Chapter 5 Coastal Engineering: Wave Prediction

Chapter 5 Coastal Engineering: Wave Prediction#

Introduction: Wave Prediction
Simulation: Wave Prediction
Self-Assessment

1. Introduction#

Descriptive alt text for accessibility — Fig. 28 **Figure 5.6 **: Wind Wave Prediction.#

🌬️ Wind-Driven Waves: Characteristics, Formation, and Modeling#

1. What Are Wind-Driven Waves?#

Wind-driven waves are surface gravity waves generated by wind blowing across the ocean, sea, or large lake surfaces. They result from energy transfer from the atmosphere to the water surface.

2. Key Characteristics#

Property	Typical Range (Fully Developed Sea)
Height	0.3 m to 15 m
Period	3 s to 15 s
Wavelength	10 m to 200 m
Speed	10 km/h to 50+ km/h

Wave height: Vertical distance from trough to crest
Wave period: Time between successive crests
Wave length: Horizontal distance between crests
Wave direction: Aligned with prevailing wind

3. Theories of Energy Transfer#

3.1 Phillips Mechanism#

Initiates wave formation via random pressure fluctuations from turbulent wind.

3.2 Miles Mechanism#

Describes exponential wave growth due to shear instability.

3.3 Wave-Wave Interactions#

Redistribute energy across wave frequencies, forming swell.

3.4 Stokes Drift & Momentum Transfer#

Wind stress induces orbital motion and net energy propagation.

4. Factors Limiting Wave Growth#

Limiting Factor	Description
Wind Speed	Must exceed wave speed for energy transfer.
Fetch	Distance over which wind blows uninterrupted.
Duration	Time wind has been blowing over the fetch.
Water Depth	Shallow water alters wave speed and causes breaking.
Wave Saturation	Fully developed sea limits further growth.

5. Models for Estimating Wind Waves#

5.1 First-Generation Models#

Empirical (e.g., Sverdrup-Munk-Bretschneider).

5.2 Second-Generation Models#

Include parameterized nonlinear interactions (e.g., SWAMP).

5.3 Third-Generation Models#

Solve full spectral energy balance:
- WAM
- SWAN
- WAVEWATCH III

5.4 Deep Learning Models#

Neural networks (e.g., Swin Transformer) trained on reanalysis data.

6. Empirical Estimation Example#

For wind speed of 37 km/h over a 139 km fetch for 10 hours:

Wave Height ≈ 1.5 m
Wavelength ≈ 33.8 m
Period ≈ 5.7 s

7. USACE Wind Wave Prediction#

7.1 Sources#

SACEM1984: Empirical breaker indices.
USACE2002: Hindcast modeling and empirical methods.

7.2 Empirical Prediction#

Based on wind speed, fetch, and duration.
Uses dimensionless curves and breaker indices.

7.3 Numerical Modeling (WIS)#

Models: WAM, WAVEWATCH III
Driven by ERA5 wind fields.
Outputs: Hm0, Tm, αp

7.4 Modeling Features#

Wind adjusted to 10 m elevation.
Includes storm systems.
Hindcast: 1980–2021

7.5 Accuracy Metrics#

Parameter	Bias	RMSE	Correlation
Hm0	±0.01–0.23 m	0.26–0.53 m	>0.90
Tm	±0.7–1.6 s	1.0–2.6 s	~0.75
Wind Speed	±1 m/s	1.5–3 m/s	~0.80

8. Comparative Table: USACE vs Other Models#

Feature / Model Aspect	USACE (WIS, FUNWAVE, CMS-Wave)	SWAN (Delft)	WAVEWATCH III (NOAA)	WAM (ECMWF)
Model Type	Phase-averaged / resolving	Phase-averaged	Phase-averaged	Phase-averaged
Primary Use	Coastal engineering, wakes	Nearshore transformation	Global forecasting	Regional/global forecasting
Wind Input	ERA5, local	User-defined	NCEP	ECMWF
Wave Processes	Wind input, dissipation, interaction	Friction, breaking	Wind input, dissipation	Wind input, dissipation
Resolution	High (nearshore)	Flexible	~0.5° global	~0.25° global
Wave Types	Wind waves, wakes, tsunami	Wind waves, swell	Wind waves, swell	Wind waves, swell
Computational Demand	Moderate to high	Moderate	High	High
User Interface	SMS, Wiki, CLI	GUI + CLI	CLI	CLI
Strengths	Coastal detail, wake modeling	Easy setup	Global coverage	Spectral accuracy
Limitations	Complex setup, limited global	Deep water limits	Nearshore detail	Coastal tuning needed

9. References#

[] generation explains initial wave formation from turbulent wind pressure, predicting linear energy growth. [] generation models exponential wave growth via shear instability and critical layer theory. [] defines a spectral shape for fully developed seas, enabling wave forecasting from wind data. [] provides empirical and numerical methods for predicting wave height and period under varied wind conditions. [of Engineers, 1984] uses breaker indices and fetch-based curves to estimate design wave forces and wave growth. Together, these works span theory, spectrum modeling, and practical design tools for wind-driven wave prediction.

