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Three scores of the sea: PM, JONSWAP and TMA

The Pierson–Moskowitz discovery (1964): the universal melody of the mature sea In 1964, Pierson and Moskowitz analysed wave records under strong, steady winds in the North Atlantic. They found that when the wind has blown long enough and over a large enough fetch, the sea reaches a state of full development in which the spectral shape becomes universal, independent of the details of the storm. They proposed the now-famous formula: \[ S_{\text{PM}}(\omega) = \alpha \,\frac{g^{2}}{\omega^{5}} \exp\!\left[-\beta \left(\frac{\omega_p}{\omega}\right)^{4}\right], \] with \(\alpha \approx 8.1 \times 10^{-3}\) and \(\beta \approx 0.74\). The crucial feature is the \(\omega^{-5}\) decay at high frequency, the so-called equilibrium range: any attempt by the wind to pump more energy there leads to breaking and dissipation. To compare seas of different sizes — for instance, one with 6-second waves and another with 20-second waves — they introduced dimensionless variables: \...
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Coffee in Edinburgh and Galilean Extremism

Yesterday, over coffee with some friends in Edinburgh, we ended up in one of those conversations that start with society and drift off anywhere. True to my style, I ended up talking about complex systems. I told them that one of the best ways to analyse them is through what I call Galilean extremism : pushing a system to an impossible case in order to understand the possible. Galileo did this brilliantly. He observed that a stone and a feather do not fall in the same way, but asked: what would happen in a perfect vacuum, without air? That impossible scenario revealed the essential: all bodies fall with the same acceleration. The impossible allowed him to make sense of the possible. In nonlinear physics , we do something similar: we go to extremes (such as infinite friction in a pendulum) to discover the hidden rules that organise behaviour. Try it yourself: Drop a light piece of paper and a coin at the same time. Then put the paper on top of the coin and drop them together — th...
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Non-dimensionalisation for mortals: waves, nonlinearity and dispersion (part 1)

  When you open a fluid mechanics or wave theory book, the first thing you meet are these monster PDEs: \[ \begin{aligned} & u_t + u u_x + w u_z = -\frac{1}{\rho_0} P_x, \\ & w_t + u w_x + w w_z = -g - \frac{1}{\rho_0} P_z, \\ & u_x + w_z = 0. \end{aligned} \] This is just momentum conservation (horizontal and vertical, without viscosity) and mass conservation (incompressibility). But in this dimensional form, the equations don’t tell us when waves will behave linearly, when nonlinearity will dominate, or when dispersion matters. That’s the whole point of non-dimensionalisation. What is non-dimensionalisation? Think of it as putting the equations on a new measuring scale. Instead of dragging metres, seconds and Pascals everywhere, we rescale everything with characteristic values (length, depth, speed…). What’s left are pure numbers that show us which physical processes are important. In wave theory, the two heroes are: Nonlinearity (\(\varepsilon\)): ampl...
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Anyone can run a model. Building a reliable framework is the challenge

When someone says to me, “I’ve learned how to run WaveWatch III” or “I can launch cases in SWASH now” , I’m often tempted to reply: “Great… and then what?”. Because running the model is just the tip of the iceberg. Those of us working in coastal numerical modelling know that opening the software, setting up a basic input, and running a simulation can be learned in a matter of days or weeks. Even generating a nice animation of the free surface can look impressive at first glance. But that alone doesn’t make you an expert. The real challenge begins when you have to build a modelling framework that’s robust, efficient, and reusable . That’s where the amateurs and the professionals part ways. It’s Not Just the Physics – It’s What You Do With It Understanding the physics behind the models — energy transfers, dispersion, friction, nonlinearity, slope effects, infragravity generation, etc. — is fundamental. But it’s not enough. Knowing the theory without knowing how to implement it is li...
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The Wee Spectrum That Grew: Tales o’ Gamma an’ the Growin’ Sea

By César Esparza Since the beginning of modern ocean and coastal engineering, one of the key questions has been: “Can we mathematically describe the state of the sea?” We know waves don’t come alone or uniformly: they arrive in groups, with varying heights, frequencies, and directions. Spectral analysis allowed us to look at the sea as a sum of sinusoidal components. But what does that spectrum actually look like? Throughout the 1950s and 60s, many researchers attempted to answer this. But it wasn’t until 1964 that Pierson and Moskowitz proposed a concrete spectral shape based on the similarity theory of Kitaigorodskii. Less than a decade later, Klaus Hasselmann and his team sharpened that curve with the now-famous JONSWAP experiment. This blog-column takes you on that journey—explaining the foundations, the derivation of the PM spectrum, the need for JONSWAP, and the intriguing role of the peak enhancement factor: the famous γ \gamma . Kitaigorodskii's Similarity Theory: Looking f...
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The invisible force that waves carry: Understanding Longuet-Higgins Radiation Stress Tensor — Part 1

Why did Michael Longuet-Higgins call it radiation stress ? At first glance, the term might sound like something out of nuclear physics — but don’t worry, nothing’s glowing here. In this context, “radiation” simply refers to how waves carry or radiate momentum forward as they travel across the sea, much like light radiates energy through space. Surface waves aren’t just wiggling water up and down — they’re transporting actual momentum. And when those waves grow, shoal, or break near the coast, that momentum doesn’t vanish — it has to go somewhere. It gets transferred into the ocean itself, pushing the water around. That internal push — a force per unit area within the fluid — is what we call stress . Hence the term: radiation stress . If you’ve ever stood waist-deep in the sea and felt a swell gently nudge you shoreward — even before it breaks — then you’ve already experienced radiation stress in action. It’s subtle, but it’s there. That invisible shove is the ocean’s response to ...
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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 ∗ = q g H m 0 3 q^* = \frac{q}{\sqrt{g H_{m0}^3}} Followed by exponential adjustments: q ∗ = a ⋅ exp ⁡ ( − b R c / H m 0 ) q^* = a \cdot \exp(-b R_c/H_{m0}) But what h...
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