Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 50

1.4.7 The Papkovich-Neuber solution
Other forms of solutions to the Navier equation can be developed. Using the dimen-
sionless Poisson's ratio σ (1.255), it can be recast in the form

1 F

μ .

2 u

∇ +

2σ ∇

( ∇· u ) =− (1.323)

1 −

2 L , it follows that 2 (

Since ∇· u =∇ ∇ ( ∇· u ) =∇ ∇ L ), and

2 u

L

1 F

μ .

∇ +

2σ ∇

=− (1.324)

1 −

Writing the vector quantity in square brackets as Φ /2μ,wehave

2 Φ

∇ =− 2 F , (1.325)

with

1
2μ

1

Φ = u +

2σ ∇

L . (1.326)

1 −

Taking the divergence of both sides of the latter relation gives

1
2μ ∇·

2 (1 − σ)

2 L .

Φ =

2σ ∇

(1.327)

1 −

Using the vector identity

2 ( R 2 Φ

∇ · Φ ) = R ·∇ + 2 ( ∇· Φ ), (1.328)

and relation (1.325), we obtain

2

F

1 1 − σ

2 ( R 2 L .

∇· Φ = ∇ · Φ ) + 2 R · = 4μ

2σ ∇

(1.329)

1 −

Thus,

2 2 μ
1

Φ

1 1

∇ L −

− σ ) R

· =

− σ ) R

· F . (1.330)

−

4 ( 1 2 ( 1 2σ

Denoting the scalar in square brackets by B ,

1

2 B =

∇

− σ) R

· F , (1.331)

2 (1

and

B

Φ

1 − 2σ

2μ

1

L = +

− σ) R

· . (1.332)

4 (1