How the Sea Tops Over: An Informal Journey Through De Ridder et al. (2024)

Introduction with Salt Spray

There’s something mesmerising about watching the sea overflow a coastal defence. It’s not just dramatic—it’s complex. In coastal engineering, predicting how much overtopping will occur isn’t just an equation: it’s a necessity. Get it wrong, and you either waste millions or end up with a structure that behaves like a sieve. And this becomes much murkier in shallow water, where waves don’t behave like they do in textbooks. This is where De Ridder et al. (2024) really makes waves.

This paper isn’t just another dataset—it’s a blend of experiments, new insights, practical formulas, and some uncomfortable truths about how little we understand the waves that matter most. Let’s dive in.

The Problem with the Usual Formulas

For years, we’ve relied on classic mean overtopping formulas like:

q∗=qgHm03q^* = \frac{q}{\sqrt{g H_{m0}^3}}

Followed by exponential adjustments:

q∗=a⋅exp⁡(−bRc/Hm0)q^* = a \cdot \exp(-b R_c/H_{m0})

But what happens when the waves are already breaking before they reach the structure? When the water’s so shallow that IG waves (infragravity) steal the spotlight? Classic formulas start falling apart. Instead of just tweaking coefficients, De Ridder and co. took a more honest route: back to the lab, with fresh eyes.

Also, many of the existing formulations suffer from what we might call overfitting syndrome. They were born in flumes, over small datasets, fine-tuned to specific lab conditions. They perform like acrobats on that narrow beam but fall flat when applied in more chaotic real-world settings. Some even show errors two to three times higher than the new models proposed here.

This is not just a theoretical detail—if your model overfits, it might give you perfect predictions in the lab, but dangerously wrong ones on the coast.

Lab-Grade Waves

They ran 104 tests using straight-sloped rubble mound breakwaters (1:20, 1:50, and 1:100), with wave flume experiments both with and without the structure. This gave them clean incident wave characterisation and overtopping measurements.

Fun fact? In some tests, up to 70% of the wave energy was in the lower frequencies. The slow stuff. The invisible push. See Fig. 5 and Fig. 6 to grasp the scale.

The Long Waves You Didn’t Know You Needed

In this study, IG waves moved from background noise to centre stage. A practical cut-off frequency was proposed:

fcutoff=fp1.5≈0.45Tm−1,0f_{cutoff} = \frac{f_p}{1.5} \approx \frac{0.45}{T_{m-1,0}}

This split the spectrum into "IG" and "HF" (high-frequency) components. Equation 21 estimates the IG energy fraction based on:

  • Toe depth

  • Iribarren number using short waves

  • Deep water khk h

Outcome? RMSE = 0.06. Pretty sharp for spectral stuff.

That 2% Wave You Can’t Ignore

In shallow water, it’s not the average wave that matters. It’s the extreme ones. The study shows that these 2% exceedance waves can be up to 40% taller than the short-wave Hm0:

H2%/Hm0,HF≈1.4H_{2\%} / H_{m0,HF} \approx 1.4

That flips the script. If you’re not accounting for this, your design might be way off. Fig. 8 says it all.

What Matters Most? Let the Forest Decide

No more guessing: they used machine learning (random forest) to assess variable importance. Spoiler: depth is not top.

Top 3 influencers:

  1. Relative crest height Rc/Hm0R_c / H_{m0}

  2. Short-wave steepness sm−1,0,HFs_{m-1,0,HF}

  3. IG wave height & asymmetry

Here’s the good news: you can predict overtopping without knowing the exact seabed, as long as your wave data is solid.

Better still, when tricky-to-measure parameters like asymmetry and skewness are removed (not practical in engineering), the model leans more on short-wave steepness and IG wave height. Even better, H2% comes through as a reliable proxy.

And here’s something to really splash about: if your data is poor, you’re better off using simpler formulas like Eq. (24). Why? Because adding complexity when your inputs are messy is like painting details on a foggy window.

New Formulas for a Messier World

Here come the stars of the paper—the new formulas.

Equation (24):

q∗=0.74exp⁡(−8.51RcHm0γfsm−1,0,HF0.32)q^* = 0.74 \exp\left(-8.51 \frac{R_c}{H_{m0} \gamma_f} s_{m-1,0,HF}^{0.32}\right)

Equation (27):

q∗=0.44exp⁡(−8.75Rc−0.38Hm0,LFHm0,HFγfsm−1,0,HF0.33)q^* = 0.44 \exp\left(-8.75 \frac{R_c - 0.38 H_{m0,LF}}{H_{m0,HF} \gamma_f} s_{m-1,0,HF}^{0.33}\right)

Equation (28):

q∗=0.15exp⁡(−9.45Rc−0.90Hm0,LFH2%γfsm−1,0,HF0.28)q^* = 0.15 \exp\left(-9.45 \frac{R_c - 0.90 H_{m0,LF}}{H_{2\%} \gamma_f} s_{m-1,0,HF}^{0.28}\right)

Message? When you include long-wave height, short-wave steepness, and H2%, you slash RMSLE from 1.34 to 0.64.

And don’t miss this point (Fig. 12 Panel B): foreshore slope and relative depth don’t explain the scatter. What matters is the wave field: its energy, its extremes, its shape. The rest is embedded within the spectrum.

 What If Your Data Is Rubbish?

They tested input noise to mimic reality:

  • Wave height uncertainty = big error.

  • Short-wave steepness and IG = more robust.

Key insight: If your data’s messy, stick to simpler, more robust formulas like Equation (24). Complexity doesn’t help if your inputs are shaky. Fig. 13 shows the sensitivity results.

What Wasn’t Included (But Matters)

This study fixed roughness and geometry. So if your structure’s different, adjust the coefficients.

Also, no multi-directional or short-crested waves were included. These could impact IG energy. The formulas seem solid, but validation for these cases is still needed.

Why Focus on Short-Wave Steepness?

A final word on why the authors chose to base steepness only on high-frequency energy:

In shallow waters, the total wave steepness (defined as sm−1,0=Hm0/(gTm−1,02)s_{m-1,0} = H_{m0} / (g T_{m-1,0}^2)) becomes misleading. It's overwhelmed by infragravity energy, no longer reflecting the real shape or impact of short waves.

Instead, a more accurate definition is used:

sm−1,0,HF=Hm0,HFgTm−1,0,HF2s_{m-1,0,HF} = \frac{H_{m0,HF}}{g T_{m-1,0,HF}^2}

This "short-wave steepness" reflects the dynamics that actually push water over the crest.

And isolating the role of IG waves? Very tricky. Their effects are tangled with wave setup, asymmetry, skewness—all hard-to-measure variables rarely used in real-world engineering.

So what De Ridder et al. did was smart: they chose predictors that engineers can actually estimate reliably, and that capture the essence of shallow water extremes. It's not about ignoring physics, it's about surfacing the useful parts.

Final Splash: The Sea Doesn’t Lie

The sea rarely behaves the way we wish it would. In shallow water, short and long waves dance together, and that dance decides if your structure gets overtopped.

De Ridder et al. didn’t just drop new formulas—they delivered a deeper way to see what truly drives overtopping.

If you’re designing coastal structures and only looking at Hm0, this paper kindly says: you’re missing the point. Read the spectrum, separate the slow from the fast. Then—and only then—run your numbers.

Key Figures to Check Out:

  • Fig. 5–6: IG energy trends

  • Fig. 8: Extremes vs Hm0

  • Fig. 11: Fitting improvements

  • Fig. 12: Scatter sources

  • Fig. 13: Sensitivity tests

Catch you on the next wave. Hopefully, it won’t top over you.
César!

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