Resonant ultrasound has found multiple applications, ranging from medical imaging to a variety of nondestructive testing techniques. Here focus is placed on nested cylinders. A series of ultrasonic measurements has been carried out in a geometry with an outer hollow steel cylinder, and a smaller, solid inner steel cylinder. In the interspace were placed several liquids. Ultrasound was introduced by a narrow contact transducer. A second set of narrow transducers was used as pickups and placed at angles ranging from 10 to 150 deg. The frequency of the ultrasound was varied from 1 to 5 MHz and showed a sequence of eight composite and sharp resonances spaced at about 0.5 MHz. The composite resonances showed a strongly skewed behavior as a function of frequency. The velocity potential has been modeled by using combinations of Hankel functions of complex argument. Convergence was achieved by using angular m's up to 150. These showed that the asymmetry of the composite resonance peaks is primarily a phase or geometrical optics effect.