Adams–Williamson equation

Theory

The two types of seismic body waves are compressional waves (P-waves) and shear waves (S-waves). Both have speeds that are determined by the elastic properties of the medium they travel through, in particular the bulk modulus K, the shear modulus μ, and the density ρ. In terms of these parameters, the P-wave speed v_p and the S-wave speed v_s are

${\displaystyle \begin{array}{cc}{v}_p& =\sqrt{\frac{K_S+(4/3)\mu }{\rho }}\\ v_s& =\sqrt{\frac{\mu }{\rho }}.\end{array}}$

These two speeds can be combined in a seismic parameter

The definition of the isentropic bulk modulus,

${\displaystyle K_S=-V\frac{dP}{dV},}$

is equivalent to

Suppose a region at a distance r from the Earth's center can be considered a fluid in hydrostatic equilibrium, it is acted on by gravitational attraction from the part of the Earth that is below it and pressure from the part above it. Also suppose that the compression is adiabatic (so thermal expansion does not contribute to density variations). The pressure P(r) varies with r as

where g(r) is the gravitational acceleration at radius r.

Combining 1,2 and 3 gives the Adams–Williamson equation:

${\displaystyle \frac{d\rho }{dr}=-\frac{\rho (r)g(r)}{\mathrm{\Phi }(r)}.}$

This equation can be integrated to obtain

${\displaystyle \ln \left(\frac{\rho }{{\rho }_0}\right)=-{\int }_{r_0}^r\frac{g(r)}{\mathrm{\Phi }(r)}dr,}$

where r₀ is the radius at the Earth's surface and ρ₀ is the density at the surface. Given ρ₀ and profiles of the P- and S-wave speeds, the radial dependence of the density can be determined by numerical integration.