488 Solutions to odd-numbered problems
Chapter 6
(1a)
a 0 = 0, a k = 0, b k = (2/kπ)(1 − cos (kπ)).
(1b)
a 0 = 0, a k = 0, b k = −cos (kπ)/kπ = (−1)
k+1
/kπ.
(3a)
−1.
(3b)
4i.
(3c)
−i.
(3d)
1.5 + 2.6i.
(7a)
πe
−iπ /2 [δ(ω − ω 0 ) − δ(ω + ω 0 )].
(9a)
a 2 σ 2
u + b 2 σ 2
v + 2abσ 2
uv .
(9b)
a
2
v
2
σ
2
u + a
2 u
2
σ
2
v + 2a
2
uvσ
2
uv .
(9c)
(a
2 /v
2 )σ
2
u + (a
2
u
2 /v
4 )σ
2
v − 2(a
2
u/v
3
)σ
2
uv .
(9d)
a
2
b
2 u
(2b−2) σ
2
u .
(11a)
v∆t/(2 cos i).
(11b)
10 km.
(11c)
(σ
2
v ∆t
2 + σ
2
∆t v
2 + σ
2
i v
2
∆t
2 tan
2 i)/4 cos
2 i.
(11d)
4 km.
Appendix
(1)
21°.
(5)
a 11 a 22 a 33 − a 11 a 23 a 32 − a 12 a 21 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 −
a 13 a 22 a 31 .
(7)
A
− =
−
−
⎛
⎝
⎜
⎞
⎠
⎟
1
2 3 1 3
5 6 1 6
/
/
/
/
.
(13a)
6xy
2 + 2x
3 + 2.
(13b)
(0, 0, 0).
(13c)
(6y
2 + 6x
2
, 12xy, 0).
(13d)
xz + x
3
y
2 + y
2 + constant.
(15)
Hint: use Eqn A.7.4.
(C-5)
| (1, 4, 2) | = 21; | (2, 3, 1) | = 14; sum = (3, 7, 3); a · b = 16;
a × b = (−2, 3, −5); θ = 21.1°.
(C-7b)
(0, −1, 1).
(C-9b)
0 7
0 1 0 3
0 5
0 5 0 5
0 8
0 4 0 2
.
.
.
.
.
.
.
.
.
.
−
−
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
(C-11b) 1, 3, 4.
(C-13b) i: ∆ = 93°, ζ = 48°.
ii: ∆ = 54°, ζ = 49°.
iii: ∆ = 87°, ζ = 148°.
iv: ∆ = 35°, ζ = 232°.
Earthquake d:
−
−
−
−
−
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 960 0 218 0 082
0 218 0 843 0 351
0 082 0 351 0 117
.
.
.
.
.
.
.
.
.
.
Earthquake e:
0 960 0 218 0 082
0 218 0 843 0 351
0 082 0 351 0 117
.
.
.
.
.
.
.
.
.
.
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
(C-7b)
0.95.
(C-9a)
Gaussian: 0.1%; Poisson: 4%.
(C-9b)
Gaussian: 0.3%; Poisson: 3%.
(C-9c)
Gaussian: 0.5%; Poisson: 2%.
(C-11a) τ = 21.8 yr; σ = 7.2 yr; Poisson: p = 37%;
Gaussian: C(1993, 1985) = 64%.
(C-11b) τ = 21.8 yr; σ = 1.5 yr; Poisson: p = 37%;
Gaussian: C(1993, 1985) = 99%.
(C-11c) τ = 25.5 yr; σ = 11.1 yr; Poisson: p = 31%;
Gaussian: C(2018, 2010) = 82%.
(C-11d) τ = 27.2 yr; σ = 14.7 yr; Poisson: p = 29%;
Gaussian: C(2028, 2020) = 74%.
Chapter 5
(1a)
0.77 m; M w = 6.8, length = 31 km.
(1b)
4.62 m; M w = 7.8, length = 240 km.
(3a)
40 mm/yr.
(3b)
125, 250, 500 yr.
(3c)
For 25%: 500, 1000, 2000 yr; for 50%: 250, 500, 1000 yr.
(3e)
M w ≈ 8.4; M 0 = 5 × 10
28 dyn-cm.
(5)
6 × 10 31 dyn-cm; M w 10.5.
(7a)
47 mW/m 2 .
(7b)
33 mW/m
2 .
(7c)
84 mW/m
2 .
(9)
~1 Ga.
(11)
−21.5 bar/°C.
(13)
vL 3 /(24κ
2 t); 28 (for v = 10 cm/yr and t = 150 Ma).
(17)
58°; 251 MPa.
(C-1b)
San Andreas: 46 mm/yr at 324°; Aleutian: 53 mm/yr at 346°.
(C-3b)
(θ, φ, |ω |) = (−63.0°, 107.4°, 0.641 °/my).
(C-3c)
Hawaii: 66 mm/yr at 299°.
