phenomenon 30.47 The three seismic phases

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1610 · p. 1025

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

depends_on phenomenon 30.43 Two bulk wave speeds ¶
- depends_on proposition 30.30 Stability bounds ¶
  - depends_on equation 30.23 eq:elast-energy-density ¶
  - depends_on theorem 5.116 Sylvester's law of inertia ¶
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on definition 5.110 Bilinear map ¶ ↺
      - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
    - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
    - proves proof ch:03-linear-algebra-representations@proof-53 ¶
  - proves proof ch:13-continuum-elasticity@proof-13 ¶
- depends_on theorem 30.40 Navier–Cauchy equations ¶
  - depends_on equation 30.21 eq:elast-hooke-isotropic ¶
  - depends_on theorem 30.21 Cauchy's equation of motion ¶
    - depends_on theorem 7.133 Gauss ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-79 ¶
    - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - depends_on definition 30.17 Traction ¶
      - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - proves proof ch:13-continuum-elasticity@proof-7 ¶
    - proves proof ch:13-continuum-elasticity@proof-9 ¶
  - proves proof ch:13-continuum-elasticity@proof-18 ¶
- proves proof ch:13-continuum-elasticity@proof-21 ¶
depends_on proposition 30.49 The Rayleigh speed of a Poisson solid ¶
- depends_on proposition 30.48 Rayleigh's secular equation ¶
  - depends_on definition 30.22 Boundary conditions of elastostatics ¶
    - depends_on definition 10.24 Well-posed problem ¶
      - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition 6.24 Metric ¶
    - depends_on theorem 30.21 Cauchy's equation of motion ¶ ↺
  - depends_on theorem 30.40 Navier–Cauchy equations ¶ ↺
  - proves proof ch:13-continuum-elasticity@proof-24 ¶
- proves proof ch:13-continuum-elasticity@proof-25 ¶
proves proof ch:13-continuum-elasticity@proof-23 ¶

Supports

Nothing declares a dependency on this node yet.

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`cites`	→	Some Problems of Geodynamics	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:1623
`cites`	→	On the propagation of earthquake motion to great distances	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:1616
`cites`	→	On Waves Propagated along the Plane Surface of an Elastic Solid	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:1621
`depends_on`	→	Two bulk wave speeds	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1624
`depends_on`	→	The Rayleigh speed of a Poisson solid	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1624
`proves`	←	ch:13-continuum-elasticity@proof-23	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1627