Que Onda con los Papers

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: \...

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...

  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...