  We give a detailed treatment of electromagnetic signals generated by
gravitational waves (GWs) in resonant cavity experiments. Our investigation
corrects and builds upon previous studies by carefully accounting for the gauge
dependence of relevant quantities. We work in a preferred frame for the
laboratory, the proper detector frame, and show how to resum short-wavelength
effects to provide analytic results that are exact for GWs of arbitrary
wavelength. This formalism allows us to firmly establish that, contrary to
previous claims, cavity experiments designed for the detection of axion dark
matter only need to reanalyze existing data to search for high-frequency GWs
with strains as small as $h\sim 10^{-22}-10^{-21}$. We also argue that
directional detection is possible in principle using readout of multiple cavity
modes. Further improvements in sensitivity are expected with cutting-edge
advances in superconducting cavity technology.
