Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 226

Applying the orthogonality relation (1.180), this expression reduces to

( n −

1)( n + m )
2 n

( n + 2)( n − m + 1)

u − Cn = m v − m t − m t − m i Ω +Ω − . (3.82)

n

n − 1 n + 1

− 1 2 n + 3

After replacing − m with m , the radial coe cient of the radial spheroidal part of the

body force per unit volume becomes

u Fn = ω

2

ρ 0 u n −

n

2 m ω Ω ρ 0 v

2 i ω Ω ρ 0 ( n t n + 1

− 1)( n − m ) ( n + 2)( n + m + 1)

t n − 1 − − . (3.83)

2 n − 1 2 n + 3

The radial coe cient of the transverse spheroidal part of the vector Ω × u is

given by

2π π ∞ k = l

dP n
d θ

1
4π n ( n +

2 n +

v − Cn = 1) m

k

l P l

( − −Ω ik v cosθ

1 ) 0 0

l = 0 k =− l

dP l
d θ

dP l
d θ

dP n
d θ

k
l

P n cosθ + u l P l P n sinθ +Ω t l + im Ω v im Ω cosθsinθ

e i ( k + m )φ d θ d φ.

mkt l P l P n cotθ −Ω (3.84)

Consider the first integral arising on the right,
k 2π
0

π

dP n
d θ

P l cosθ e i ( k + m )φ d θ d φ. (3.85)

0

On integrating by parts, it becomes

⎝ ⎠

k 2π
0

π

dP l
d θ

π
0 −

m

− k P l P n cosθ

P l sinθ P n e i ( k + m )φ d θ d φ 2π k δ cosθ −

0

⎝ ⎠

k π
0

dP l
d θ

m
−

P l sinθ − P n d θ, = 2π k δ cosθ (3.86)

where the first term on the left vanishes for m = − k = 0, and for m = − k 0it

also vanishes, since P l ( P n ( ± 1) = ± 1) = 0. This combines with the second integral

arising on the right of (3.84),

− k π
0

dP l
d θ

m

P n cosθ d θ, − 2π m δ (3.87)