2. Simulation#

🌬️ Wind-Driven Wave Height Estimator — Empirical Comparison Tool#

This Jupyter Notebook provides a comparative visualization of estimated significant wave heights using multiple empirical models based on wind conditions. It empowers learners to explore how wind speed, fetch length, and wind duration influence the formation of surface gravity waves in open water bodies.

💡 Purpose#

The simulation helps answer questions like:

How do different empirical formulas estimate wave height?
How do wind fetch and duration influence wave growth?
How do rapid changes in wind speed affect offshore conditions?

⚙️ How It Works#

Defines wave height estimation methods:
- Simple empirical: proportional to ( V^2 )
- Duration-limited: includes time-limited growth
- Fetch-limited: includes distance over which wind acts
- NOAA rule of thumb: quick linear scale of wave height
Allows user to adjust wind speed (V), fetch (F), and duration (D)
Computes height estimates and displays comparison bar chart

🎛️ Inputs#

Parameter	Meaning
`V` (m/s)	Wind speed
`F` (m)	Fetch distance — length of water exposed to wind
`D` (hr)	Duration of wind blowing

Models Implemented#

The following empirical formulas are used to estimate wave height ( H ):

Simple Empirical:
$$ H = 0.025 \cdot V^2 $$ A basic quadratic fit for quick estimation.
Duration Limited:
$$ H = 0.0146 \cdot D^{5/7} \cdot V^{9/7} $$ Appropriate when wave growth is constrained by how long the wind blows.
Fetch Limited:
$$ H = 0.0163 \cdot \sqrt{F / 1000} \cdot V $$ Models cases where the fetch is the limiting factor for wave development.
NOAA Rule of Thumb:
$$ H = 0.3 \cdot V $$ A quick linear approximation often used in engineering estimates.

🔍 Units:
( V ) = Wind speed (m/s),
( F ) = Fetch length (m),
( D ) = Wind duration (hr)

🎛 Interactive Controls#

Use the sliders below to adjust:

Wind Speed (5–50 m/s)
Fetch Length (1,000–500,000 m)
Wind Duration (1–24 hours) Each setting dynamically updates the chart, helping you compare how wave height predictions vary by method.

📊 Output Visualization#

The bar chart shows estimated wave heights (in meters) from each model for the selected wind conditions. Values are labeled above each bar to enhance clarity and comparison.

🧭 How to Interpret#

Higher wind speeds → consistently higher wave heights
Fetch-limited vs duration-limited: shows whether spatial or temporal constraints dominate
Helps understand method sensitivity and idealized estimates
Useful for rapid planning, initial design screening, and educational purposes

This tool makes empirical wave theory accessible and visually comparable — helpful for coastal engineers, planners, and students.

3. Simulation#

🌬️ Wind-Generated Wave Height Estimator — Empirical model Comparison Tool#

🧠 What It Is#

An interactive visualization tool that compares multiple empirical formulas for estimating wind-driven wave height based on:

Wind speed (V)
Fetch distance (F)
Storm duration (D)

⚙️ How It Works#

Implements simple mathematical formulas to estimate wave height using different assumptions:
- Simple empirical: basic velocity-based relation
- Duration-limited: depends on how long the wind blows
- Fetch-limited: depends on horizontal water exposure
- NOAA estimate: rule-of-thumb formula
Uses ipywidgets to create sliders for input control
Plots results in a comparison bar chart using matplotlib

🎛️ Inputs#

Parameter	Meaning
`Wind Speed (V)`	Speed of wind (m/s)
`Fetch (F)`	Distance over which wind acts (m)
`Duration (D)`	Length of time wind blows (hours)

📊 Outputs#

Bar chart comparing wave height estimates (in meters) across methods
Value labels above each bar for quick readability
Chart title displays current input values for context

🧭 How to Interpret#

Wave height increases with wind speed and fetch
Different methods may yield significantly different estimates depending on input conditions
Helps highlight the sensitivity of wave growth to time and distance
Useful for preliminary wave modeling, design screening, and student exploration

This tool offers a clear, adjustable view of how waves grow under wind—across simplified empirical approaches.

