ANALYSIS OF RANDOM WAVES
EN330 – Probs & Stats with Ocean Applications
Department of Ocean Engineering
United States Naval Academy
Annapolis, MD 21402
6
5
4
3
η (ft)
2
1
0
-1
-2
-3
-4
-5
-6
0
10
20
30
40
50
t (s)
60
70
80
90
100
Random Wave Analysis
• Defining wave height on monochromatic,
Regular Waves is straightforward.
4
3
2
η (ft)
1
0
-1
-2
-3
-4
0
10
20
30
40
50
t (s)
60
70
80
90
100
Random Wave Analysis
• With Random Waves it is a bit more complex .
6
5
4
3
η (ft)
2
1
0
-1
-2
-3
-4
-5
-6
0
10
20
30
40
50
t (s)
60
70
80
90
100
Zero-Crossing Analysis
1. Identify Zero Down-Crossings in Random
Wave Record.
2. Identify Maximum Crest and Minimum Trough
in Each Interval.
3. Calculate Individual Wave Heights, Hi, as:
H i = ηcrest i − η trough i
4. Calculate Wave Statistics.
Zero-Crossing Analysis
1. Identify Zero Down-Crossings in Random
Wave Record.
6
5
4
3
η (ft)
2
1
0
-1
-2
-3
-4
-5
-6
0
10
20
30
40
50
t (s)
60
70
80
90
100
Zero-Crossing Analysis
2. Identify Maximum Crest and Minimum Trough
in Each Interval.
6
5
4
3
η (ft)
2
1
0
-1
-2
-3
-4
-5
-6
0
10
20
30
40
50
t (s)
60
70
80
90
100
Zero-Crossing Analysis
3. Calculate Individual Wave Heights, Hi, as:
H i = ηcrest i − η trough i
6
5
4
H1 = 0.8 ft - (-0.9 ft)
3
η (ft)
2
H1 = 1.7 ft
1
0
-1
-2
H 2 = 2.4 ft - (-3.0 ft)
-3
H 2 = 5.4 ft
-4
-5
-6
0
10
20
30
40
50
t (s)
60
70
80
90
100
and so on
Zero-Crossing Analysis
4. Calculate Wave Statistics.
1
H=
N
H
s
N
∑H
i =1
i
= H 1 = mean of 1/3 largest waves in a record
3
(note - round to an integer # of waves if N/3 is not an integer)
H rms =
1
N
N
2
H
∑ i
i =1
Distribution of η
Water Surface Elevations follow a Gaussian Distribution.
Probability Density Function, PDF
PDF = p(η ) =
0.5
CDF = area under curve
PDF
0.4
1
e
σ 2π
1 η 
−  
2 σ 
2
where η = water surface elevation of interest
PDF
0.3
σ = standard deviation of water surface elevation
0.2
Cumulative Density Function, CDF
POE
0.1
CDF = P(η ) =
0.0
-3
-2
-1
0
1
2
3
1
 η 
erf
1
+



2
 σ 2 
η/σ
Probability of Exceedance, POE
POE = 1 − CDF
NOTE: Use standard normal distribution tables that follow to calculate CDF and POE
Distribution of η
Water Surface Elevations follow a Gaussian Distribution.
Calculate Standard Normal Variable, x
η
x=
σ
Cumulative Density Function, CDF or Φ ( x)
look up Φ ( x) from following tables
Note - only positive values of x are tabled but
Φ (- x) = 1 − Φ ( x) and POE ( x) = 1 − Φ ( x)
NOTE: Use standard normal distribution tables that follow to calculate CDF and POE
Standard Normal Dist. Tables
Standard Normal Dist. Tables
Standard Normal Dist. Tables
Distribution of H
Wave Heights follow a Rayleigh Distribution.
Probability Density Function, PDF
PDF = p(H ) = 2
1.0
CDF = area under curve
0.8
H
e
2
H rms
 H
−
 H rms



2
PDF
PDF
0.6
Cumulative Density Function, CDF
0.4
CDF = P(η ) = 1 − e
POE
0.2
 H
−
 H rms



2
0.0
0.0
0.5
1.0
1.5
H/Hrms
2.0
2.5
Probability of Exceedance, POE
POE = 1 − CDF
Distribution of H
Wave Heights follow a Rayleigh Distribution.
Approximate Relations for Rayleigh Dist.
H = H avg ≈
H s = H 1 ≈ 2 H rms
3
π
H rms
2
≈ 4σ ≈ 4 mo
H 1 ≈ 1.8 H rms
10
H
1
100
≈ 2.36 H rms
H max ≈ ln N H rms
NOTE: mo is the integral of (or area under) the energy spectrum S(f) vs f
Basic Sampling Theory
Lowest Frequency that can be Resolved, ∆f
1
∆f =
sampling duration
Highest Frequency that can be Resolved, f n
sampling frequency, f s
f n = Nyquist Frequency =
2