Problem 4. The sound waves in an organ pipe (with a close end at 1 = 0 and an open end at r = L (say) are governed by the second order equation y" :r) + o’y(x) = 0 with boundary conditions y(0) = 0 and y(L) = 0 and where we take a to be constant (and related to the speed of sound in air, the width of the pipe and so on). 1. Show that the general solution takes the form y= A cos ax + B sinar 2. Show that the boundary conditions lead to A = 0 and the eigenfunctions (2n-1) yn () = B sin n=1,2,3... 21 3. Sketch the first three eigenfunctions on the interval 0 SSL to convince yourself that the answer makes sense.