4 Simulation#

USACE Wave Prediction Tool (CEM Dimensional Analysis)#

This notebook provides an interactive tool to estimate significant wave height (Hs) and peak wave period (Tp) using fetch- and duration-limited empirical formulas derived from the U.S. Army Corps of Engineers Coastal Engineering Manual (CEM).

Objective#

To calculate and visualize wave height development under different wind conditions by considering:

Wind speed (V) at 10 meters elevation
Fetch length (F) — the uninterrupted distance over which wind blows
Wind duration (t) — the time wind has been blowing at a constant speed and direction

Theoretical Framework#

The predictions are based on non-dimensional empirical relationships fitted to observed data from the Shore Protection Manual (SPM), expressed as:

Non-dimensional fetch:
$$ \hat{X} = \frac{g \cdot F}{V^2} $$
Non-dimensional duration:
$$ \hat{T} = \frac{g \cdot t}{V} $$

The empirical formulas for non-dimensional wave height ($( \hat{H} $)) and non-dimensional period ($( \hat{T}_p $)) are:

Ĥ = 0.283 * tanh(0.0125 * X̂)
T̂p = 1.2 * tanh(0.077 * X̂**0.25)

import numpy as np
import matplotlib.pyplot as plt
from ipywidgets import interact, FloatSlider, IntSlider

g = 9.81  # gravitational acceleration (m/s²)
def nondim_wave_height_fetch(X_hat):
    return 0.283 * np.tanh(0.0125 * X_hat)

def nondim_wave_period_fetch(X_hat):
    return 1.2 * np.tanh(0.077 * X_hat**0.25)

def nondim_wave_height_duration(T_hat):
    return 0.283 * np.tanh(0.0125 * T_hat)

def nondim_wave_period_duration(T_hat):
    return 1.2 * np.tanh(0.077 * T_hat**0.25)

# Constants
g = 9.81  # gravitational acceleration
A = 0.283
B = 0.0125

def fetch_limited_wave(V, F):
    X_hat = g * F / V**2
    H_hat = nondim_wave_height_fetch(X_hat)
    T_hat = nondim_wave_period_fetch(X_hat)
    Hs = H_hat * V**2 / g
    Tp = T_hat * V / g
    return Hs, Tp

def duration_limited_wave(V, t):
    T_hat = g * t / V
    H_hat = nondim_wave_height_duration(T_hat)
    T_hat_val = nondim_wave_period_duration(T_hat)
    Hs = H_hat * V**2 / g
    Tp = T_hat_val * V / g
    return Hs, Tp

def fetch_limited_wave_heightspm(V, F):
    """Fetch-limited wave height (SPM)"""
    gF_V2 = g * F / V**2
    return A * np.tanh(B * gF_V2) * V**2 / g

def duration_limited_wave_heightspm(V, t):
    """Duration-limited wave height (SPM)"""
    gt_V = g * t / V
    return A * np.tanh(B * gt_V) * V**2 / g

def plot_usace_wave(V=15, F_km=50, t_hr=3):
    F = F_km * 1000  # km → m
    t = t_hr * 3600  # hr → s
    Hf, Tf = fetch_limited_wave(V, F)
    Hd, Td = duration_limited_wave(V, t)
    Hs = min(Hf, Hd)
    Tp = Tf if Hf < Hd else Td
    print(f"🌬️ Wind Speed: {V:.1f} m/s")
    print(f"🌊 Fetch-Limited Hs: {Hf:.2f} m, Tp: {Tf:.2f} s")
    print(f"⏱️ Duration-Limited Hs: {Hd:.2f} m, Tp: {Td:.2f} s")
    print(f"✅ Selected Hs: {Hs:.2f} m, Tp: {Tp:.2f} s")
    # Plotting
    fetch_range = np.linspace(1, 200, 300)
    Hs_fetch = [fetch_limited_wave(V, f * 1000)[0] for f in fetch_range]
    plt.figure(figsize=(10, 5))
    plt.plot(fetch_range, Hs_fetch, label="Fetch-Limited Hs", color='navy')
    plt.axhline(Hd, color='orange', linestyle='--', label=f'Duration-Limited Hs = {Hd:.2f} m')
    plt.axhline(Hs, color='green', linestyle='-.', label=f'Selected Hs = {Hs:.2f} m')
    plt.xlabel("Fetch (km)")
    plt.ylabel("Significant Wave Height Hs (m)")
    plt.title("USACE Wave Height Prediction (CEM Dimensional Analysis)")
    plt.grid(True, linestyle='--', alpha=0.6)
    plt.legend()
    plt.tight_layout()
    plt.show()
    spm_FetchlimitedHs=fetch_limited_wave_heightspm(V, F)
    spm_DurationlimitedHs=duration_limited_wave_heightspm(V, F)
    
