Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 61

where the integration variable in the last expression has been changed to u = cosθ.

Thus,

C 2π
0

1 2π 1

1

u 2 du cos 2

C 2 λ + μ
λ + μ

T 1 = dud φ − − φ d φ

0 0 0

2π 1

2 C λ +

−

2μ
λ + μ

u 1

u

+ u du cos 2

− φ d φ

1

0 0

2π 1

C λ +

2μ
λ + μ

u ) 2 du cos 2 + (1 − φ d φ

0 0

2π 1

C λ +

2μ
λ + μ

1 − u

−

u dud φ.

(1.421)

1 +

0 0

The evaluation of T 1 then requires the evaluation of the five elementary integrals,

2π 2π

1
2 (1

cos 2 I 1 = φ d φ = + cos 2φ) d φ

0 0

2π
0 = π,

1
2 φ +

1
4 sin 2φ

= (1.422)

3 u 3

1 1

0 =

1

u 2 du

1 2

3 ,

I 2 = − = u − (1.423)

0

1 1

2 1
0

1

u (1 − u ) 1

I 3 = du = (1 − u ) du −

u du

+ du

1 + u 1 +

0 0 0

u

1
0 =

1 3

2 − 2 u 2

= u − − 2 log(1 + u ) + 2log2, (1.424)

− u ) 3

1

1
0 =

1
3 ( 1

1
3 ,

− u ) 2 du =−

I 4 =

( 1

(1.425)

0

1

2 1
0

1

−

1 u 1

I 5 =

+ u du

=

+ u du

− du

1 1

0 0

u

1
0 =

= 2 log(1 + u ) − 2log2 − 1. (1.426)

Then,

3
2 −

2log2 C 2 λ + μ

2
3 −

2 C λ +

2μ
λ + μ · π ·

T 1 = C · 2π −

λ + μ · π ·

C λ +

2μ
λ + μ · π ·

1
3 −

C λ +

2μ
λ + μ ·

2π · 2log2 1 = + − 0. (1.427)

Consideration of the integrals over φ shows that T 2 0. The extra displace-

ment fields thus make no contribution to the traction on the hemisphere centred on
the point of application of the tangential force.

= T 3 =