What is Radial and Thickness Mode Vibration in a Piezo Electric Ceramic Disc Transducer?

There is an excellent white paper from piezotechnologies.com which is probably worth a read if you are new to working with piezoelectric devices. Quoting from the section on resonance frequencies:

If an AC voltage is applied to a piezoceramic disc with a diameter several times its thickness, two series of resonances may be observed. The resonance at the lowest frequency corresponds to motion in the largest dimension of the ceramic, which in this case is in the radial direction. Often, several harmonics will also be seen at higher frequencies.

The first thickness resonance appears quite different from the radial harmonics that are found just below it. The impedance drops to a much lower value and the span in frequency is far wider. The thickness mode frequency often appears much “rougher” than the radial mode. This is because the harmonics of the radial mode are excited by the strong thickness mode.

So, the radial mode is lower in frequency because it has a larger dimension. Since it is also the lowest frequency resonance it is much cleaner because it does not coincide with any other harmonics. See the figure below.

The axial (thickness) mode on the other hand has a stronger resonance since the piezo-crystal is designed to be driven in that direction. In addition, it overlaps with some of the higher harmonics of the radial mode and therefore has a 'dirtier' appearance. See the figure below.

The full spectrum showing the narrower, weaker radial mode and its resonances together with the axial mode puts the relative broadness and strength of the axial mode in perspective.