    print("spm_FetchlimitedHs=",spm_FetchlimitedHs)
    print("spm_DurationlimitedHs=",spm_DurationlimitedHs)
interact(
    plot_usace_wave,
    V=FloatSlider(value=15, min=5, max=100, step=1, description="Wind Speed (m/s)"),
    F_km=FloatSlider(value=50, min=1, max=200, step=1, description="Fetch (km)"),
    t_hr=IntSlider(value=3, min=0, max=24, step=1, description="Duration (hr)")
)   

<function __main__.plot_usace_wave(V=15, F_km=50, t_hr=3)>

5 Simulation#

Interactive Wave Spectrum Comparison: PM vs JONSWAP#

This Jupyter Notebook demonstrates the fundamental differences between the Pierson–Moskowitz (PM) and JONSWAP wave spectra—two widely used models to describe the energy distribution of sea surface waves generated by wind.

Learning Objectives#

Understand the theoretical foundations of the PM and JONSWAP spectra
Explore how wave energy is distributed over frequency for varying wind speeds
Examine the effect of spectral peakedness through the JONSWAP γ (gamma) parameter
Interpret real-time spectral plots using interactive sliders

Theoretical Background#

Pierson–Moskowitz (PM) Spectrum#

The PM spectrum models a fully developed sea where waves have grown under consistent wind over time and space. It assumes the sea is in equilibrium and is controlled primarily by wind speed ( U_{10} ) at 10 meters above the surface.

The spectral shape is defined by:

\[ S_{PM}(f) = \alpha \frac{g^2}{(2\pi)^4} f^{-5} \exp\left[-1.25 \left( \frac{f_p}{f} \right)^4 \right] \]

🌐 JONSWAP Spectrum#

The JONSWAP spectrum extends PM by introducing a peak enhancement factor ( \gamma ) to account for young, developing seas. It narrows the spectral peak and adds asymmetry to the distribution.

The expression is:

\[ S_{JONSWAP}(f) = S_{PM}(f) \cdot \gamma^{\exp \left[ - \frac{(f - f_p)^2}{2 \sigma^2 f_p^2} \right]} \]

Where:

( f_p ): peak frequency, inversely related to wind speed
( \gamma ): spectral peakedness (default ≈ 3.3)
( \sigma ): frequency-dependent bandwidth (0.07 if ( f < f_p ), else 0.09)

Interactive Controls#

Use the sliders below to change:

Wind speed ( U_{10} ) (in m/s): influences peak frequency and total energy
Gamma (γ): controls the sharpness and intensity of the JONSWAP peak

The simulation will regenerate and plot:

The PM spectrum (solid line)
The JONSWAP spectrum (dashed line)

All curves are plotted against frequency in Hz with spectral energy density in m²/Hz.

Output Interpretation#

Higher wind speeds shift the peak to lower frequencies (longer waves)
Larger γ values sharpen the JONSWAP spectrum, representing more “peaky” wave energy
The difference between spectra highlights stages of sea development (from growing to fully developed)

💡 Use this tool to explore real-world sea state dynamics, inform spectral wave modeling, or just to build intuition about wave growth under varying wind conditions.

# 🔹 Cell 1: Imports
import numpy as np
import matplotlib.pyplot as plt
from ipywidgets import interact, FloatSlider, FloatRangeSlider
import warnings
warnings.filterwarnings('ignore')
# 🔹 Cell 2: Spectrum Functions
g = 9.81
alpha = 0.0081
sigma_low = 0.07
sigma_high = 0.09

# Pierson–Moskowitz Spectrum
def pm_spectrum(f, U10):
    fp = g / (2 * np.pi * 0.83 * U10)
    return (alpha * g**2 / (2 * np.pi)**4) * f**(-5) * np.exp(-1.25 * (fp / f)**4)

