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A sympathetic vibration is an induced resonant vibration. All objects have some natural frequency at which they vibrate (most have many such frequencies when one considers harmonics). External vibrations at this frequency will be more readily absorbed, and the object will begin to vibrate "in sympathy" with the inducing energy. An object that absorbs sufficient vibrational energy will shake apart. The term "sympathetic vibration" has generally given way to the more accurate and general term "resonance."
In 2267, Kirk and Spock employed this principle on Capella IV to delay their pursuers. They used a pair of tuned communicators to induce resonance in a rock face, causing it to fracture, and creating a rockslide. Several of their pursuers were killed or injured and the pass they hadd followed into a canyon was sealed, forcing their pursuers to detour to another entrance. (TOS: "Friday's Child")
In 2371, a band of Nausicaan raiders broke into the central museum by using an ultrasonic generator to create a sympathetic vibration in the webbing and shatter it. Odo later used a covariant oscillator to try and shatter a crystal that he believed was encasing an individual whom he believed to be Kira Nerys. (DS9: "Heart of Stone")