Understanding Random Waves in Breakwater Hydraulic Modelling

 

From Pierson–Moskowitz to JONSWAP — A Practical Explanation for Young Engineers

When engineers design coastal structures such as breakwaters, seawalls, and harbour protection systems, they rarely test them against a single perfect wave.

Real oceans do not produce regular waves.

Instead, they generate random wave fields composed of thousands of interacting wave components.

Because of this, modern hydraulic modelling uses wave spectra rather than individual waves.

A classic experimental study on this topic is the work of Kloppman and Van der Meer, who investigated random wave behaviour in front of reflective coastal structures using laboratory wave flumes. 

Their research shows how wave spectra change near structures and why engineers must carefully measure incident and reflected waves when testing breakwaters.

This article explains the core ideas behind that research in a practical way.

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1 The difference between regular waves and random waves

In basic wave theory courses, we usually begin with a simple wave:

$$𝜂(𝑥,𝑡)=𝑎\cos (𝑘𝑥-𝜔𝑡)$$

Where:

• $𝑎$ = wave amplitude

• $𝑘$ = wave number

• $𝜔$ = angular frequency

• $𝑥$ = distance

• $𝑡$ = time

This represents a perfect sinusoidal wave.

However, the ocean is not composed of a single sine wave.

Instead, the sea surface is better described as a superposition of many waves with different frequencies and amplitudes.

Mathematically,

$$$$𝜂(𝑥,𝑡)=∑𝑖=1𝑁𝑎𝑖cos⁡(𝑘𝑖𝑥−𝜔𝑖𝑡+𝜙𝑖)

This means the water surface is the sum of many components.

Instead of tracking every wave individually, engineers describe the wave field using spectral energy distribution.

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2 What is a wave spectrum?

A wave spectrum describes how wave energy is distributed across frequencies.

The spectrum function is written as

$$$$𝑆(𝑓)

Where

• $𝑓$ = frequency

• $𝑆(𝑓)$ = wave energy density at that frequency

The total wave variance becomes

$${𝑚}_0={\int }_0^{\mathrm{\infty }}𝑆(𝑓)𝑑𝑓$$

The significant wave height is related to this variance:

$${𝐻}_{𝑠}=4\sqrt{{𝑚}_0}$$

This is the fundamental relationship used in both numerical wave models and hydraulic laboratories.

The experimental work of Kloppman and Van der Meer used this spectral framework to analyze wave fields in front of reflective structures.

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3 The Pierson–Moskowitz spectrum

The Pierson–Moskowitz spectrum represents a fully developed sea, meaning the wind has blown long enough for waves to reach equilibrium.

It is defined as

$${𝑆}_{𝑃𝑀}(𝑓)=\frac{𝛼{𝑔}^2}{(2𝜋{)}^4{𝑓}^5}\exp (-𝛽{\left(\frac{{𝑓}_{𝑝}}{𝑓}\right)}^4)$$

Typical constants:

$$𝛼=0.0081$$
$$𝛽=0.74$$

Where:

• ${𝑓}_{𝑝}$ = peak frequency

• $𝑔$ = gravity

This spectrum produces a smooth energy curve.

Physically this means

• energy spreads over a wider range of frequencies

• waves are less concentrated around the peak.

This behaviour was also observed in laboratory measurements where broad spectra damp standing-wave oscillations near reflective structures.

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4 The JONSWAP spectrum

The JONSWAP spectrum modifies the Pierson–Moskowitz spectrum to represent fetch-limited seas, such as the North Sea or Arabian Gulf.

It introduces a peak enhancement factor.

The spectrum becomes

$${𝑆}_{𝐽}(𝑓)={𝑆}_{𝑃𝑀}(𝑓){𝛾}^{\exp [-\frac{(𝑓-{𝑓}_{𝑝}{)}^2}{2{𝜎}^2{𝑓}_{𝑝}^2}]}$$

Where

$$𝛾\approx 3.3$$

This parameter sharpens the spectral peak.

Typical values

$$𝜎=\{\begin{array}{ll}0.07& 𝑓\le {𝑓}_{𝑝}\\ 0.09& 𝑓>{𝑓}_{𝑝}\end{array}$$

Physically this means:

• wave energy is concentrated around the peak frequency

• wave groups become stronger

• wave heights fluctuate more intensely.

The hydraulic experiments showed that JONSWAP spectra produce clearer standing wave patterns near reflective structures than Pierson–Moskowitz spectra.

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5 Why random waves create standing patterns near breakwaters

When waves hit a reflective structure, such as a vertical wall or breakwater, they reflect back toward the sea.

The incident and reflected waves interact.

Linear theory shows the total wave elevation becomes

$$𝜂(𝑥,𝑡)=𝑎\cos (𝑘𝑥-𝜔𝑡)+𝑎𝑅\cos (𝑘𝑥+𝜔𝑡+𝜙)$$

Where

• $𝑅$ = reflection coefficient

This produces nodes and antinodes, forming a standing wave pattern.

Laboratory experiments measured these variations using wave gauges placed along the flume.

The measurements confirmed that

• nodes occur where destructive interference happens

• antinodes occur where wave energy concentrates.

The experiments also showed that the standing pattern is strongest near the structure and gradually fades offshore.

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6 Hydraulic modelling experiment

The study performed tests in a glass-walled wave flume approximately

• 45 m long

• 1 m wide

A piston-type wave generator produced random waves.

More than 30 wave gauges were used to measure the spatial variation of the wave field.

Two reflective structures were tested:

1. vertical wall

2. rubble mound breakwater

Measurements showed

• wave spectra change significantly near reflective structures

• nodes and antinodes form in the significant wave height

• the distance between these oscillations increases offshore.

These results match predictions from linear wave interference theory.

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7 Why this matters for breakwater design

Understanding spectral waves is critical because

1️⃣ Breakwaters experience random waves, not regular waves.

2️⃣ Wave reflection can amplify local wave heights.

3️⃣ Standing wave patterns affect:

• armour stability

• toe scour

• overtopping behaviour.

Hydraulic modelling therefore uses random wave spectra such as JONSWAP or Pierson–Moskowitz to realistically reproduce ocean conditions.

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8 Key takeaway for young coastal engineers

If you remember only three ideas, remember these:

1. Real seas are random.
Engineers must model waves using spectra.

2. JONSWAP and Pierson–Moskowitz describe how wave energy is distributed.

3. When waves meet structures, reflection creates standing wave patterns that strongly influence hydraulic performance.

Understanding these ideas is the first step toward mastering breakwater hydraulic modelling.

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Conclusion

Hydraulic modelling remains one of the most powerful tools in coastal engineering.

By combining

• spectral wave theory

• laboratory wave generation

• precise measurements of reflection and interference

engineers can understand how real seas interact with coastal structures.

The experiments discussed here demonstrate that even complex random wave fields can be interpreted using relatively simple theoretical principles.

This combination of theory and physical modelling continues to guide the design of modern breakwaters around the world.

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References

Kloppman, G., and Van der Meer, J. W.
Random Wave Measurements in Front of Reflective Structures.
Journal of Waterway, Port, Coastal, and Ocean Engineering.