# JONSWAP Spectrum
def jonswap_spectrum(f, U10, gamma):
    fp = g / (2 * np.pi * 0.83 * U10)
    S_pm = pm_spectrum(f, U10)
    sigma = np.where(f <= fp, sigma_low, sigma_high)
    exponent = -((f - fp)**2) / (2 * sigma**2 * fp**2)
    peak_enhancement = gamma ** np.exp(exponent)
    return S_pm * peak_enhancement
# 🔹 Cell 3: Plot and Calculate Hs from Both Spectra
def plot_and_estimate(U10=15.0, gamma=3.3, freq_range=(0.05, 1.5)):
    f_min, f_max = freq_range
    f = np.linspace(f_min, f_max, 500)

    # Compute spectra
    S_pm = pm_spectrum(f, U10)
    S_jonswap = jonswap_spectrum(f, U10, gamma)

    # Integrate spectra to get Hs
    m0_pm = np.trapz(S_pm, f)
    Hs_pm = 4 * np.sqrt(m0_pm)

    m0_js = np.trapz(S_jonswap, f)
    Hs_js = 4 * np.sqrt(m0_js)

    # Plot spectra
    plt.figure(figsize=(10, 6))
    plt.plot(f, S_pm, label=f"PM Spectrum (Hs = {Hs_pm:.2f} m)", linewidth=2)
    plt.plot(f, S_jonswap, '--', label=f"JONSWAP Spectrum (γ = {gamma:.1f}, Hs = {Hs_js:.2f} m)", linewidth=2)
    plt.xlabel("Frequency (Hz)")
    plt.ylabel("Spectral Energy Density $S(f)$ (m²/Hz)")
    plt.title(f"Wave Spectra Comparison at U10 = {U10:.1f} m/s")
    plt.legend()
    plt.grid(True, linestyle='--', alpha=0.6)
    plt.tight_layout()
    plt.show()

# 🔹 Cell 4: Interactive Sliders
interact(
    plot_and_estimate,
    U10=FloatSlider(value=15, min=5, max=30, step=1, description="Wind Speed (m/s)"),
    gamma=FloatSlider(value=3.3, min=1.0, max=7.0, step=0.1, description="Gamma (γ)"),
    freq_range=FloatRangeSlider(value=(0.05, 1.5), min=0.01, max=2.0, step=0.01, description="Freq Range (Hz)")
)

<function __main__.plot_and_estimate(U10=15.0, gamma=3.3, freq_range=(0.05, 1.5))>

#Defining the Spectrum based on observation data and then estimating the significante wave height

6 Simulation#

Interactive Estimation of Wave Spectra and Significant Wave Height (Hs)#

This notebook compares two fundamental ocean wave spectral models—Pierson–Moskowitz (PM) and JONSWAP—to investigate how wind speed, frequency range, and spectral peakedness affect wave energy distributions and significant wave height estimations.

–

Purpose#

Explore how:

Wind speed at 10 meters height ((U_{10}))
JONSWAP peak enhancement factor (( \gamma ))
Frequency range of interest influence wave spectra shape and total wave energy.

Models Implemented#

PM Spectrum: Represents a fully developed sea state assuming equilibrium under steady wind. $$ S_{PM}(f) = \alpha \cdot \frac{g^2}{(2\pi)^4} f^{-5} \exp\left[-1.25 \left( \frac{f_p}{f} \right)^4 \right] $$
JONSWAP Spectrum: Enhances the PM spectrum to represent growing seas using a peakedness factor ( \gamma ).

\[ S_{JONSWAP}(f) = S_{PM}(f) \cdot \gamma^{\exp \left( - \frac{(f - f_p)^2}{2 \sigma^2 f_p^2} \right)} \]

Interactive Inputs#

Use the sliders to control:

Wind Speed (m/s): Adjusts the peak frequency and energy content
Gamma (γ): Defines the sharpness of the spectral peak (JONSWAP only)
Frequency Range (Hz): Allows you to focus on a specific slice of the spectrum

Outputs#

A line plot comparing the PM and JONSWAP spectra across frequencies
Real-time calculation and display of:
- $( H_s $) from PM spectrum: $( H_s = 4 \sqrt{m_0} $)
- $( H_s $) from JONSWAP spectrum The area under each curve is numerically integrated to yield the zeroth moment ( m_0 ), which is then used to estimate the significant wave height.

