\rho \left(E - 9K\right)}}$ $\sqrt{\tfrac{\lambda}{2 \nu \rho} - \tfrac{\lambda}{\rho}}$ $\sqrt{\tfrac{\mu}{\rho}}$ $\sqrt{\tfrac{E}{2\rho(1+\nu)}}$ $\sqrt{-\tfrac{3K (2\nu - 1)}{2\rho (\nu + 1)}}$
Velocity ratio
$\Gamma=\,$
$\frac{V_\mathrm{P}}{V_\mathrm{S}}$ $\sqrt{\tfrac{\lambda+2\mu}{\mu}}$ $\sqrt{\tfrac{3E + 3\lambda + X}{2E}}$ $\sqrt{\tfrac{E - 4 \mu}{E - 3 \mu}}$ $\sqrt{\tfrac{\tfrac43 \lambda - 2 K}{\lambda - K}}$ $\sqrt{\tfrac{K+\frac43\mu}{\mu}}$ $\sqrt{\tfrac{E + 3K}{E}}$ $\sqrt{\tfrac{2\nu - 2}{2\nu - 1}}$ $\sqrt{\tfrac{2\nu - 2}{2\nu - 1}}$ $\sqrt{\tfrac{2\nu - 2}{2\nu - 1}}$ $\sqrt{\tfrac{2\nu - 2}{2\nu - 1}}$
1st Lamé parameter
$\lambda=\,$
$\rho (V_\mathrm{P}^2 - 2V_\mathrm{S}^2)$ $\lambda$ $\lambda$ $\tfrac{\mu(E-2\mu)}{3\mu-E}$ $\lambda$ $K-\tfrac{2\mu}{3}$ $\tfrac{3K(3K-E)}{9K-E}$ $\lambda$ $\tfrac{2 \mu \nu}{1-2\nu}$ $\tfrac{E\nu}{(1+\nu)(1-2\nu)}$ $\tfrac{3K\nu}{1+\nu}$
Shear modulus
$\mu=\,$
$\rho V_\mathrm{S}^2$ $\mu$ $\tfrac{E - 3\lambda + X}{4}$ $\mu$ $\tfrac{3(K-\lambda)}{2}$ $\mu$ $\tfrac{3KE}{9K-E}$ $\tfrac{\lambda(1-2\nu)}{2\nu}$ $\mu$ $\tfrac{E}{2(1+\nu)}$ $\tfrac{3K(1-2\nu)}{2(1+\nu)}$
Young's modulus
$E=\,$
$\tfrac{\rho V_\mathrm{S}^2 (3V_\mathrm{P}^2 - 4V_\mathrm{S}^2)}{V_\mathrm{P}^2 - V_\mathrm{S}^2}$ $\tfrac{\mu(3\lambda + 2\mu)}{\lambda + \mu}$ $E\$ $E$ $\tfrac{9K(K-\lambda)}{3K-\lambda}$ $\tfrac{9K\mu}{3K+\mu}$ $E\$ $\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}$ $2\mu(1+\nu)\,$ $E\$ $3K(1-2\nu)\,$
Bulk modulus
$K=\,$
$\rho (V_\mathrm{P}^2 - \tfrac43 V_\mathrm{S}^2)$ $\lambda+ \tfrac{2\mu}{3}$ $\tfrac{E + 3\lambda + X}{6}$ $\tfrac{E\mu}{3(3\mu-E)}$ $K$ $K$ $K$ $\tfrac{\lambda(1+\nu)}{3\nu}$ $\tfrac{2\mu(1+\nu)}{3(1-2\nu)}$ $\tfrac{E}{3(1-2\nu)}$ $K$
Poisson's ratio
$\nu=\,$
$\tfrac{V_\mathrm{P}^2 - 2V_\mathrm{S}^2}{2(V_\mathrm{P}^2 - V_\mathrm{S}^2)}$ $\tfrac{\lambda}{2(\lambda + \mu)}$ $\tfrac{- E - \lambda + X}{4\lambda}$ $\tfrac{E}{2\mu}-1$ $\tfrac{\lambda}{3K-\lambda}$ $\tfrac{3K-2\mu}{2(3K+\mu)}$ $\tfrac{3K-E}{6K}$ $\nu$ $\nu$ $\nu$ $\nu$
P-wave modulus
$M=\,$
$\rho V_\mathrm{P}^2$ $\lambda+2\mu\,$ $\tfrac{E - \lambda + X}{2}$ $\tfrac{\mu(4\mu-E)}{3\mu-E}$ $3K-2\lambda\,$ $K+\tfrac{4\mu}{3}$ $\tfrac{3K(3K+E)}{9K-E}$ $\tfrac{\lambda(1-\nu)}{\nu}$ $\tfrac{2\mu(1-\nu)}{1-2\nu}$ $\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}$ $\tfrac{3K(1-\nu)}{1+\nu}$