A simple analysis predicts the Helmholtz and higher modes of cylindrical resonators. Formulas for the quarter-wave tube, the closed tube, and the classic Helmholtz resonator are special cases. It is found that the classic Helmholtz formula begins to lose accuracy for resonator lengths L≳λ/16. A corrected Helmholtz equation is given as an approximation to the exact result. The analysis requires values of inside and outside orifice end corrections. Rayleigh’s model of a piston in an infinite wall and Ingard’s calculations for a piston radiating into a tube give satisfactory results when the theory is compared to experiments. Resonator experiments were performed with k0L in the range π/8 to 3π/8 and orifice to cavity diameter ratios from 0.1 to 0.8. Subject Classification: 20.40.