This document contains equations related to fluid mechanics and wave theory. Some key equations include: 1) Equations for velocity profiles in boundary layers and free surface waves. 2) Equations describing linear wave theory including wave celerity, wave frequency, and potential functions for wave propagation. 3) Equations for hydrostatic pressure and continuity. 4) Equations describing forces on structures like drag and inertia.

1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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