📘 Insight: A higher gamma value leads to a narrower, more peaked spectrum—often representing developing wind seas. The PM spectrum is smoother and broader, idealized for fully developed conditions.

import pandas as pd
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import requests

alpha = 0.0081
sigma_low = 0.07
sigma_high = 0.09

# Example: Station 41001 (East of Cape Hatteras)
station_id = '41001'
url = f'https://www.ndbc.noaa.gov/data/realtime2/{station_id}.txt'

# Read the data, skipping header rows
df = pd.read_csv(url, delim_whitespace=True, skiprows=[1], na_values='MM')

# Rename columns for clarity
df.columns = ['YY', 'MM', 'DD', 'hh', 'mm', 'WDIR', 'WSPD', 'GST', 'WVHT', 'DPD',
              'APD', 'MWD', 'PRES', 'ATMP', 'WTMP', 'DEWP', 'VIS', 'PTDY','TIDE'][:len(df.columns)]

df.rename(columns={'YY': 'year', 'MM': 'month', 'DD': 'day', 'hh':'hour', 'mm':'minute'}, inplace=True)
# Combine date columns into a timestamp
#df['timestamp'] = pd.to_datetime(df[['YY', 'MM', 'DD', 'hh', 'mm']])
df['timestamp'] = pd.to_datetime(df[['year', 'month', 'day', 'hour', 'minute']])
df.set_index('timestamp', inplace=True)

# Preview wave height and dominant period
print(df[['WVHT', 'DPD']].dropna().head())
# Use most recent record
latest = df[['WVHT', 'DPD']].dropna().iloc[0]
Hs_obs = latest['WVHT']
Tp_obs = latest['DPD']
print(f"Observed Hs: {Hs_obs:.2f} m, Tp: {Tp_obs:.2f} s")

def pm_spectrum(f, U10):
    fp = g / (2 * np.pi * 0.83 * U10)
    return (alpha * g**2 / (2 * np.pi)**4) * f**-5 * np.exp(-1.25 * (fp / f)**4)

def jonswap_spectrum(f, U10, gamma=3.3):
    S_pm = pm_spectrum(f, U10)
    fp = g / (2 * np.pi * 0.83 * U10)
    sigma = np.where(f <= fp, sigma_low, sigma_high)
    enhancement = gamma ** np.exp(-((f - fp)**2) / (2 * sigma**2 * fp**2))
    return S_pm * enhancement

# Estimate wind speed from PM inverse formula
g = 9.81
U10_pm = (Tp_obs * g) / (2 * np.pi * 0.83)

# Estimate peak frequency
fp = 1 / Tp_obs

# Print estimated parameters
print(f"Estimated from NOAA data:")
print(f"  Hs = {Hs_obs:.2f} m")
print(f"  Tp = {Tp_obs:.2f} s")
print(f"  fp = {fp:.3f} Hz")
print(f"  Estimated U10 (PM) = {U10_pm:.2f} m/s")

# Frequency range
f = np.linspace(0.05, 1.5, 500)
# Compute spectra
S_pm = pm_spectrum(f, U10_pm)
S_jonswap = jonswap_spectrum(f, U10_pm, gamma=3.3)
# Integrate to get Hs from spectra
m0_pm = np.trapz(S_pm, f)
m0_js = np.trapz(S_jonswap, f)
Hs_pm = 4 * np.sqrt(m0_pm)
Hs_js = 4 * np.sqrt(m0_js)
# Plot
plt.figure(figsize=(10, 6))
plt.plot(f, S_pm, label=f"PM (Hs ≈ {Hs_pm:.2f} m)", lw=2)
plt.plot(f, S_jonswap, '--', label=f"JONSWAP γ=3.3 (Hs ≈ {Hs_js:.2f} m)", lw=2)
plt.axhline(0, color='black', lw=0.5)
plt.xlabel("Frequency (Hz)")
plt.ylabel("Spectral Density $S(f)$ (m²/Hz)")
plt.title(f"{station_id} Buoy Spectral Comparison\nObserved Hs: {Hs_obs:.2f} m, Tp: {Tp_obs:.2f} s, U10 pm: {U10_pm:.2f} m/s")
plt.legend()
plt.grid(True, linestyle='--', alpha=0.6)
plt.tight_layout()
plt.show()

                     WVHT  DPD
timestamp                     
2025-07-21 22:50:00   1.5  7.0
2025-07-21 22:20:00   1.3  7.0
2025-07-21 21:50:00   1.4  7.0
2025-07-21 21:20:00   1.4  7.0
2025-07-21 20:50:00   1.4  7.0
Observed Hs: 1.50 m, Tp: 7.00 s
Estimated from NOAA data:
  Hs = 1.50 m
  Tp = 7.00 s
  fp = 0.143 Hz
  Estimated U10 (PM) = 13.17 m/s

../_images/424bef48e94aed22011e01225f899ca83123c8818ab7d14817d6b5c7ab5ad752.png

7. Self-Assessment#

🌊 Wave Spectrum Simulation — Quiz Questions#

🧠 Quiz 1: Gravitational Acceleration#

Question: What is the value of gravitational acceleration $ g $ used in the spectrum functions?

