486 Solutions to odd-numbered problems
Solutions to selected odd-numbered
problems
Note that in many problems (as in reality), the solution varies
depending on the interpretation of the data or the assumptions
used.
Chapter 2
(1)
R 12 = 0, T 12 = 1.
(3a)
(3, −2, 5).
(3b)
(2, 1, 3).
(3c)
(5, 3, 0)/ 14.
(5a)
σ 1 = 2, σ 2 = 0, σ 3 = −2; n
(1) = (1, 1, 0)/ 2, n
(2) = (0, 0, 1),
n (3) = (1, −1, 0)/ 2.
(5b)
τ = 2. Planes have normals (1, 0, 0) and (0, 1, 0).
(7b)
−150 kbar.
(7c)
D
.
=
−
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
0
2 1
2 5 3
1 3 5
(7d)
450 km.
(9)
2%.
(11a)
6
6
0
0
0
6
2
2
0
2
6
4
λ
µ
λ
µ
µ
µ
λ
µ
.
+
+
+
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(11b)
18λ + 16µ.
(13a)
[(λ + 2µ)/ρ]
1/2 .
(13b)
(µ /ρ)
1/2 .
(15)
3 .
(17a)
L = 11.25 km/s; M = 6.18 km/s.
(17b)
L /M = 1.82.
(19a)
0.8 km, 8 km, 800 km.
(19b)
0.000125 s, 0.125 s, 12.5 s; 8000 Hz, 8 Hz, 0.08 Hz.
(21)
i 2 = 13°; i 3 = 17°; i c = 37°.
(23a)
For the i 1 = 0° wave: i 2 = 0°, l 1 = 2 km, l 2 = 2 km, T = 3.3 s. For
the i 1 = 30° wave: i 2 = 49°, l 1 = 2.3 km, l 2 = 3.0 km, T = 4.3 s.
(23b)
For the i 1 = 0° wave: s 1 = (0, 1) s/km, | s 1 | = 1/v 1 , s 2 = (0, 2/3) s/km,
| s 2 | = 1/v 2 . For the i 1 = 30° wave: s 1 = (0.5, 3 /2) s/km,
| s 1 | = 1/v 1 , s 2 = (0.5, 0.44) s/km, | s 2 | = 1/v 2 .
(25a)
Ψ I = B 1 exp [i(ωt − k x x + k x r β z)],
Ψ R = B 2 exp [i(ωt − k x x − k x r β z)],
Φ R = A 2 exp [i(ωt − k x x − k x r α z)].
(25b)
σ
µ
xz
z x
x
z
=
+
−
⎛
⎝
⎜
⎞
⎠
⎟ = ,
2
0
2
2
2
∂
∂ ∂
∂
∂
∂
∂
2
2
Φ
Ψ
Ψ
σ zz = λ
µ
∂
∂
∂
∂
∂
∂
∂
∂ ∂
2
2
2
2
2
2
2
2
0
Φ
Φ
Φ
Ψ
x
z
z
x z
.
+
⎛
⎝
⎜
⎞
⎠
⎟ +
+
⎛
⎝
⎜
⎞
⎠
⎟ =
2r α A 2 + (1 − r β
2 )B 1 + (1 − r β
2
)B 2 = 0,
A 2 (λ + λr
2
α + 2µr
2
α ) − 2µr β B 1 + 2µr β B 2 = 0.
(25c)
B 2 /B 1 = −1, A 2 /B 1 = 0.
(25d)
| |
| |
| |
| |
u
u
SR
SI
B
B
;
=
2
1
| |
| |
| |
| |
u
u
PR
SI
A
B
.
=
β
α
2
1
0
0
SR
SI
B
B
,
=
2
2
1
2
0
0
PR
SI
A
B
.
=
η
η
α
β
2
2
1
2
(27a)
For ScS: j slab = 26°, j surf = 4°. For ScSp: i slab = 50°, i surf = 20°.
(27b)
50 km.
(29)
ω 1 = 0.58, ω 2 = 1.16.
(33)
(3/2)(5 cos
2 θ sin θ − sin θ).
(35a)
2591 s.
(35b)
4.4 km/s.
(35c)
8.3 km/s, 13.8 km/s, 74.5 km/s.
(35d)
c = 5.36 km/s, 5.01 km/s, 4.05 km/s; λ = 11,440 km, 1312 km,
307 km.
(37a)
∆ω /ω observed: 0.059.
(37b)
∆ω /ω predicted: 0.037.
(C-3)
For P waves at the CMB, T mc = 0.975, R mc = −0.025,
T cm = 1.025, R cm = 0.025, 0 R /0 I = 0.0006, 0 T /0 I = 0.9994.
For S waves at the CMB, T mc = 2, R mc = 1, 0 R /0 I = 1, 0 T /0 I = 0.
Chapter 3
(1)
α 0 = 5.7 km/s, α 1 = 7.8 km/s, h 0 = 23 km.
(3a)
α c = 6.7 km/s, α m = 8.2 km/s.
(3b)
3.1 km.
(3c)
5.4 km.
(5)
α c = 6.5 km/s, α m = 8 km/s, dip = 4°, h u = 50 km, h d = 30 km.
