Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 234

respectively. For n ≥ 1, the first two equations of the first system have a non-trivial

solution for n 1 δα = 0, which implies that α = 0. For α = 0, the last two equations

give the condition for a non-trivial solution as

4 ( n − 1)( n + 2)μ(λ + μ)

− δ = = 0. (3.147)

λ + 2μ

Thus, the first system has the non-trivial solution A 1,0 = A 3,0 only for α = 0, n = 1,

when n 1. This degree-one displacement field is special, since it describes a rigid-

body displacement with respect to the centre of mass, that is involved in any
exchange of linear momentum. The second system gives A 5,0

≥

= 0, except when

α = 0. This degree-zero, radial spheroidal deformation field is special, since

the governing system degenerates to fourth order. The special cases of n

0, n =

= 0and

n = 1 will be considered separately later. Let us now consider the case where n ≥ 1.

1, the homogeneous first and third equations of the system (3.139) may
be used to eliminate unknowns. The third equation gives

For ν ≥

1
μ

A 1,ν =− η A 3,ν + A 4,ν − 1 . (3.148)

Eliminating A 1,ν in the first equation with this relation yields

A 2,ν − 1 =− q 1 (η) A 3,ν + q 2 (η) A 4,ν − 1 , (3.149)

with

= n 1 3) λ + q 1 (η) + η(η + 2η(η + 1)μ, (3.150)

3) λ q 2 (η) = 2 (η + 1) + (η +

μ .

(3.151)

These two relations allow the elimination of A 1,ν and A 2,ν − 1 in the second equation,
giving the new left side

− p 1 (η) A 3,ν + p 2 (η) A 4,ν − 1 , (3.152)

with

= η n 1 3) λ +

2η η

2 n 1 μ, 3η −

2

p 1 (η) + η(η + + (3.153)

3) λ p 2 (η) = 2η(η + 3) − n 1 + η(η +

μ ,

(3.154)

and in the fourth equation, giving the new left side

− r 1 (η) A 3,ν + r 2 (η) A 4,ν − 1 , (3.155)

with

= n 1 3) λ + r 1 (η) + η(η + 2 ( n 1 + η − 1)μ, (3.156)

3) λ r 2 (η) = η + 5 + (η +

μ .

(3.157)