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1. u10 = uz
10
z
1
7
2. t = 1609
u10
3. ur = 0.71u1.23
10
4. ∂u
∂x
+ ∂v
∂y
+ ∂w
∂z
= 0
5. Du
Dt
= X − 1
ρ
∂p
∂x
, etc
6. Du
Dt
= X − 1
ρ
∂p
∂x
+ ϑ 2
u, etc
2
≡ ∂2
∂x2 + ∂2
∂y2 + ∂2
∂z2
7. wx = 1
2
∂v
∂z
− ∂w
∂y
etc
8. u = ∂φ
∂x
= ∂ψ
∂y
w = ∂φ
∂z
= −∂ψ
∂x
9. 2
φ = 0
10. 1
2
(u2
+ v2
+ w2
) + p
ρ
+ gz + ∂φ
∂t
= 0
11. η = −1
g
∂φ
∂t z=0
∂η
∂t
= ∂φ
∂z z=0
12. φ = gH
2ω
cosh k(d+z)
cosh κd
sin (κx − ωt)
13. η = H
2
cos (κx − ωt)
κ = 2π
L
14. ω2
= gκtanh κd
ω = 2π
T
15. φ = πH
κT
cosh κ(d+z)
sinh κd
sin (κx − ωt)
16. c = gT
2π
tanh κd
c = L
T
17. d
L0
= d
L
tanh 2π d
L
L0 = gT2
2π
1

18. u = πH
T
cosh κ(d+z)
sinh κd
cos (κx − ωt)
19. w = πH
T
sinh κ(d+z)
sinh κd
sin (κx − ωt)
20. ˙u = 2π2H
T2
cosh κ(d+z)
sinh κd
sin (κx − ωt)
21. ˙w = −2π2H
T2
sinh κ(d+z)
sinh κd
cos (κx − ωt)
22. ξ = −H
2
cosh κ(d+z)
sinh κd
sin (κx − ωt)
23. ζ = −H
2
sinh κ(d+z)
sinh κd
cos (κx − ωt)
24. η = P+γz
γκz
´C
´C = 1
25. kz = cosh κ(d+z)
cosh κd
C = L
T
ω2
= gκ tanh κd
26. E = γH2
8
27. CG = nC = 1
2
1 + 2κd
sinh 2κd
C
C = L
T
28. P = ECG
29. HS = 4
√
m0
30. c = g (H + d)
31. u = CN
1+cos (Mz
d )cosh (Mx
d )
[cos (Mz
d )+cosh (Mx
d )]
2
32. w = CN
sin (Mz
d )sinh (Mx
d )
[cos (Mz
d )+cosh (Mx
d )]
2
33. E = 8
3
√
3
γH
3
2 d
3
2
34. FD = 1
2
CDρAu |u|
35. FI = CM ρ∀ ˙u
36. Kc = umaxT
D
2

37. Re = umaxD
ϑ
38. FL = 1
2
CLρAu |u|
39. Fx = C |Vn| Vnx + Kanx, ..etc
40. |Vn| = Vx
2
+ Vy
2
+ Vz
2
− [cxVx + cyVy + czVz]2
1
2
41.


Vnx
Vny
Vnz

 =


1 − cx
2
−cxcy −cxcz
−cxcy 1 − cy
2
−cycz
−cxcz −cycz 1 − cz
2




Vx
Vy
Vz


similarly for anx, ..
42. cx = sin φcos θ; cx = sin φsin θ; cz = cos φ
43. FTD = CDρD
32κ
(ωH)2 2κ(d+z)+sinh 2κ(d+z)
sinh2
κd z=η
|cos θ| cos θ
44. FTI = CM ρπD2
4
(Hω2
)
[sinh κ(d+z)]z=η
2κsinh κd
sin θ
45. MTD = CDρD
64κ2 (ωH)2 2κ(d+z)sinh 2κ(d+z)−cosh 2κ(d+z)+2[κ(d+z)]2
+1
sinh2
κd z=η
|cos θ| cosθ
46. MTI = ρCM
2κ2
πD2
4
Hω2
sinh κd
κ(d+z)sinh κ(d+z)−cosh κ(d+z)+1
sinh κd
z=η
sin θ
47. Fm = φmρgCDH2
D
48. Mm = αmρgCDH2
D
49. w = CM
CD
D
H
50. ∆F = 2ρgHaA(κa)
κa
cosh κ(d+z)
cosh κd
cos (ωt − δ)
51. F = 2ρgHadA(κa)
κa
tanh κd
κd
cos (ωt − δ)
52. M = 2ρgHad2 A(κa)
κa
κd sinh κd+1−cosh κd
(κd)2
cosh κd
cos (ωt − δ)
53. ∆F = π
8
ρgHκD2 cosh κ(d+z)
cosh κd
Cm (κa) cos (ωt − δ)
54. F = π
8
ρgHD2
tanh κdCm (κa) cos (ωt − δ)
55. M = π
8
ρgH D2
κ
κd sinh κd+1−cosh κd
cosh κd
Cm (κa) cos (ωt − δ)
56. Fmax = π2ρHLD2Cm(κa)
4T2
3

