Hexagonal 2D arrays of Au nanorods support discrete plasmon resonance modes at visible and near-infrared wavelengths when coupled with light at normal incidence (k(z)). Reflectance spectra of nanorod arrays mounted on a thin Au baseplate reveal multiple resonant attenuations whose spectral positions vary with nanorod height and the dielectric medium. Simulations using 3D finite-element method calculations reveal harmonic sets of longitudinal standing waves in cavities between nanorods, reminiscent of acoustic waves generated by musical instruments. The nodes and antinodes of these quarter-wave plasmon modes are bounded, respectively, at the base and tips of the array. The number of harmonic resonances and their frequencies can be adjusted as a function of nanorod height, diameter-spacing ratio, and the refractive index of the host medium. Dispersion relations based on these standing-wave modes show strong retardation effects, attributed to the coupling of nanorods via transverse modes. Removal of the metal baseplate is predicted to result in resonant transmission through the Au nanorod arrays, at frequencies defined by half-wave modes within the open-ended cavities.