海洋工程环境学(1,1)

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∞
∞
均方波高
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均方根波高
∞
H rms = 2 2∑ S (ω n )ω = 2 2m 0
n =1
m 0 = ∫ S (ω )dω
0
∞
零阶谱矩
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(4) 线性变换系统
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S Y (ω ) = H (ω ) S X (ω )
2
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(5) 实用海浪谱
B S (ω ) = 5 exp 4 ω ω A
H 2H p(H ) = 2 exp H rms H rms
9
2
累计概率分布函数 (Cumulative probability distribution function )
P (H ) =
H
∫
0
H p (H )dH = 1 exp H rms
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实用海浪谱
1. 2. 3. 4. 5. Pierson-Moscowitz (1964) Spectrum (P-M Spectrum) ITTC (1987) Spectrum with Double Parameters (ISSC Spectrum) JONSWAP (1973) Spectrum Bretschneider (1959) Spectrum Darbyshir (1952) Spectrum
∞ ∞
2
π H rms = 0.8862H rms = 4
P (H p ( H )= max. ) = 0.39
H p ( H )= max. =
1 2
H rms = 0.71H rms ,
H rms p (H p ( H = max.) ) = 0.86,
13
均方根波高
H rms = H rms ,
17
雷利概率密度函数 (Rayleigh probability density function )
H 4H p (H ) = 2 exp 2 H HS S
H P (H ) = 1 exp 2 H S
2
2
累计概率分布函数 (Cumulative probability distribution function )
1 H = N
∑ Hi
i =1
N
TZ =
1 N
∑ TZi
i =1
N
4
(2) 均方根值
H rms = 1 N
∑
i =1
N
H i2
TZ,rms =
1 N
∑
i =1
N
2 TZi
5
(3) 1/n平均值
1 = N n H ∑ j
j =1 N n
H1 n
有义波高 (n=3)
H S = H1 3
1 = N 3
逆独立的离 散形式
A R (τ ) = ∑ cos(ω nτ ) n =1 2
∞
2 n
随机相位 的 概率密度函数
R (τ ) = ∑ S (ω n ) cos(ω nτ )ω
n wk.baidu.com1
∞
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2 An S (ω n ) = 2ω
2 Hn 2 An = 2 S (ω n )ω = 4
2 2 H rms = ∑ H n = 8∑ S (ω n )ω n =1 n =1
自相关函数函数特征
T
R(0) ≥ R(τ )
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(3) Wiener-Khintchine 定理
正定 理
S (ω ) =
∫ R(τ ) cos(ωτ )dτ π
2
0
∞
∞
逆定 理
R (τ ) = ∫ S (ω ) cos(ωτ )dω
0
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频率为ω n 波在时间序列相隔 τ 的自相关函数
2 An Rn (τ ) = E [η n (t )η n (t + τ )] = ∫ [η n (t )η n (t + τ )] p( )d = cos(ω nτ ) 2 0 2π
T 2
T
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宽带过 程
窄带过 程
正弦信 号
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谱函数特征
S (ω ) ≥ 0
ω = ωP :
S (ω P ) = max
S (0) = 0 & S (∞ ) → 0
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(2) 自相关函数
1 R(τ ) = lim ∫ η (t )η (t + τ )dt T →∞ T 0
> R(τ ) = 0 <
∞
H rms p (H S ) = 0.38,
P (H S ) = 0.86
15
最大波高
H1 N 1 P (H 1 N ) = exp H rms
2
1 = N
H 1 N = LnN H rms
1 = lnN H S 2
H max = H 1 N =
∞
∫
H1 N
p (H ) H dH 1 N
∑
j =1
N 3
H j
6
概率特征 1.1.2 概率特征 (1) 平稳的(stationary)各态历经的(ergodic) 随机过程
H+ ≈ H
TZ ≈T Z
beating pattern
∑η i ≈ 0
i
H (t 1 ) ≈ H (t 2 )
7
窄带过程
8
(2) 概率密度函数
雷利概率密度函数 (Rayleigh probability density function )
H rms p (H rms ) = 0.74,
P(H rms ) = 0.63
14
有义波高
Hu
P (H u ) =
∫
0
Hu p (H )dH = 1 exp H rms
2 = 3
H u = 1.05H rms
p (H ) HS = ∫ H dH = 2 H rms , 13 Hu
= 0
2
H p ( H )= max. =
1 2
H rms = 0.71H rms ,
P (H p ( H )= max. ) = 0.39
12
平均波高
H 2H H = E (H ) = ∫ Hp(H )dH = ∫ H 2 exp H rms H rms 0 0
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(6)方向谱
S (ω , θ ) = S (ω )G (θ )
∫ π G(θ )dθ = 1
π
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方向谱
1. Denis and Pierson (1953) 2. Longuet-Higgins, Cartwright and Smith (1961) 3. Cote (1962) 4. Conner (1980) 5. ITTC and ISSC
10
2
(3) 特征波
零波高
H = 0,
p(0) = 0,
P(0 ) = 0
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最大概率密度波高
p(H ) = max.,
H d d 2H p (H ) = 2 exp dH dH H rms H rms
H rms p (H p ( H = max.) ) = 0.86,
16
最大波高
Number of waves N 100 1 000 10 000 100 000
Exceed probability
PE (H 1 N )
The lower limit of maximum wave height H 1 N / H S 1.52 1.86 2.15 2.40
10-2 10-3 10-4 10-5
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概率特征 概率特征 1.1.3 谱特征 (1) 谱函数
η (t ) = ∑η (ω n ) = ∑ A(ω n ) cos(ω n t + n )
n =1 n =1 ∞ ∞
1 lim ∫ A 2 (t , ω n + ω )dt T →∞ T 0
1 S (ω ) = lim ω → 0 ω 1 lim T → ∞ T ∫ A (t, ω n + ω )dt 0
波浪理论及其工程应用
1. 随机波浪理论 2. 波浪运动的数学模型 3. 线性波理论 4. 非线性波理论 5. 波浪理论的工程应用
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1. 随机波浪理论 1.1 短期统计特征 1.2 长期统计特征 1.3 设计波
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1. 随机波浪理论 1.1 短期统计特征
连续子样
[H i ;TZi ]
离散子样
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1.1.1平均特征 (1) 平均值