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 osciliating functions, which propagate in a simple way. Consider waves on the surtace of long and narrow circular channel. The surtace elevation can be written as h(x,t)=n=N/2N/21Anei(hntkidntr). Here kn=2n/L is the wavenumber with L the length of the channel (remember, the channel is circular, so the domain is periodic), and (kn) the frequency at which a wave with wavenumber km oscillates, x is the position along channel (measured from an arbitrary point) and t is time. For water waves we have (kk)=knkngkntanh(knH) where g is the gravitational acceleration and H 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) coelficients An are calculate from measurements of the surface at t=0. Then the equation above can be used to determine the shape of the surface at any later time. Write a Matlab function hOut = evolvelave ( hIn,9,H,L,t) that takes a vector hIn of measurements of surface elevation at time 0 and return a vector hOut containing the surface elevation at an arbitrary later time t. H and g are the depth of the channel and the acceleration of gravit respectively and L 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 hIn is even. Use your function to plot hOut vs X when X and hIn are given by [X,hIn]=profi le ( ) and t=10, g=9.81,H=5,L=5. Submit your code to Function 3 Code to call your function 2