We have seen how non-dispersive waves propagate on tensioned cables. But there are other systems that are governed by the same equation. So, we must expect the same kind of behavior. For instance in acoustics, we had the equation for the evolution of pressure when it depended only on one coordinate. So, the equation was, using the simplified notation for the space and time derivative, pddot -c^2 p"=0. This is exactly the same as that of the cable, except that the variable is here the pressure. If we converted this to an equation on density, or on displacement, we would have the same equation. We will therefore have the same solutions -- non-dispersive waves that propagate at the velocity c. But now this is sound propagation. And you can hear me properly because the waves are non-dispersive -- the signal is propagated without any change. The wave velocity c here is given, if you remember well, by c= sqrt (dF/drho) where F gives the state equation relating pressure and density. The velocity is directly dependent on the way the fluid reacts to a change of pressure. In air the velocity of sound in air is about 300 m/s, and in water about 1500 m/s. We also had the equation when the pressure depended on all three space coordinates. What can happen then ? For instance if there is a local perturbation somewhere, a source of pressure like a bird singing. How does the perturbation evolve with time and space ? Let us rewrite the equation assuming that the pressure only depends on the distance to the bird p(x,y,z,t)=p(r,t). Using the Laplace operator in spherical coordinate the equation reads simply pddot-c^2/r^2(r^2p')'=0. Well, this is not the cable equation. This is no surprise because as you go away from the bird you know that sound decreases. But still the sound itself does not seem to be modified much as you go way from the bird, so we must have some kind of non-dispersive waves. Here is the equation for p, again If we try a change of variable p(r,t)=pi(r,t)/r then the equation on pi reduces to Pi ddot-c^2 pi"=0, which is the cable equation! This means that what is non-dispersive is not the evolution of pressure but the evolution of pressure once we have corrected by the variation with distance, that goes like 1/r. For instance, the D'alembert solution reads p(r,t)=1/r(F(r-ct)+G(r+ct)). This is why you can hear the bird as it sings, and without deformation of its song, but with a decrease as you go away. This illustration here shows you the propagation of the harmonic wave that goes towards the positive r, corresponding to the F function above. And you can see the 1/r decrease as you get away from the source. Let us move to some another case of wave motion in fluids. We also wrote down the equations for the evolution of a free surface of a fluid. As you remember, we had a condition on the free surface that related the evolution of pressure with the change in height of the fluid. That had to be associated with an equation to be solve in the whole fluid domain. Actually these equations can be much simplified in the limit of what we shall call shallow waters. What does shallow mean? 10cm? 1m? 1mm? None of this of course, it just means when the depth of the water H is small compared to the wavelength of the perturbations we are discussing, say lambda. Of course the perturbation of the surface needs to be small versus the depth, too, in order to remain in the linear domain. Here is a sketch of that situation. The depth is H, the displacement is y(x,t). The relation at the free surface is p - rho g y=0. We assume that the horizontal velocity of the fluid does not depend much on the vertical axis and that vertical velocities are small versus other quantities. The momentum balance in the x direction relates the acceleration rho du/dt with the horizontal gradient of pressure -dp/dx. From the relation at the free surface and the momentum balance in the vertical direction, this gradient is just rho g dy/dx. So, rho du/dt = -rho g dy/dx. The mass balance needs also to be considered. Let us do this in an integral form, I mean integrated over the whole depth. The time variation of mass per unit length in x is rho d(H+y)/dt. The variation in x of the flux of mass is rho d/dx (H+y) u. The sum should be zero, as the variation of flux should result in a variation height. This simplifies into rho dy/dt +rho H du/dx = 0. when we neglect the higher order term in y and u. We may now combine the momentum balance and the mass balance and we have simply yddot- gH y"=0. This is equation that governs the evolution in space and time, in dimension 1 in x, of the vertical displacement of the free surface. Look at this. This is exactly the same equation as the tensioned cable and as the sound propagation in 1D. And of course we are going to have the same solutions, with the D'alembert general solution, the dispersion relation, and so on. The waves will be non-dispersive. What is interesting is the value of the velocity of waves, c=sqrt (gH). It depends on gravity, of course, but also on the depth H. For instance, for 10cm of depth the velocity is about 1m/s. But this also applies to propagation of tsumani waves in the ocean. This may seem surprising, because oceans are deep, not shallow. But the assumption of shallow water corresponds to the condition that the depth is small compared to the wave length of the perturbation. This is the case of tsunami which are very very large scale perturbations. Let us compute an order of magnitude of the velocity propagation of the terrible 2004 tsunami. From the original source, near Sumatra Island to the Maldives islands. The typical depth of the ocean there is about 4000 m. That would give a velocity of the surface waves of about 200 m/s, which is very fast. In 2004 The Maldives islands, which are about 2800 km away, were reached about 3 hours after the original earthquake. This gives an average velocity of the waves of 260 m/s. As you can wee we have a rather good approximation of the velocity with this simple model. Moreover, we can understand why the energy of the earthquake is transported so far with devastating effects. It is because the waves are essentially non-dispersive. As a consequence, apart from the radial decrease, similarly to acoustics, the waves are undeformed. And the consequence is terrible, all the energy reaching a given point in a short time. From these two examples, sound and tsunamis, you understand that the very simple equation yddoy-c^2y"=0, and its equivalent in higher dimensions, is applicable in many domains. Non-dispersive waves are simple and efficient in carrying energy from one place to another, and are not very difficult to compute. But as you know they will only exist when the equations relating the time and space evolutions are somewhat equivalent in time and space. And this is not so common. Next, you will learn to address the case of dispersive waves.
