An approximate theory is presented for torsional modes in a solid cylinder having a larger‐diameter central section. Modes in a narrow frequency range just above the cutoff of each branch are ‘‘trapped’’ such that, in long samples, the amplitudes decay exponentially with distance from the central section. To confirm the theory, measurements were performed on aluminum alloy cylinders using noncontacting electromagnetic‐acoustic transduction. Vibrational amplitudes of a trapped mode measured as a function of position along the length of a sample are in good agreement with theoretical calculations. The resonant frequencies of three samples with different dimensions also closely match the theory for trapped modes. Additional generally weak resonances that are observed may be associated with torsional modes that vary sinusoidally along the entire length of the samples.