The Resonant Wine Barrel · AI Generator #GEN-37AF7DE3

Physics

Problem

A wine barrel maker discovers that when she taps the side of an empty cylindrical barrel, it produces a clear musical note. The barrel has radius R=0.40  mR = 0.40\;\text{m} and height h=1.20  mh = 1.20\;\text{m}. When tapped, the fundamental radial breathing mode is excited, where the circular cross-section oscillates uniformly in radius while maintaining its circular shape. In this breathing mode, the air pressure varies as p(r,t)=p0+AJ0(αrR)cos⁡(ωt)p(r,t) = p_0 + A J_0\left(\frac{\alpha r}{R}\right) \cos(\omega t), where J0J_0 is the zeroth-order Bessel function, α=2.405\alpha = 2.405 is the first zero of J0J_0, rr is the radial distance from the barrel's axis, and the boundary condition requires that the radial velocity of air at the barrel wall matches the wall's oscillation. The measured frequency of this breathing mode is f=142  Hzf = 142\;\text{Hz}. Using this information, determine the speed of sound in the air inside the barrel. Take the speed of sound in air at standard conditions to be v0=343  m/sv_0 = 343\;\text{m/s}.

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