8.91 m/s²
10.0 m/s²
9.81 m/s²
9.8 m/s²

🧠 Quiz 2: Purpose of PM Spectrum#

Question: What is the purpose of the Pierson–Moskowitz (PM) spectrum function?

To model wave spectra for fully developed seas
To calculate wind speed at different altitudes
To simulate the effect of gamma on wave height
To determine the frequency range of ocean waves

🧠 Quiz 3: Role of $ \alpha $ in PM Spectrum#

Question: In the PM spectrum formula, what does $ \alpha $ represent?

A constant related to wave energy
The gravitational acceleration
The wind speed at 10 meters above sea level
The frequency of ocean waves

🧠 Quiz 4: Role of $ \sigma $ in JONSWAP Spectrum#

Question: What is the role of $ \sigma $ in the JONSWAP spectrum?

It determines the width of the frequency peak
It represents the gravitational acceleration
It scales the wave energy density
It defines the wind speed at 10 meters above sea level

🧠 Quiz 5: Enhancement Factor $ \gamma $#

Question: What is the enhancement factor $ \gamma $ used for in the JONSWAP spectrum?

To amplify the energy near the peak frequency
To calculate the gravitational acceleration
To determine the wind speed at 10 meters
To scale the frequency range

🧠 Quiz 6: Significant Wave Height $ H_s $#

Question: How is the significant wave height $ H_s $ calculated from the wave spectrum?

$ H_s = 4 \sqrt{m_0} $
$ H_s = 2 \sqrt{m_0} $
$ H_s = m_0 / 4 $
$ H_s = m_0 \times 4 $

🧠 Quiz 7: Zeroth Moment $ m_0 $#

Question: What does the zeroth moment $ m_0 $ of the wave spectrum represent?

The total energy of the wave spectrum
The peak frequency of the wave spectrum
The width of the frequency peak
The enhancement factor of the wave spectrum

🧠 Quiz 8: Default Wind Speed $ U_{10} $#

Question: What is the default value of wind speed $ U_{10} $ in the interactive plot?

15 m/s
10 m/s
20 m/s
5 m/s

🧠 Quiz 9: Purpose of `interact` Function#

Question: What is the purpose of the interact function in the code?

To create interactive sliders for user input
To calculate the wave spectrum
To plot the wave spectra
To integrate the wave spectrum

❓ Question 1: Gravitational Acceleration#

What is the value of gravitational acceleration ( g ) used in the code?

10.0 m/s²
8.91 m/s²
9.81 m/s²
9.8 m/s²

❓ Question 2: Purpose of `nondim_wave_height_fetch(X_hat)`#

What does the function nondim_wave_height_fetch(X_hat) calculate?

Non-dimensional wave height based on fetch
Non-dimensional wave period based on fetch
Dimensional wave height based on fetch
Dimensional wave period based on fetch

❓ Question 3: Formula for ( X_{\hat} )#

What is the formula used to calculate ( X_{\hat} ) in the fetch-limited wave function?

( X_{\hat} = \frac{g \cdot F}{V^2} )
( X_{\hat} = \frac{F}{g \cdot V^2} )
( X_{\hat} = \frac{V^2}{g \cdot F} )
( X_{\hat} = \frac{g \cdot V^2}{F} )

❓ Question 4: Output of `duration_limited_wave(V, t)`#

What does the function duration_limited_wave(V, t) return?

Significant wave height and wave period based on duration
Significant wave height and wave period based on fetch
Non-dimensional wave height and wave period based on duration
Non-dimensional wave height and wave period based on fetch

❓ Question 5: Role of Constants A and B#

What do the constants A = 0.283 and B = 0.0125 represent in the SPM formulas?

Empirical coefficients for wave height scaling
Wind speed and fetch conversion factors
Gravity and time scaling parameters
Plotting thresholds for wave height

❓ Question 6: Purpose of `plot_usace_wave`#

What is the purpose of the plot_usace_wave function?

To plot fetch-limited and duration-limited wave heights and select the final wave height
To calculate only fetch-limited wave height and period
To calculate only duration-limited wave height and period
To interactively adjust wind speed without plotting

❓ Question 7: Units Conversion in `plot_usace_wave`#

How are fetch and duration converted before calculations?

Fetch: km → m, Duration: hr → s
Fetch: m → km, Duration: s → hr
Fetch: km → s, Duration: hr → m
No conversion is applied

❓ Question 8: Final Selection Logic#

How is the final wave height ( H_s ) selected in the plot_usace_wave function?

