theorem 31.94 Gravity–capillary waves on a free surface

open in the book · parts/03-classical-mechanics/14-fluid-dynamics.tex:3208 · p. 1078

Rests on

depends_on theorem 31.74 The interface dispersion relation ¶
- depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
  - depends_on corollary 31.13 Incompressible flow ¶
    - depends_on theorem 31.12 Conservation of mass ¶
      - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - proves proof ch:14-fluid-dynamics@proof-5 ¶
    - proves proof ch:14-fluid-dynamics@proof-6 ¶
  - depends_on equation 10.24 eq:pde-poisson-equation ¶
  - depends_on proposition 7.136 Properties of conservative fields ¶
    - depends_on definition 7.135 Conservative field ¶
      - depends_on equation 7.124 eq:ana-nabla-gradient ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - depends_on theorem 7.132 Stokes ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
      - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on theorem 7.131 Green ¶
      - proves proof ch:05-real-analysis@proof-78 ¶
    - proves proof ch:05-real-analysis@proof-80 ¶
  - proves proof ch:14-fluid-dynamics@proof-14 ¶
- depends_on theorem 31.24 Bernoulli ¶
  - depends_on equation 31.4 eq:fluid-lamb-identity ¶
  - depends_on theorem 31.22 Euler's equations of motion ¶
    - depends_on definition 31.21 Ideal fluid ¶
      - depends_on definition 31.1 Fluid ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶ ↺
    - depends_on theorem 30.21 Cauchy's equation of motion ¶
      - depends_on theorem 7.133 Gauss ¶
      - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - proves proof ch:13-continuum-elasticity@proof-9 ¶
    - proves proof ch:14-fluid-dynamics@proof-11 ¶
  - proves proof ch:14-fluid-dynamics@proof-12 ¶
- proves proof ch:14-fluid-dynamics@proof-35 ¶
proves proof ch:14-fluid-dynamics@proof-47 ¶

Supports

depends_on phenomenon 31.96 The solitary wave ¶
depends_on phenomenon 31.95 The observed dispersion of water waves ¶

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
`depends_on`	→	The interface dispersion relation	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3217
`depends_on`	←	The solitary wave	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3310
`depends_on`	←	The observed dispersion of water waves	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3257
`proves`	←	ch:14-fluid-dynamics@proof-47	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:3220