Chapter 6
(1a)
a 0 = 0, a k = 0, b k = (2/kπ)(1 − cos (kπ)).
(1b)
a 0 = 0, a k = 0, b k = −cos (kπ)/kπ = (−1)
k+1
/kπ.
(3a)
−1.
(3b)
4i.
(3c)
−i.
(3d)
1.5 + 2.6i.
(7a)
πe
−iπ /2 [δ(ω − ω 0 ) − δ(ω + ω 0 )].
(9a)
a 2 σ 2
u + b 2 σ 2
v + 2abσ 2
uv .
(9b)
a
2
v
2
σ
2
u + a
2 u
2
σ
2
v + 2a
2
uvσ
2
uv .
(9c)
(a
2 /v
2 )σ
2
u + (a
2
u
2 /v
4 )σ
2
v − 2(a
2
u/v
3
)σ
2
uv .
(9d)
a
2
b
2 u
(2b−2) σ
2
u .
(11a)
v∆t/(2 cos i).
(11b)
10 km.
(11c)
(σ
2
v ∆t
2 + σ
2
∆t v
2 + σ
2
i v
2
∆t
2 tan
2 i)/4 cos
2 i.
(11d)
4 km.
Appendix
(1)
21°.
(5)
a 11 a 22 a 33 − a 11 a 23 a 32 − a 12 a 21 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 −
a 13 a 22 a 31 .
(7)
A
− =
−
−
⎛
⎝
⎜
⎞
⎠
⎟
1
2 3 1 3
5 6 1 6
/
/
/
/
.
(13a)
6xy
2 + 2x
3 + 2.
(13b)
(0, 0, 0).
(13c)
(6y
2 + 6x
2
, 12xy, 0).
(13d)
xz + x
3
y
2 + y
2 + constant.
(15)
Hint: use Eqn A.7.4.
(C-5)
| (1, 4, 2) | = 21; | (2, 3, 1) | = 14; sum = (3, 7, 3); a · b = 16;
a × b = (−2, 3, −5); θ = 21.1°.
(C-7b)
(0, −1, 1).
(C-9b)
0 7
0 1 0 3
0 5
0 5 0 5
0 8
0 4 0 2
.
.
.
.
.
.
.
.
.
.
−
−
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
(C-11b) 1, 3, 4.
(C-13b) i: ∆ = 93°, ζ = 48°.
ii: ∆ = 54°, ζ = 49°.
iii: ∆ = 87°, ζ = 148°.
iv: ∆ = 35°, ζ = 232°.
Earthquake d:
−
−
−
−
−
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 960 0 218 0 082
0 218 0 843 0 351
0 082 0 351 0 117
.
.
.
.
.
.
.
.
.
.
Earthquake e:
0 960 0 218 0 082
0 218 0 843 0 351
0 082 0 351 0 117
.
.
.
.
.
.
.
.
.
.
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
(C-7b)
0.95.
(C-9a)
Gaussian: 0.1%; Poisson: 4%.
(C-9b)
Gaussian: 0.3%; Poisson: 3%.
(C-9c)
Gaussian: 0.5%; Poisson: 2%.
(C-11a) τ = 21.8 yr; σ = 7.2 yr; Poisson: p = 37%;
Gaussian: C(1993, 1985) = 64%.
(C-11b) τ = 21.8 yr; σ = 1.5 yr; Poisson: p = 37%;
Gaussian: C(1993, 1985) = 99%.
(C-11c) τ = 25.5 yr; σ = 11.1 yr; Poisson: p = 31%;
Gaussian: C(2018, 2010) = 82%.
(C-11d) τ = 27.2 yr; σ = 14.7 yr; Poisson: p = 29%;
Gaussian: C(2028, 2020) = 74%.
Chapter 5
(1a)
0.77 m; M w = 6.8, length = 31 km.
(1b)
4.62 m; M w = 7.8, length = 240 km.
(3a)
40 mm/yr.
(3b)
125, 250, 500 yr.
(3c)
For 25%: 500, 1000, 2000 yr; for 50%: 250, 500, 1000 yr.
(3e)
M w ≈ 8.4; M 0 = 5 × 10
28 dyn-cm.
(5)
6 × 10 31 dyn-cm; M w 10.5.
(7a)
47 mW/m 2 .
(7b)
33 mW/m
2 .
(7c)
84 mW/m
2 .
(9)
~1 Ga.
(11)
−21.5 bar/°C.
(13)
vL 3 /(24κ
2 t); 28 (for v = 10 cm/yr and t = 150 Ma).
(17)
58°; 251 MPa.
(C-1b)
San Andreas: 46 mm/yr at 324°; Aleutian: 53 mm/yr at 346°.
(C-3b)
(θ, φ, |ω |) = (−63.0°, 107.4°, 0.641 °/my).
(C-3c)
Hawaii: 66 mm/yr at 299°.