57. Mmax = ρgHLD2
Cm (κa) κd tanh κd+sech κd−1
16
58. Fd = 1
16
γ 1 + 2κd
sinh κd
Hi
2
+ Hr
2
− Ht
2
59. X = 1
T
T
0
x1 (t) dt
60. σx
2
= 1
T
T
0
x2
(t) dt
61. SX (f) = 4
ω
0
Rx (τ) cos 2πfτ dτ
62. X (f) =
T
0
X1 (t) .e−ı2πft
dt
63. S (f) = 1
T
E [{X∗
(f) X (f)]
64. p (X) = 1
σX
√
2π
e
−(X−X)2
[2σ2
x]
65. mn =
∞
0
fn
S (f) df
Hs = 4
√
m0
Tav = m0
m2
t = 1 −
m2
2
m0m4
66. S (ω) = αg2
ω5 exp −β ω0
ω
4
α = 0.0081; β = 0.74; ω0 = g
u
67. u =
Hsg
√β
α
1
2
√
2
68. S (f) = αg2
(2π)4
f5 exp −
´β
f4
´β = 0.74 g
2πH
4
; α = 0.0081
69. S (ω) =
´´αg2
ω5 exp −
´´β ω0
ω
4
γ
exp −
(ω−ω0)2
2ω2
0σ2
´´α = 0.066 gF
u2
−0.22
;
´´β = 1.25
ω0 = 2.84 gF
u2
−0.33
; γ = 3.3; σ = 0.08
70. S (f) = 5H2
s
16f0
1
f
f0
5 exp −5
4
f
f0
−4
4

71. S (ω) = 0.214H2
s exp − (ω−ω0)2
0.065(ω−ω0+0.26)
1
2
72. RXY (τ) = 1
T
T
0
X (t) Y (t + τ) dt
73. SXY (f) =
∞
0
RXY (τ) e−12πfτ
dτ
74. RFF (τ) = C2
σ4
uG Ruu(τ)
σ2
u
+ K2
Raa (τ)
G (r) = 8
π
r + 4
3π
r3
+ ...
75. SFF (f) = 8
π
C2
σ2
uSuu (f) + K2
Saa (f)
76. Suu (f) = 4π2
f2 cosh2
κ(d+z)
sinh2
κd
Sηη (f)
77. Saa (f) = 4π2
f2
Suu (f)
78. p (H) = H
4σ2
n
e
− H2
8σ2
n
79. P (H) = 1 − e
− H2
8σ2
n
80. H =
√
2πση
81. Hrms = 2
√
2ση
82. H1
3
= 4ση
H 1
10
= 5.08ση; H 1
100
= 6.67ση
83. E [Hmax] = 0.705Hs
√
lnN
84. p (T) = 2.7T3
T
4 exp −0.675 T
T
4
85. p (T) = 1 − exp −0.675 T
T
4
86. P (ζ) = 1
2
[1 + ζ2
]
−3
2
ζ =
(T−T)
ϑ
; T = m0
m1
ϑ = 1 −
m2
1
m0m2
87. E = 1 − e(− L
Tr
)
5

88. P (Hs) = exp [−exp [−α (Hs − u)]]
α = π
(6σ2
Hs)
1
2
; u = Hs − 0.5772
α
Hs = allı Hsi all
Wı
W
σ2
Hs = allı H2
si all
Wı
W
− Hs
2
89. P (Hs) = 1 − exp − Hs−A
B
C
90. P (Hs) = exp − A−Hs
B
C
91. P (Hs) = 1√
2π
Hs
h=0
1
ch
exp −1
2
ln h−B
C
2
dh
92. P (Hs) = exp − Hs
B
−C
93. u |u| = 8
π
σuu
94. σ2
F = 8
π
C2
σ4
u + K2
σ2
a
95. H
L
= (0.142) tanh 2πd
L
96. β =
H0
L0
m2
NI = 1√
β
= tan θ
Hi
L0
97. H = 1
2n
C0
C
1
2
H0 = KsH0
98. H = 1
2n
C0
C
1
2 b0
b
1
2
H0 = KsKrH0
Kr = cos α0
cos α
1
2
99. C0
sin α0
= C1
sin α1
100. Tn = (4πLB)
ngtanh πdn
L0
1
2
101. Tn = 8πLB
g(2n−1)tanh πd
2LB
(2n−1)
1
2
6

102. pc = γH
cosh κd
+ γd
at z = −d
103. pt = − γH
cosh κd
+ γd
at z = −d
104. h0 =
πH2
i
L
coth 2πd
L
105. ´F = rF F
rF = b
yc
2 − b
yc
106. ´M = rM M
rM = b
yc
2
3 − 2 b
yc
107.
´´F = (1 − rF ) F
108.
´´M = (1 − rM ) M
´´M2 =
´´M1 − b (1 − rF ) F
109. Pm = 101r Hb
LD
ds
D
(D + ds)
D = ds + mLds
110. FT = 1
2
γ Hb
2
+ ds ds + Hb
2
+ 1
3
PmHb
111. MT = 1
6
γ ds + Hb
2
3
+ 1
3
PmHbds
112. ´FT = rF FT
113. ´MT = dsFT − (ds + a) (1 − rF ) FT
114. FT = γ
2
dbhc + 1
2
γ (ds + hc) [ds + hc]
hc = 0.78Hb
115. MT = γ
2
dbhc ds + hc
2
+ 1
6
γ(ds + hc)3
116. FT = 1
2
γdbhc 1 − X1
X2
4
+ 1
2
γh2
c 1 − X1
X2
2
117. MT = 1
4
γdbh2
c 1 − X1
X2
4
+ 1
6
γh3
c 1 − X1
X2
3
7

118. Fnet = Fsin2
α
´Fnet = Fnetsin2
θ
119. W = γrH3
KD(Sr−1)3 cot α
120. W50 = γrH3
KRR(Sr−1)3 cot α
121. W = γrH3
N3
s (Sr−1)3
122. uz = u10
z
10
1
12
123. FD = 1
2
CDρApu2
z
F = 0.0473CDApu2
z
124. uz = us(d+z)
d
125. uz = us
d+z
d
1
7
8

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