Que Onda con los Papers

In our first part, we explored how Fourier’s idea of breaking down periodic functions into sine waves helps us understand complex patterns. Now, let’s continue the journey with something very physical: heat flowing through a metal ring. What starts as a warm patch eventually spreads out. But how exactly? The answer naturally leads us from Fourier series to one of the most important tools in mathematics: the Fourier Transform. The Setup: Heat Around a Ring Let’s imagine a thin metal ring with uneven temperature — hotter in some places, cooler in others. We denote the initial temperature distribution as \( f(x) \), where \( x \in [0,1] \), and the function is periodic: \( f(0) = f(1) \). Over time, the heat spreads out according to the heat equation : $$ \frac{\partial u}{\partial t} = \frac{1}{2} \frac{\partial^2 u}{\partial x^2}, $$ where \( u(x,t) \) is the temperature at position \( x \) and time \( t \), and we’ve taken the thermal diffusivity \( \kappa = 1/2 \) just to simpl...

  A throwback to fix the present Some old papers you read out of curiosity… and end up feeling guilty for not reading them sooner. One of those is the 1980 classic by Ottensen Hansen, Gravesen and colleagues: Correct reproduction of group-induced long waves . Sounds heavy, but it’s actually a brilliant little gem that shows—plain and simple—how many wave flumes have been getting the sea wrong for decades. And no, it’s not just a vintage issue. The paper set out to answer one very specific question: Can we properly reproduce group-induced wave effects in a lab flume like they happen at sea? And here’s the twist: they weren’t just looking at the surface. They wanted to see whether we could also reproduce what happens inside the water column —flow velocities, pressures, and especially the long, slow wave that accompanies grouped wave trains. Spoiler: it’s not as easy as adding a few sine waves. Why wave grouping matters Ever noticed how sometimes the sea is calm...

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 set...

  Over two centuries ago, a French mathematician named Joseph Fourier dared to propose a revolutionary idea: that any periodic function —even those with discontinuities or without derivatives—could be represented as an infinite sum of sines and cosines. This proposal not only transformed physics and mathematics forever but also triggered a storm of criticism from the scientific community of his time. Fourier was suggesting something that many of his contemporaries considered mathematical heresy: that “ugly” functions could be decomposed into “beautiful” ones—smooth, infinitely differentiable sines and cosines. Even if a function had a jump or a sharp corner, it could still be “drawn” as an infinite sum of gentle waves. Today, these ideas are fundamental to physics, engineering, data science, and even digital music. But figuring out how to make sense of an infinite sum of oscillatory functions was anything but simple. The central question: How do we represent complicate...