The theory has been developed for vibrations in long thin rods of piezoelectric barium titanate with the electric field parallel to the length. The theory shows how the coupling coefficient k23 and the piezoelectric constant d33 are related to a measurement of the fundamental resonant and antiresonant frequencies. The antiresonant frequency in this mode of vibration is controlled by the physical properties of the material in exactly the same way that these properties control the resonant frequencies in other modes. An experiment was devised, and the results of the experiment verified this result.

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