Hi, my name is Michael Benzaquen and I am a CNRS researcher here at Polytechnique. Today I will tell you an inspiring story about ship wakes that attracted our interest a couple of years ago. Anyone could have noticed that the wake behind a moving disturbance at the air-water interface seems to be quite universal. This is: it looks almost the same regardless of the nature and speed of the disturbance itself, be it a large cargo ship, a sail boat, or a small duckling. The first person who took real interest in this matter is Lord Kelvin who stated in 1887 that the wake angle of a disturbance moving along a straight line at constant speed was independent of the velocity and approximately equal to 19.5 degrees. This result contrasts with Mach's result on supersonic pressure waves in non-dispersive media for which the wake angle scales as the inverse of the disturbance velocity. Kelvin's constant wake angle can actually be explained through the dispersive nature of water waves. Let's go through the geometric construction of Kelvin's wake. Consider a disturbance travelling with constant velocity V along a straight line from point A to point B during time interval Delta t such that AB = V Delta t. An essential ingredient comes from noticing that the wave pattern is stationary in the frame of reference of the moving disturbance such that one may write that the circular frequency that frame of reference omega'= 0. Then, a simple Doppler shift allows to relate omega' to omega (the circular frequency in the absolute frame of reference) through omega' = omega - k . V where k is the wave vector. Introducing the phase velocity c = omega/ k this equation becomes c = V cos theta where theta is the angle between the wave vector k and the direction of motion. At this point one may say that: if the waves generated at point A had traveled at their phase velocity c, they would have reached the blue circle at time t. But as you know waves do not travel at their phase velocity but rather at their group velocity. In the particular case of water waves the group velocity is precisely half the phase velocity such that the waves have only reached the red circle at time t. Reproducing this construction with different time intervals Delta t yields the wake enveloppe whose half-angle can be easily determined to be precisely the arcsin (1/3) which is approximately 19.5 degrees. But all I have said so far is textbook physics. For over a century no one doubted these results, until 2013 when two colleagues from Orsay university published a very inspiring study based on Google Earth images. They measured many wake angles from these Google Earth images, as you can see here. and they plotted them as function of the Froude number which is none other than the dimensionless velocity of the moving disturbance. They found that beyond a certain Fr number the wake angle decreases and scales as the inverse of the velocity, very much like a Mach angle. This result really tossed a pebble into the pond. So, a century after Lord Kelvin, and a few months after the colleagues in Orsay we got interested in the problem. We were able to show both numerically and analytically that actually two angles can be identified in the wake: First, the angle delimiting the wake, always constant and equal to Kelvin's prediction, and Second, the angle of the highest waves that indeed decreases as the one over the velocity, consistent with the observations of my colleagues. To wrap up, here is a photo where both angles can be clearly identified. This is it folks, a nice example on how modern technology can bring us to question century old results and discover new physics. But bear in mind that Kelvin was always right, his theory only dealt with wave crests regardless of their amplitudes!