World Library  
Flag as Inappropriate
Email this Article

Adams–Williamson equation

Article Id: WHEBN0028974704
Reproduction Date:

Title: Adams–Williamson equation  
Author: World Heritage Encyclopedia
Language: English
Subject: Structure of the Earth, Earthquake duration magnitude, Doublet earthquake, Coordinating Committee for Earthquake Prediction, Fault (geology)
Publisher: World Heritage Encyclopedia

Adams–Williamson equation

The Adams–Williamson equation, named after L. H. Adams and E. D. Williamson, is a relation between the velocities of seismic waves and the density of the Earth's interior. Given the average density of rocks at the Earth's surface and profiles of the P-wave and S-wave speeds as function of depth, it can predict how density increases with depth. It assumes that the compression is adiabatic and that the Earth is spherically symmetric, homogeneous, and in hydrostatic equilibrium. It can also be applied to spherical shells with that property. It is an important part of models of the Earth's interior such as the Preliminary reference Earth model (PREM).[1][2]


Williamson and Adams first developed the theory in 1923. They concluded that "It is therefore impossible to explain the high density of the Earth on the basis of compression alone. The dense interior cannot consist of ordinary rocks compressed to a small volume; we must therefore fall back on the only reasonable alternative, namely, the presence of a heavier material, presumably some metal, which, to judge from its abundance in the Earth's crust, in meteorites and in the Sun, is probably iron."[1]


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 vp and the S-wave speed vs are

\begin{align} v_p &= \sqrt{\frac{K+(4/3)\mu}{\rho}} \\ v_s &= \sqrt{\frac{\mu}{\rho}}. \end{align}

These two speeds can be combined in a seismic parameter

\Phi = v_p^2-\frac{4}{3}v_s^2 = \frac{K}{\rho}.






The definition of the bulk modulus,

K = -V\frac{dP}{dV},

is equivalent to

K = \rho\frac{dP}{d\rho}.






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

\frac{dP}{dr} = -\rho(r)g(r),






where g(r) is the gravitational acceleration at radius r.[1]

If Equations 1,2 and 3 are combined, we get the Adams–Williamson equation:

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

This equation can be integrated to obtain

\ln\left(\frac{\rho}{\rho_0}\right) = -\int_{r_0}^r \frac{g(r)}{\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.[1]



  • Poirier, Jean-Paul (2000). Introduction to the Physics of the Earth's Interior. Cambridge Topics in Mineral Physics & Chemistry.  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.