(11)
24,000,000.
Solutions to selected odd-numbered
problems
Note that in many problems (as in reality), the solution varies
depending on the interpretation of the data or the assumptions
used.
Chapter 2
(1)
R 12 = 0, T 12 = 1.
(3a)
(3, −2, 5).
(3b)
(2, 1, 3).
(3c)
(5, 3, 0)/ 14.
(5a)
σ 1 = 2, σ 2 = 0, σ 3 = −2; n
(1) = (1, 1, 0)/ 2, n
(2) = (0, 0, 1),
n (3) = (1, −1, 0)/ 2.
(5b)
τ = 2. Planes have normals (1, 0, 0) and (0, 1, 0).
(7b)
−150 kbar.
(7c)
D
.
=
−
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
0
2 1
2 5 3
1 3 5
(7d)
450 km.
(9)
2%.
(11a)
6
6
0
0
0
6
2
2
0
2
6
4
λ
µ
λ
µ
µ
µ
λ
µ
.
+
+
+
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(11b)
18λ + 16µ.
(13a)
[(λ + 2µ)/ρ]
1/2 .
(13b)
(µ /ρ)
1/2 .
(15)
3 .
(17a)
L = 11.25 km/s; M = 6.18 km/s.
(17b)
L /M = 1.82.
(19a)
0.8 km, 8 km, 800 km.
(19b)
0.000125 s, 0.125 s, 12.5 s; 8000 Hz, 8 Hz, 0.08 Hz.
(21)
i 2 = 13°; i 3 = 17°; i c = 37°.
(23a)
For the i 1 = 0° wave: i 2 = 0°, l 1 = 2 km, l 2 = 2 km, T = 3.3 s. For
the i 1 = 30° wave: i 2 = 49°, l 1 = 2.3 km, l 2 = 3.0 km, T = 4.3 s.
(23b)
For the i 1 = 0° wave: s 1 = (0, 1) s/km, | s 1 | = 1/v 1 , s 2 = (0, 2/3) s/km,
| s 2 | = 1/v 2 . For the i 1 = 30° wave: s 1 = (0.5, 3 /2) s/km,
| s 1 | = 1/v 1 , s 2 = (0.5, 0.44) s/km, | s 2 | = 1/v 2 .
(25a)
Ψ I = B 1 exp [i(ωt − k x x + k x r β z)],
Ψ R = B 2 exp [i(ωt − k x x − k x r β z)],
Φ R = A 2 exp [i(ωt − k x x − k x r α z)].
(25b)
σ
µ
xz
z x
x
z
=
+
−
⎛
⎝
⎜
⎞
⎠
⎟ = ,
2
0
2
2
2
∂
∂ ∂
∂
∂
∂
∂
2
2
Φ
Ψ
Ψ
σ zz = λ
µ
∂
∂
∂
∂
∂
∂
∂
∂ ∂
2
2
2
2
2
2
2
2
0
Φ
Φ
Φ
Ψ
x
z
z
x z
.
+
⎛
⎝
⎜
⎞
⎠
⎟ +
+
⎛
⎝
⎜
⎞
⎠
⎟ =
2r α A 2 + (1 − r β
2 )B 1 + (1 − r β
2
)B 2 = 0,
A 2 (λ + λr
2
α + 2µr
2
α ) − 2µr β B 1 + 2µr β B 2 = 0.
(25c)
B 2 /B 1 = −1, A 2 /B 1 = 0.
(25d)
| |
| |
| |
| |
u
u
SR
SI
B
B
;
=
2
1
| |
| |
| |
| |
u
u
PR
SI
A
B
.
=
β
α
2
1
0
0
SR
SI
B
B
,
=
2
2
1
2
0
0
PR
SI
A
B
.
=
η
η
α
β
2
2
1
2
(27a)
For ScS: j slab = 26°, j surf = 4°. For ScSp: i slab = 50°, i surf = 20°.
(27b)
50 km.
(29)
ω 1 = 0.58, ω 2 = 1.16.
(33)
(3/2)(5 cos
2 θ sin θ − sin θ).
(35a)
2591 s.
(35b)
4.4 km/s.
(35c)
8.3 km/s, 13.8 km/s, 74.5 km/s.
(35d)
c = 5.36 km/s, 5.01 km/s, 4.05 km/s; λ = 11,440 km, 1312 km,
307 km.
(37a)
∆ω /ω observed: 0.059.
(37b)
∆ω /ω predicted: 0.037.
(C-3)
For P waves at the CMB, T mc = 0.975, R mc = −0.025,
T cm = 1.025, R cm = 0.025, 0 R /0 I = 0.0006, 0 T /0 I = 0.9994.
For S waves at the CMB, T mc = 2, R mc = 1, 0 R /0 I = 1, 0 T /0 I = 0.
Chapter 3
(1)
α 0 = 5.7 km/s, α 1 = 7.8 km/s, h 0 = 23 km.
(3a)
α c = 6.7 km/s, α m = 8.2 km/s.
(3b)
3.1 km.
(3c)
5.4 km.
(5)
α c = 6.5 km/s, α m = 8 km/s, dip = 4°, h u = 50 km, h d = 30 km.
(11)
24,000,000.