Professor Simen Ellingsen of the Norwegian University of Science and Technology recently found the solution to the controversy. Source: Physical Review Letters Ship Wakes: Kelvin or Mach Angle? Fr represents the Froude number, which directly varies with respect to the velocity of the object. Figure 6 shows the effect of higher speeds on the wake angle alpha. Their numerical simulations seem to suggest that at higher speeds, boats would produce smaller wake angles. They created new mathematical models to describe the narrower wakes through analyzing the images and making measurements of the hull lengths, wake angles, the velocities, and assuming that an object of length B cannot produce wavelengths greater than B. Raubad and Moisy discovered that even without all of these circumstances, some ships still had smaller wake angles. Before their discovery, narrower wakes have been spotted, but scholars stated that they were due to unique circumstances such as shallow waters, turbulence, and more. While looking at Google Earth images of ships, two French physicists Marc Rabaud and Frédéric Moisy, saw that some of the wake angles were not 19.47 degrees, that is, they did not conform to the prediction that Lord Kelvin made in 1887. Using trigonometry, it can be shown that sin(theta) = x/3x = ⅓ => arcsin(⅓) = theta = 19.47 degrees. Constructing tangent lines to each individual wave, joining them with point A, and representing where the waves are when the boat is at point B with the red dots as indicated in the figure, the sequence of red dots forms a circle with radius AB/4.Ĭonstructing a tangent line to the red circle, as shown in figure 5, yields a right triangle with the side opposite to theta with length x, and the side along AB with length 3x. This is shown in figure 4, where each circle from point A represents a wave moving at phase velocities of 10, 20, 30 … 99.99 percent the speed of the object. If point B is the location of the object traveling across the body of water, then multiple constructions can be formed with point A representing the epicenter of all waves emitted from it, relative to point B. Source: University of Hannover in Germany The red dots represent the phase velocity and the shape of the wake. Geometric representation of a water wake, showing the individual propagating waves. Using this relation, and the fact that water waves disperse, a geometric representation of the wake can be generated, such as the one in figure 3. From the expression of group velocity and the given expression of phase velocity, it can be determined that v_g = v_phase / 2, that is, the group velocity is ½ of the phase velocity. The group velocity is the velocity of the overall shape of the wake. Its equation is v_phase = sqrt(g*lambda/2pi). The phase velocity is the velocity of the individual wave crests. However, it can be assumed that the water is sufficiently deep enough to not be a factor and since the surface tension only affects short waves, it is negligible. On what factors do the velocity of the waves depend upon? Certainly, the wavelength lambda, the depth of the water H, surface tension sigma, the gravitational force g, and the density of water rho are all factors. This is the fundamental reason why water wakes differ from other phenomena, such as a Mach cone, which is the pressure wave produced by bodies moving faster than the speed of sound. That is, the longer the wave, the faster it propagates. While light and sound waves propagate with the same velocity, 3 * 10^8 and 348 m/s respectively, the velocity of water waves depends on their wavelength. However, the divergent waves are of more interest, since they are directly related to theta. There are two types of waves in a wake: transverse waves and divergent waves, as depicted in figure 2. In order to derive theta, first, one must learn the essential concepts and vocabulary in order to do so. This article will attempt to present this solution as well as some controversies regarding the Kelvin angle. Crawford, a professor of physics at the University of California at Berkeley, found theta using only elementary mathematics - specifically geometry and trigonometry. Using rigorous mathematics, he determined in 1887 that the angle theta (Kelvin angle) that the wake fans out is always the same, regardless of the object and its speed, is approximately 19.47 degrees. The famous physicist Lord Kelvin noticed an interesting fact about the wakes. One might also note that the wake patterns, regardless of whether it is being produced by a duck or even a cruise ship, all seem to have a similar shape. While watching boats travel by or a duck on a pond, one might wonder how the astonishing wake patterns emerge, such as the ones depicted in figure 1.
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