Hello. You have already learned how surface waves propagate, and we have seen such waves in a real experimental situation - the wave flume. You have also seen that adding boundaries creates wave reflexions, so that in finite domains, waves can combine at some frequencies to create an eigenmode. In the wave flume, we prevented wave reflections by putting a sort of beach. What happens if we remove the beach? Look at this animation. We indeed identify at some frequencies oscillations of strong amplitude that are probably due to the existence of an eigenmode. Let us try to identify eigenmodes in this case. Firstly, we need to define the problem we want to solve. All the basic equations have already been defined in a previous video. Let me recall them. In the fluid the laplacian of the pressure equals zero. At the top of the domain, z = h and we are at the free surface. Here, the pressure satisfies d^2p/dt^2 + g dp/dz = 0. At the left and right boundaries of the domain, the velocity component along x of the fluid vanishes. If we make use of the momentum conservation in the fluid, we can show that this is equivalent to impose that the pressure gradient along x vanishes at these boundaries. Similarly, the vertical component of the velocity vanishes at the bottom where z = 0. This system of equation will now be solved by looking at the pressure in the form of a product of functions of x only, z only and t only. This is what we call the variable separation principle. Introducing a solution in this form into the local fluid equation laplacian of p = 0 and dividing the resulting equation by a times b times c leads to an simple equation involving a function of x only and a function of z only. Next, we can put the function of x and z on different sides of the equal sign. Since both sides of the equation depend on different variables, it has to be constant. We choose to call this constant k squared. Let us first solve the equation for a. It can be written as a'' = -k^2 a. The x derivative of the pressure vanishes at x = 0 and l. This imposes that the derivative of the a vanishes at x equals 0 and l. There exists an infinite set of solutions of this equation with these boundary conditions. It is a_n = cos(kx) with k = n pi / l, n being a positive interger. And now for the second equation governing b(z). It can be written in the form b'' = k^2 b. There is here one boundary condition at the bottom. It is dp/dz = 0. Expressed for b, this imposes b' = 0 at z = 0. The solution of this equation is b = cosh(kz), k being unchanged. The pressure as we know it at the moment is written here. There is one last function c that is undetermined, and there is one boundary condition that has not been used yet. It is boundary condition at the free surface. Inserting this form of p into the free surface equation gives this formula c'' cos(kx) cosh(kh) + c g k cos(kx) sinh(kh) = 0. Let us reorder this differential equation and write it in this particular form. The solution of this equation is the sum of a cosine and a sinus function. The constants A and B in this equation depend on the initial conditions. If we now put all these solutions together, we can write the solution in the form of an inifine sum of products of a function of space and a function of time. In this product, the function phi_n(x,z) on the left is the eigenmode number n, and the functions on the right correspond to an harmonic evolution at the frequency omega_n. Note that the relationship we have obtained that links the frequency to the wavenumber is actually the dispersion relation of the system. If you wonder why it is different than the shallow water and deep water dispersion relation you have found before, I suggest you to do the following: firstly, have in mind that the value of k_n h measures the ration between the water deepness and the wavelength. Consider then that k_n h is small. You will find that the dispersion relation becomes that of the shallow water case! If you do the opposite and consider large values of k_n h, you will find the dispersion relation in deep water. The eigenmode equation is written again here. Additionnally, you can find on the right the relationship between the water elevation and the pressure at the free surface. This relation has been shown in a previous video. Using this relation we can plot the evolution of the free surface deformation corresponding to each mode as function of time. It is here represented for modes 1 2 3 and 10. Let us now compare these modes with what is observed in a real bassin. Here you can see what happens if the frequency of the beater is selected such that the water displays large amplitude resonant oscillations. This is for instance the mode. In conclusion, we have calculated in this video the eigenfrequencies and eigenmodes of a 2-dimensionnal bassin. The knowledge of this dynamics has a great importance for instance in the understanding of the influence of the water dynamics in tanks on the vehicle dynamics. When calculating the eigenfrequencies, we have identified a dispersion relation more general than the previous ones valid for shallow and deep water. And finally we have seen the kinematics associated with each eigenmode we calculated, and how it compares with the real dynamics of the bassin.