Hello. In the previous videos you have seen that different media can be classified after the behaviour of the propagation velocity of perturbation waves. If this velocity is constant, the medium was said non dispersive. In contrary, if the wave propagation velocity depends on the frequency or wavenumber, the medium was said dispersive. In this video, we will se how the propagation of waves can be influenced by boundaries in the medium, and how a non-dispersive media can bear dispersive waves by the presence of these boundaries. For that purpose, the case of acoustical waves propagating in pipes will be addressed. The particular waves that will exist in such geometries are called guided waves, because they can only propagate in the direction of the axis of the pipe. Pipes are then called waveguides. This kind of problem has many applications in the industry where pipes are present, and where noise is produced somewhere along the pipes. Waveguides are also the main part of any musical wind instrument! But guided waves are not only found in the acoustics domain. Electromagnetic waves in an optic fiber are also guided waves. Here on the right, there are images of metallic pipes used to guide electromagnetic waves. This allows to transport electromagnetic signals in a very efficient way with very low interferences. Here is our simple model of a waveguide. It consists of a fluid domain bounded by four walls at x = 0 and L_x and y = 0 and L_y. The domain is still considered infinite in the z direction. Inside this fluid domain, the medium is considered to be a fluid of standard properties, so that the pressure satisfies the classical 3D wave equation shown here on the right. In the second equation, the Laplace operator is developed in cartesian coordinates. Let us now invoke the variable separation principle. A solution of these equations is sought in the form of a product of a function of t and z only, a function of x only and a function of y only. If we divide all terms by c squared P f g, we obtain a simpler equation. The three terms in this equation depend on different variables. z and t only, x only, and y only. We start our resolution from this particular form of the wave equation. First of all, by puting the term f'' over f on the left of this equation, and all other terms on the right, we obtain something that on the left, depends only on x, and only on the other variables on the right. Since it is verified for any value of all these variables, it has to be constant. Let us call this constant k_x^2. Next, the same process can be done for g, introducing another constant that we choose to call k_y^2. We can now make use of the boundary conditions, which impose that the normal velocity vanishes at the walls. In order to obtain a boundary condition for the pressure, we need to introduce a widely used fundamental equation in fluids - the momentum conservation in non viscous fluid at small velocities. This equation states that the local acceleration in the fluid equals minus the gradient of the pressure. Hence, at the boundaries, imposing that the normal velocity vanishes is equivalent to impose that the normal gradient of p vanishes. The consequence is that at x=0 and Lx, the derivative of f with respect to x vanishes. There exist an infinite set of solutions of the equation governing f with these boundary conditions. It is f = A cos(k_x x) with k_x = m pi / L_x and m a positive integer. The same can be done for g. The solutions of the equation with boundary conditions imposing dg/dx being zero at y=0 and y=L_y are g = B cos(k_y y) with k_y = n pi / L_y and n a positive integer. Here again, there is an infinite set of solutions because n can take values between 0 and infinity. If we now group all these solutions together and write again the acoustic wave equation, we can show that the pressure capital P, which is only a function of z and t, satifies the equation written here on the right. In this equation there are infinite possibilities for k_x^2 and k_y^2. If we express in details their values, we show that in the pipe, the pressure, noted here P_mn, satisfies a wave equation. With an additionnal term, very similar to the term we had studied the string on an elastic foundation. So, if we summarize now, we have obtained an infinite set of pressure wave families propagating along z. Each familly (m,n) has its own propagation equation, which is exactly the equation governing the displacement of a cable on a spring foundation! The equation is written here again with a spring foundation stiffness b_mn. It has been shown in the previous video this week that the phase velocity of the waves in such media depends on the frequency. The medium is hence said dispersive. Moreover, it was shown that below a frequency omega_c = sqrt( b_mn ), the is no possible propagation of the waves. Just above this critical frequency, the phase velocity tends to infinity. There is a very particular familly of waves�. It is the (0,0) familly. For this familly, the additionnal stiffness term vanishes. And the wave propagation becomes a very standard wave equation. For this familly, the waves that travel along the pipe are plane waves and the medium is non dispersive. The velocity is c - the velocity of sound. One can then wonder what is the lowest cutoff frequency of a given pipe. This is a fundamental question, because below this frequency, only the (0,0) familly can exist. In other words, below the lowest cutoff frequency, only non dispesive plane waves travel at the speed of sound. To calculate this frequency we have to know which is the largest width of the pipe, L_x or L_y. Let us consider for instance that it is L_y. The smallest cutoff frequency is then omega_c01 which equals pi c / L_y. Expressed in terms of a frequency in hertz, it writes F_(c_min) = c / 2 L_y. This result is important, because generally waveguides are used at frequencies below the cutoff frequency. For instance in musical instruments, L_y is of the order of a centimeter. The cutoff frequency is hence of the order of seventeen thousand hertz, which is almost the highest frequency at which the human ear is sensitive. It is hence only important to consider plane and non dispersive waves in musical instruments pipes! It is time to conclude and summarize our results. We have considered propagation of waves in bounded domains in the form of pipes, called waveguides. We have shown that depending on the frequency, different famillies of waves can propagate. In general, these waves are dispersive, but below the cutoff frequency c / 2 L_y, only plane non dispersive waves can propagate. Next week, we will see that adding additionnal boundaries at the begining and at the end of the pipe constrains even more the famillies of waves that can exist. We will learn that this will select stationnary waves, called eigenmodes. But this is another story!