Adams–Williamson equation
Encyclopedia
The Adams–Williamson equation, named after L. H. Adams and E. D. Williamson, is a relation between the velocities of seismic wave
s 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).
properties of the medium they travel through, in particular the bulk modulus
K, the shear modulus μ, and the density
ρ0. In terms of these parameters, the P-wave speed vp and the S-wave speed vs are
These two speeds can be combined in a seismic parameter
The definition of the bulk modulus,
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.
If Equations , and are combined, we get the Adams–Williamson equation:
This equation can be integrated to obtain
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.
Seismic wave
Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano that imparts low-frequency acoustic energy. Many other natural and anthropogenic sources create low amplitude waves commonly referred to as ambient vibrations. Seismic waves...
s and the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of the Earth's interior. Given the average density of rocks at the Earth's surface and profiles of the P-wave
P-wave
P-waves are a type of elastic wave, also called seismic waves, that can travel through gases , solids and liquids, including the Earth. P-waves are produced by earthquakes and recorded by seismographs...
and S-wave
S-wave
A type of seismic wave, the S-wave, secondary wave, or shear wave is one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves....
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
Hydrostatic equilibrium
Hydrostatic equilibrium or hydrostatic balance is the condition in fluid mechanics where a volume of a fluid is at rest or at constant velocity. This occurs when compression due to gravity is balanced by a pressure gradient force...
. 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) and shear waves (S-waves). Both have speeds that are determined by the elasticElasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
properties of the medium they travel through, in particular the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...
K, the shear modulus μ, and the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
ρ0. In terms of these parameters, the P-wave speed vp and the S-wave speed vs are
These two speeds can be combined in a seismic parameter
The definition of the bulk modulus,
is equivalent to
Suppose a region at a distance r from the Earth's center can be considered a fluid in hydrostatic equilibrium
Hydrostatic equilibrium
Hydrostatic equilibrium or hydrostatic balance is the condition in fluid mechanics where a volume of a fluid is at rest or at constant velocity. This occurs when compression due to gravity is balanced by a pressure gradient force...
, 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
Thermal expansion is the tendency of matter to change in volume in response to a change in temperature.When a substance is heated, its particles begin moving more and thus usually maintain a greater average separation. Materials which contract with increasing temperature are rare; this effect is...
does not contribute to density variations). The pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
P(r) varies with r as
where g(r) is the gravitational acceleration
Gravitational acceleration
In physics, gravitational acceleration is the acceleration on an object caused by gravity. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass....
at radius r.
If Equations , and are combined, we get the Adams–Williamson equation:
This equation can be integrated to obtain
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.