Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 235

The system (3.139) is then reduced to
⎝ ⎠

01 q 1 (η) − q 2 (η)

00 p 1 (η) p 2 (η)

10 η −

−

x ν = b ν . (3.158)

1/μ

00 − r 1 (η) r 2 (η)

The determinant of the coe cient matrix of the reduced system (3.158) is

=

− p 1 (η) p 2 (η)

p 2 (η) r 1 (η) − p 1 (η) r 2 (η). (3.159)

− r 1 (η) r 2 (η)

The coe cient matrix is singular for

p 1 (η)μ r 2 (η) = r 1 (η)μ p 2 (η), (3.160)

or

η { 2η η

3η 2 n 1 μ ( η + 5 ) μ

2

+ η ( η + 3 ) 3 ) λ +

( η +

n 1 } λ + + −

n 1 + η(η +

1 μ η(η + n 1 μ .

(3.161)

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

2 cancel in this condition, and those in μ 2 and λμ each yield the

The terms in λ

quartic equation

4 3 2

η + 6η + [11 − 2 n ( n + 1)]η + 6 [1 − n ( n + 1)]η

+ ( n − 2) n ( n + 1)( n + 3) = 0, (3.162)

with n 1 replaced by n ( n + 1). In turn, the quartic factors into the product of quad-

ratics,

η

1) η 3)

2 2

+ 3η − ( n − 2)( n + + 3η − n ( n + = 0. (3.163)

Finally, this expression factors to

(η − n + 2)(η − n )(η + n + 1)(η + n + 3) = 0. (3.164)

The eigenvalues of the matrix, M n , in the first system (3.139), are thenη =− ( n + 3)

and η =− n .

Much more directly, the eigenvalues of the matrix, N n , in the second system
(3.143), are η =−

( n + 1),aswellasη = n − 2andη =

( n +

2 ) and η = n − 1.

Now consider how the power series solutions propagate for higher values of
ν ≥ 1 with n ≥ 2. Then, from (3.146), A 1,0 = A 3,0 = A 5,0 = 0.