While working with a piezoelectric element, I noticed some high frequency oscillations at 6kHz. I can't figure out why the piezo element has a resonant frequency. This is unlikely to be noise as there are setups that mention the self resonant frequency of the piezoelectric element. Where does the resonance come from? I'm guessing that it's the internal mechanical resonance?
Most practical piezoelectric elements in use today are piezoceramics and the physics of these materials have been carefully studied, modeled and engineered to predict how they will behave for use as either sensors or actuators; transducers in general.
American PiezoCeramics (APC) has been thoughtful enough to post on their website a summary modes of piezoceramics elements. If you link to that page you'll see that indeed there are multiple modes to consider that depend on the shape and dimensions of the element as well as the particular material.
The frequency you are seeing is probably a mode (resonance) being excited across another dimension in the element. You should be able to use the chart to explain the 6kHz. If I recall there are 'tricks' you can employ like mounting the element in epoxy that can isolate the specific mode you are trying to excite and attenuate others.