What if just one wave is enough to cause disaster? (part 1)
This is the first post in a series where I review key papers on extreme coastal flooding — whether caused by rogue waves, storm surges, or sea level anomalies. And what better place to start than with a study that asks a blunt but crucial question: what happens when a single wave does all the damage?
The paper: what’s it about?
The study is titled Distribution of individual wave overtopping volumes at rubble mound seawalls by Koosheh et al. (2022). The authors set out to understand how wave-by-wave overtopping behaves on rubble mound structures and, more importantly, how to estimate the maximum individual overtopping volume, \( V_{\text{max}} \).
Most guidelines — like EurOtop or the Coastal Engineering Manual — focus on average overtopping discharge \( q \). But here’s the problem: coastal defences don’t usually fail due to average conditions. They fail when one single wave exceeds everything we expected. That’s the kind of event this paper zooms in on.
Experimental setup: small scale, high precision
The authors conducted 135 tests in a 22.5-metre-long wave flume at Griffith University. The models were two-layer rubble mound seawalls with impermeable cores and slopes of 1:1.5 and 1:2. All tests were done under deep water conditions (i.e., \( h/H_{m0} > 3 \)) and with non-breaking waves.
Wave generation and sensors
- Wavemaker: piston-type, generating irregular JONSWAP spectra (\( \gamma = 3.3 \)).
- Incident wave detection: 3 capacitance wave gauges near the toe (WG1–WG3), using the Mansard & Funke method to separate incident and reflected waves.
- Overtopping detection: 2 wave gauges on the crest (WG4 at the seaward edge and WG5 at mid-crest). They used a threshold-crossing algorithm with a delay window to identify actual overtopping and ignore splashes or false positives.
- Volume measurement: A sealed acrylic collection box behind the crest received overtopped water via a chute. Inside, 2 more wave gauges (WG6 and WG7) measured water level changes. A stilling wall helped stabilise the readings. Sudden jumps in water level = individual overtopping events.
This allowed them to calculate how many waves overtopped \( (N_{\text{ow}}) \) and the volume of each one — not just the average.
So... what’s wrong with using Weibull?
The Weibull distribution is widely used to model overtopping volumes. But Koosheh et al. point out several issues:
- Small volumes skew the shape of the distribution.
- Trimming the dataset (e.g. top 10% or 30%) is arbitrary.
- Most importantly, Weibull tends to underestimate the actual \( V_{\text{max}} \).
To address this, they tested a weighted method where every volume is considered — but larger ones carry more statistical weight. It's more honest with the data and more robust.
The highlight: a simple formula that beats them all
After comparing Weibull, Exponential, and empirical fits, the authors proposed a clean, elegant formula that came out on top:
\( V^*_{\text{max}} = 0.0125 \cdot N_{\text{ow}}^{0.23} \cdot (q^*)^{0.28} \)
where:
- \( V^*_{\text{max}} \): dimensionless maximum overtopping volume,
- \( N_{\text{ow}} \): number of overtopping waves,
- \( q^* = \dfrac{q}{\sqrt{g H_{m0}^3}} \): dimensionless mean overtopping discharge,
- \( q \): mean overtopping discharge [m³/s/m],
- \( H_{m0} \): significant wave height [m],
- \( g \): gravitational acceleration [≈9.81 m/s²].
Despite its simplicity, this formula had the lowest bias and scatter across all test cases. Sometimes simple really is better — especially when it’s grounded in real data.
Why does it matter?
Because underestimating \( V_{\text{max}} \) could lead to critical failures in coastal infrastructure. And overestimating it could drive up costs unnecessarily.
This study provides a well-tested, easy-to-use tool that improves on traditional models — particularly for impermeable rubble mound seawalls in deep, non-breaking wave conditions.
What are the limitations?
As always, this isn’t a magic bullet for all situations. The formula is currently valid only for:
- Impermeable rubble mound structures
- Slopes of 1:1.5 and 1:2
- Deep water conditions: \( \dfrac{h}{H_{m0}} > 3 \)
- Non-breaking waves
It hasn’t yet been tested for:
- Structures with a crown wall,
- Gentler slopes (e.g. \( \cot \alpha > 2 \)),
- Shallower water conditions or wave breaking cases.
One last interesting point: even if two storms have identical wave spectra, the actual sequence of waves can lead to different overtopping extremes. That “realisation variability” is still an open field for future research.
Final thoughts
At the end of the day, what this paper really shows is that you don’t need an overly complicated model to get solid predictions. With good data, smart thinking, and a bit of humility, the authors found that a simple equation — one you can write on a napkin — actually works better than the traditional ones. And that’s a big deal.
They built it from the ground up: real tests, real waves, real volumes. No assumptions pulled from thin air. They also remind us that in coastal design, it’s not about the average wave — it’s about that one wave that pushes your structure past its limit. That’s the one that matters.
So yeah, one wave can still mess everything up. But thanks to this work, we’ve got a much better way of spotting it before it hits.
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