In a previous video you have obtained simple laws for the phase velocity of waves at the surface of a liquid. Our goal is now to verify that experimentally. For this purpose, laboratory measurements will be performed with the help of the wave flume of ENSTA Paristech that you can see in action here. So, what does this experimental setup look like in details? The main part consists of a rectangular tank with transparent borders. At one end of the tank, there is a beater, whose displacement can be imposed in realtime by sending a varying voltage at its inlets. A time-varying voltage signal is created by a LFO. This signal is sent to an amplifier, which feeds the beater. When the beater is activated, it creates waves trains that propagate along the tank. At the other end of the tank, we have placed something that looks like a beach. This particular shape is designed to minimize wave reflections. Now that we are able to force the water surface to create waves, we need something to measure them! To this end, we have two water height sensors. Each sensor consists of two metal wires placed in water. The electrical resistance between these two wires depends directly on the water height. To measure electrical resistance, we use a wheatston bridge, which converts a time varying resistance into a time varying voltage. Finally, here are two voltage signals observed with an oscilloscope when a wave is travelling and passing through the sensors. If we have previously calibrated the sensors, we know the relationship between resistance and water height, and we are able to translate these voltage signal into functions of time describing the evolution of water height at two particular points in the tank. Our goal is now use this setup to verify the theoretical dispersion relation you have obtained in a previous video. So, what did your calculations show? They have shown that two main regimes can be identified. The first one, called shallow water regime, is valid when the wavelength is large compared to the depth of the tank. In that limit, the phase velocity is constant and equals sqrt(g h). The medium is hence non dispersive. Conversely, the second regime is valid when the wavelength is small compared to the depth. In this case the phase velocity equals sqrt( g lambda / 2 pi ). It depends on the wavelength. The medium is hence dispersive. The experiment hence consists of a tank filled with water of height h with a beater on the right moving such that its angle equals theta_0 cos( omega t ). There are hence three parameters we can control: water height h, the frequency of the beater omega and its amplitude theta_0. As a first approach, we will consider that the amplitude is not an important parameter and keep it constant, but it has to be noted that at large amplitudes, the validity of the linear equations we have studied up to now could become questionnable. We hence have two parameters, h and omega. For our experiment, we will fix h to 26 cm and vary omega. For each value of omega, we measure the wavelength of the produced waves by measuring the distance at which the two sensors have to be in order to measure in phase surface displacements. The phase velocity is then computed as c_phi= lambda omega / 2 pi. This phase velocity is here plotted as function of the wavelength. Additionnally, we can plot what is predicted in the deep water regime. The experimental points clearly follow the theoretical curve, except at large wavelengths. If we also plot what is predicted by the shallow water regime, we see that when the wavelength becomes large compared to the water height, the measurements indeed seem to follow the shallow water theoretical prediction. One can also note that there is an apparently worse agreement at large wavelengths. This is due to an increased difficulty to properly impose very low forcing frequencies and measure wavelengths that become of the order of the tank length. In summary, we have seen here how we can experimentally verify the validity of the simple dispersion relations you have studied for water waves. Also, please remember that forcing a system at variable frequency and measuring its response in order to characterize its properties is a very classical approach that will be used again in future videos.