A pan flute uses a set of open cylindrical pipes to produce musical notes. One particular pipe of length L0=0.30m is tuned to its fundamental frequency. A player wishes to raise the pitch of this same physical pipe by a musical interval of a perfect fifth (frequency ratio 3:2) without shortening the pipe. They propose partially sealing the bottom of the pipe with a removable plug of wax so that the pipe behaves as closed at a point located a distance x above the original bottom end. End corrections may be neglected and the speed of sound is taken as v.
(1) Part 1: (a) Write expressions for the fundamental frequencies of the pipe when it is (i) completely open at both ends and (ii) effectively closed at a distance x from the bottom.
(2) Part 2: (b) Using the requirement that the closed configuration produce a note one perfect fifth above the open-pipe note, derive an equation relating x to the original length L0.
(3) Part 3: (c) Solve the equation from part (b) to find the numerical value of x for L0=0.30m.
(4) Part 4: (d) Discuss qualitatively how sensitive the resulting pitch is to small errors in the placement of the wax plug, citing your reasoning.