The minimum of fetch-limited and duration-limited ( H_s )
The maximum of fetch-limited and duration-limited ( H_s )
The average of both ( H_s ) values
Always the fetch-limited ( H_s )

❓ Question 9: Interactive Controls#

Which parameters are controlled by sliders in the interactive plot?

Wind speed, fetch (km), and duration (hr)
Wave height, wave period, and gravity
Fetch, duration, and plotting resolution
Only wind speed

❓ Question 10: SPM Wave Height Functions#

What do fetch_limited_wave_heightspm and duration_limited_wave_heightspm compute?

Dimensional wave height using SPM formulas
Non-dimensional wave period using SPM
Wind speed and fetch ratios
Interactive plot labels

Chapter 5 Coastal Engineering: Wave Prediction

Contents

Chapter 5 Coastal Engineering: Wave Prediction#

1. Introduction#

🌬️ Wind-Driven Waves: Characteristics, Formation, and Modeling#

1. What Are Wind-Driven Waves?#

2. Key Characteristics#

3. Theories of Energy Transfer#

3.1 Phillips Mechanism#

3.2 Miles Mechanism#

3.3 Wave-Wave Interactions#

3.4 Stokes Drift & Momentum Transfer#

4. Factors Limiting Wave Growth#

5. Models for Estimating Wind Waves#

5.1 First-Generation Models#

5.2 Second-Generation Models#

5.3 Third-Generation Models#

5.4 Deep Learning Models#

6. Empirical Estimation Example#

7. USACE Wind Wave Prediction#

7.1 Sources#

7.2 Empirical Prediction#

7.3 Numerical Modeling (WIS)#

7.4 Modeling Features#

7.5 Accuracy Metrics#

8. Comparative Table: USACE vs Other Models#

9. References#

2. Simulation#

🌬️ Wind-Driven Wave Height Estimator — Empirical Comparison Tool#

💡 Purpose#

⚙️ How It Works#

🎛️ Inputs#

Models Implemented#

🎛 Interactive Controls#

📊 Output Visualization#

🧭 How to Interpret#

3. Simulation#

🌬️ Wind-Generated Wave Height Estimator — Empirical model Comparison Tool#

🧠 What It Is#

⚙️ How It Works#

🎛️ Inputs#

📊 Outputs#

🧭 How to Interpret#

4 Simulation#

USACE Wave Prediction Tool (CEM Dimensional Analysis)#

Objective#

Theoretical Framework#

5 Simulation#

Interactive Wave Spectrum Comparison: PM vs JONSWAP#

Learning Objectives#

Theoretical Background#

Pierson–Moskowitz (PM) Spectrum#

🌐 JONSWAP Spectrum#

Interactive Controls#

Output Interpretation#

6 Simulation#

Interactive Estimation of Wave Spectra and Significant Wave Height (Hs)#

Purpose#

Models Implemented#

Interactive Inputs#

Outputs#

7. Self-Assessment#

🌊 Wave Spectrum Simulation — Quiz Questions#

🧠 Quiz 1: Gravitational Acceleration#

🧠 Quiz 2: Purpose of PM Spectrum#

🧠 Quiz 3: Role of \( \alpha \) in PM Spectrum#

🧠 Quiz 4: Role of \( \sigma \) in JONSWAP Spectrum#

🧠 Quiz 5: Enhancement Factor \( \gamma \)#

🧠 Quiz 6: Significant Wave Height \( H_s \)#

🧠 Quiz 7: Zeroth Moment \( m_0 \)#

🧠 Quiz 8: Default Wind Speed \( U_{10} \)#

🧠 Quiz 9: Purpose of interact Function#

❓ Question 1: Gravitational Acceleration#

❓ Question 2: Purpose of nondim_wave_height_fetch(X_hat)#

❓ Question 3: Formula for ( X_{\hat} )#

❓ Question 4: Output of duration_limited_wave(V, t)#

❓ Question 5: Role of Constants A and B#

❓ Question 6: Purpose of plot_usace_wave#

❓ Question 7: Units Conversion in plot_usace_wave#

❓ Question 8: Final Selection Logic#

🧠 Quiz 9: Purpose of `interact` Function#

❓ Question 2: Purpose of `nondim_wave_height_fetch(X_hat)`#

❓ Question 4: Output of `duration_limited_wave(V, t)`#

❓ Question 6: Purpose of `plot_usace_wave`#

❓ Question 7: Units Conversion in `plot_usace_wave`#