We have seen how dispersive waves can propagate on tensioned cables with elastic foundations. That was quite different from non-dispersive waves! Let us see now how this works for some other models that certainly cannot give non-dispersive waves. For instance, you remember that the equation governing the dynamics of bending beams was m d^2y/dt^2 + EI d^4y/dx^4 = 0. I can rewrite this equation as yddot+B^2 y^(4)=0, where I have divided both terms by m and defined B as B^2= EI/m. Here the (4) exponent means a fourth derivative in x. What is the dispersion relation? Simple. I assume that y goes like e^(k x-omega t) and I get D(omega,k) = omega^2-B^2 k^4=0 From this, I can derive the phase velocity c_phi=omega / k=Bk and the group velocity c_g = d omega/d k = 2Bk. Just twice the phase velocity! This result is interesting. Here are the two velocities as functions of the wavenumber k. First, the group velocity is larger than the phase velocity, unlike what we had before. Second, the large wave numbers, will go faster than the small ones. Why? Well, because of the fourth order derivative in space, short wavelength will give a very level of elastic deformation energy. What does this change? Well for instance, when I impact a beam, the high frequency content of the perturbation will travel faster than the low frequency content. The resulting motion somewhere on the beam will first be made of the high frequency content of the impact, then later, of the low frequency content. That was for beams. Let us go back to the general case of surface waves. Remember that, we had a formulation of the problem that only included the pression as variable. We solved the limit where the depth was very small, that was shallow water. The waves were non-dispersive. Let us consider now the case of deep waters. What is deep water? It is when the depth H is very large in comparison with the wave length of interest lambda. But the displacement should still remain small, versus the wavelength now. For instance, the waves created by a ship in the ocean will have a wave length of less than the ship length. If the ship length is 100 m and the ocean 5000 m deep then we are in the infinite depth approximation. But if the ship enters a harbor where the depth is 30 m, we certainly are not. We should use the shallow water model. So, here is the problem we have to solve to find the waves with an infinite depth, the condition for pressure on the surface and the condition that needs to be satisfied everywhere. Let us look for the dispersion relation. The pressure should be in the form of the real part of a function of z, say phi(z) times e^i(k x - omega t). Using the second equation, we have a condition on phi which is - k^2 phi + phi'' = 0. That gives you a solution with exponential growth and another with exponential decay as you in down. Of course, we should only retain the latter one, because we don't want to have an infinite pressure. Now let us include this result in the condition at the surface z=0. You can derive the dispersion relation, quite simply. It is just omega^2 - gk = 0. The waves are dispersive, the phase velocity is sqrt of g/k and the group velocity one half of this. Imagine a wave with a wave length of lambda = 10m. It would go at a phase velocity of c = sqrt(g/k) = sqrt (g lambda/2 pi) =about 4m/s. Now, if you have a boat of 10m she will create many waves, but the longest wave length will be about its size, 10m. These waves will go at about 4 m/s. And if the boat goes faster that 4m/s, she will overpass her own waves. This is called the hull speed. Once you go over, the wave resistance decreases, but the hull speed is a kind of limit speed for sailing boats. This is why longer boats have a larger limit speed. Let us compare the shallow water case and the deep water case. For shallow, waters the wave velocity was sqrt(gH). Here in deep water, it is sqrt (g lambda/2 pi). The relevant length of shallow water was the depth H. The relevant length now is not the depth because there is no depth, but the wave length which is the only length scale. For shallow water, the constant length H that made the waves non-dispersive. And for deep waters the wavelength lambda made them dispersive. The motion of the water is also quite different On the left you have the motion corresponding to the shallow water model. On the right to the deep water model. You can see here that in deep water motion does not penetrate much more than a wavelength of depth. Let us summarize all we have now on waves. We have seen that in all these simple systems cables, beams, compressible fluids, surface waves we had wave propagation, but that it happened quite differently. Some systems sustained non-dispersive wave propagation. Some dispersive. And in the dispersive cases, there was of course as many dispersion relations as models. This somehow concludes the lectures of our first week. But you will find other videos with additional material, research topics and experiments, because waves make a fascinating topic, both in fluids and solids. Next week you will see what happens when we take into account boundaries. When waves reflect!