We want to explore the dynamics of simple continuous systems. We just build models for cables and for beams, but we also want to see how this works in fluids. We know that fluids behave dynamically as continuous systems. For instance, sound propagates under water when whales sing, and waves propagate at the surface of the sea. Let us talk first about acoustics sound propagation. What does this mean? Imagine a fluid at rest with a pressure P_0 and the density rho_0. It may be air as around me or water as around the whale. What happens if that state of the fluid is slightly changed by my voice, as you can hear, or by the sound of the whale? For the sake of simplicity, let us consider a state of the fluid that only depends on the coordinate x and on time. A model in dimension one again. The pressure reads capital P = P_0 + p(x,t), where p is the perturbation in pressure. Similarly, the density rho reads rho = rho_0 + r(x,t), where r is the perturbation in density. The fluid velocity reads capital U = 0 + u(x,t) because, the velocity at rest is of course just zero here. How are these quantities related? Well, we know that the mass balance in one dimension in a fluid reads, d rho/dt + d/dx(rho U) = 0. Here, this simplifies a lot if we consider only the small terms are the leading order. This means that, considering r and u are small, we can neglect their products. The mass balance becomes dr/dt + rho_0 du/dx = 0. Let us do the same thing for the momentum balance in one dimension, rho dU/dt + rho dU/dx U = - dP/dx. Using again the expansions for density, velocity, and pressure, the only remaining terms are rho_0 dU/dt = - dp/dx. Finally, we have a relation between the pressure P and the density rho called the state equation. Say P = F(rho). These relation still holds when the fluid is pertubed. So that we can write as an approximation because of the small variations P_0 + p = F(rho_0) + r dF/d rho at 0. Here, the subscript 0 means at rest. The rest state satisfies P naught equals F(rho_0). Hence, the only remaining terms are p = r dF/d rho at 0. It tells us that pressure and density variations are proportional. The coefficient dF / d rho at 0 is always positive, and we can call it c^2. We can now combine these three equations: mass, momentum, and state, and eliminate r and u. The equation in p reads d^2p/dt^2 - c^2 d^2p/dx^2 = 0. It is the equation for acoustics in our simple one-dimensional framework. This is a very simple equation, and it is actually the same as that for the tension cable. We can do even more. If the quantity is p, r, and u, I know it may dependent on all three coordinates: x, y, z. The same derivation leads to the equation for 3D acoustics, d^2p/dt^2 - c^2 delta p = 0, where delta is the Laplace operator. So, we have a model for acoustics in a fluid. But the most evident and unbound continuously deforming free medium, where we see waves is certainly the sea. A fluid with a free surface. How can we model this? This is a very old problem that has fascinated scientists over centuries. Here is the simplest model we can build. Imagine as I have a fluid at rest in the x,z-space, let the pressure at rest be P_0, which of course, varies in z because of gravity. z equals 0, the fluid is in contact with another fluid at constant pressure. That would typically be the atmospheric pressure. At the free surface the pressure is equal to a constant pressure, say, the atmospheric pressure P_0(0) = P_atm. When the free surface deforms with y of x and t, as sketched here, the pressure inside the fluid varies. It becomes the sum of the original pressure, the hydrostatic pressure P_0 + a small variation p. What changes now is that the condition of contact with the atmosphere is satisfied at the point where the free surface is, which means P = P_atm at the point of it, y(x,t). I can expand this condition considering the displacement y is small and that P is made of P_0 and the perturbation P. This reads, P_0(0) + y dP_0/dz + p(0,t) = P_atm. Because P_0(0) = P_atm. I have no relation that states that the perturbation in pressure at the surface is related to the motion of the surface, namely y dP_0/dz + p = 0. This can be simplified because we know that the pressure field P_0 at rest is the hydrostatic pressure, and so, dP_0/dz = - rho g. So, the condition at the original free surface is now just p - rho g y = 0. What does this mean? It means that any vertical motion of the surface y is opposed by an increasing pressure p. It is just like a spring. So, having gravity and a free surface is equivalent to a mattress of springs on the free surface opposing motion. Surprising. The condition we have derived is a relation between the vertical displacement of the interface and the corresponding variation in pressure. Can we say more than this? Yes, certainly. We also know the momentum balance equation in the fluid. We know that the acceleration is related to the pressure gradient in the same direction. If we apply this adversary interface, we have rho d^2y/dt^2 = -dp/dz. By combining these two equations, the condition at the free surface can be expressed in terms of pressure only d^2p/dt^2 + g dp/dz = 0. Does this equation look like the others? Not exactly, because we have here only a condition at the interface and we have not taken into account the equations that govern the motion inside the domain. For an incompressible fluid, we know that perturbations in pressure around 0 satisfy delta p = 0, which complements the equation above. What we have here is a bit different from the previous models. We have to solve a Laplace problem in the fluid domain with a condition at the interface that mixes time and space derivatives. Let us summarize. We have at hand models for two fluid systems that deforms continuously in time and space. For acoustics and for surface waves. These equations govern the evolutions in time and space of the perturbation in pressure p, both contain a second order derivative in time d^2/dt^2. This is related to inertia of course, but this is not exactly inertia in the equations because the variable here is pressure, not displacement. The two equations also include space derivatives or the pressure. In acoustics, this is a second order derivative in space, but the equation for surface waves is actually combined with another equation that has to be solved in the whole fluid domain. We may now gather all our models for solids and fluids, cables, beams, acoustics, surface waves. In all, we have time and space derivatives that are related. This means, that an evolution in space will result in an evolution in time. That is exactly what we're looking for how deformations and motions are coupled in a continuous system. Let us now try and solve these equations to see what happens. What kind of waves are we going to see in cables or beams in acoustics or at the surface of a fluid? Are they similar? How do they move? This is what you're going to learn next.
