PHYSICS · Subjective · Fermi University Physics · Oscillations, Waves & Acoustics · Vibrating bodies, musical and architectural acoustics · Vibrating bodies, musical and architectural acoustics
Bending, Ringing, Breaking and Flowing, Ch Vibrations of rods, beams and plates
Problem
A uniform rectangular bar, free at both ends, has its first-overtone-to-fundamental frequency ratio at the standard free-free value f2/f1=2.756. Wanting a marimba-like ratio of 4, an apprentice argues: "A xylophone maker tunes the ratio by undercutting the bar, and undercutting removes material and makes the bar less stiff. So if I just plane the whole bar down uniformly, to say three-quarters of its original thickness along its entire length, the bar becomes floppier everywhere, the fundamental drops, and the ratio should move up towards 4." Is the apprentice right? Support your answer with the scaling of f1 and f2 with a uniform thickness reduction, and state what should be done instead.