Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 52

Taking the Laplacian of both sides of equation (1.336), and using relation (1.325),
the Galerkin vector is found to obey

F

4 G

∇ =−

− σ .

(1.339)

1

This equation can also be obtained by direct substitution of the representation of the
displacement field (1.335) by the Galerkin vector into the Navier equation (1.323).
From (1.335), we find that

G ) , (1.340)

1 − σ

μ ∇

1
2μ ∇

1 − σ

μ ∇

1
2μ ∇

2 u 4 G 2 4 G 2 (

∇ = − ∇ ( ∇· G ) = − ∇ ∇·

and

2 G )

G )

1 − σ

μ ∇

1
2μ ∇

2 (

∇ ( ∇· u ) = ∇· ∇ − ∇ ∇·

(

G ) .

1 − 2 σ
2μ ∇

2 (

= ∇ ∇· (1.341)

Substitution of these expressions into (1.323) gives

1 − σ

μ ∇

F
μ ,

4 G

=− (1.342)

confirming equation (1.339). The Galerkin vector for Kelvin's problem of a force
F 0 concentrated at (ξ 1 ,ξ 2 ,ξ 3 ) in an infinite medium is

λ + μ
4π(λ +

G =

2μ) R F 0 .

(1.343)

1.4.9 The Boussinesq problem

Kelvin's problem may be specialised to the case where the concentrated force is
applied at ( x 1 , x 2 ,0), on the surface x 3 = 0, and entirely in the x 3 -direction. The

Galerkin vector then has the form G

=

AR 0 e 3 , with A an arbitrary constant, and

where R 0 is the radius to the field point,

x 1 −

x 1 2 + x 2 − x 2 2 x 3 . R 0 = + (1.344)