It is often very interesting how ideas evolve in Math. Fourier used the technique that now bears his name to study the heat equation. As it happens, the same technique is also very useful when studying wave phenomena, which are mathematically described by a very different type of equation (more on that towards the end of the semester). For now, suffice to say that the theory tells us that in a given medium which supports waves (say, the surface of a body of water), a generic wave can be seen as a sum of simple oscillating functions, which propagate in a simple way. Consider waves on the surface of long and narrow circular channel. The surface elevation can be written as Here is the wavenumber with the length of the channel (remember, the channel is circular, so the domain is periodic), and the frequency at which a wave with wavenumber oscillates, is the position along channel (measured from an arbitrary point) and is time. For water waves we have where is the gravitational acceleration and the depth of the channel. Of course, we interested in the real part of that expression, but by now you should not be afraid to handle complex values. The (complex) coefficients are calculate from measurements of the surface at . Then the equation above can be used to determine the shape of the surface at any later time. Write a Matlab function that takes a vector of measurements of surface elevation at time 0 and return a vector containing the surface elevation at an arbitrary later time . and are the depth of the channel and the acceleration of gravit respectively and the length of the channel. The input vector contains a great number of points, so you must use fast algorithms. Assume that the size of is even. Use your function to plot hOut vs when and are given by [ and , , , . Submit your code to