The Adams–Williamson equation, named after Leason H. Adams and E. D. Williamson, is an equation used to determine density as a function of radius, more commonly used to determine the relation between the velocities of

seismic wave A seismic wave is a wave of acoustic energy that travels through the Earth. It can result from an earthquake, volcanic eruption, magma movement, a large landslide, and a large man-made explosion that produces low-frequency acoustic energy ...

s and the

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

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 __NOTOC__ In seismology and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of elastic wave and are one of the two main types of elastic body waves, so named because th ...

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 In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...

. 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).

History

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."

Theory

The two types of seismic body waves are compressional waves (

P-waves A P wave (primary wave or pressure wave) is one of the two main types of elastic body waves, called seismic waves in seismology. P waves travel faster than other seismic waves and hence are the first signal from an earthquake to arrive at any ...

) and shear waves (

S-waves __NOTOC__ In seismology and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of elastic wave and are one of the two main types of elastic body waves, so named because th ...

). Both have speeds that are determined by the elastic properties of the medium they travel through, in particular the

bulk modulus The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other moduli describ ...

''K'', the

shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackre ...

''μ'', and the

''ρ''. In terms of these parameters, the P-wave speed ''v''_p and the S-wave speed ''v''_s are :

\begin 
v_p &= \sqrt \\
v_s &= \sqrt.
\end

These two speeds can be combined in a seismic parameter

The definition of the bulk modulus, :

K = -V\frac,

is equivalent to Suppose a region at a distance ''r'' from the Earth's center can be considered a fluid in

, 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 Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kinetic ...

does not contribute to density variations). The

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

''P''(''r'') varies with ''r'' as where ''g''(''r'') is the

gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...

at radius ''r''. If Equations , and are combined, we get the Adams–Williamson equation: :

\frac = -\frac.

This equation can be integrated to obtain :

\ln\left(\frac\right) = -\int_^r \fracdr,

where ''r''₀ is the radius at the Earth's surface and ''ρ''₀ is the density at the surface. Given ''ρ''₀ and profiles of the P- and S-wave speeds, the radial dependence of the density can be determined by numerical integration.

References

{{DEFAULTSORT:Adams-Williamson Equation Structure of the Earth Geophysics Ordinary differential equations