Seismology: •  seismos: earthquake •  logos: knowledge

Seismology:

•  Study of earthquakes

•  Study of interior of the Earth

Needed: wave propagation

Seismic wave propagation: Propagation of small displacements (perturbations) from equilibrium position.

Continuum mechanics: Mechanics of a continuous medium -> Properties are continuous and differentiable functions

Newton’s 2nd law F=ma

In a continuous, homogeneous medium:

Force per unit volume: f(x,t) Mass per unit volume, density: ρ (constant) Displacement: u(x,t)

We therefore have:

Two classes of forces: •  Body forces, proportional to volume •  Surface forces (acting on a surface), proportional to surface area: tractions

Tractions

Traction on surface with normal n:

with summation convention

Normal stresses: ?

Shear stresses: ?

Principal stresses σ1 , σ2 , σ3: eigenvalues of stress tensor σ

Principal stress axes: corresponding eigenvectors

Now we can finally derive the equation of motion …

Summing the three terms, adding the body force component, and equating this net force to the density times this component of the acceleration yields

Now we will derive the strain tensor which describes the deformation due to differential motion within the body.

Although we are interested in deformation that distorts the body, there can also be a rigid body translation or a rigid body rotation, neither of which produces deformation. To distinguish these effects, we add and subtract and then separate it into two parts