  A gauge-invariant formulation for the gravitational wave equations is
presented. Using this approach, weak, plane wave solutions in a vacuum are
derived in various theories. These include general relativity with two modes of
polarization with helicity $\pm 2$, Yang's theory with three modes of
polarization with helicity $\pm 2$ and $0$, and so-called "general metric
theories" with six modes of polarization with helicity $\pm 2$, $\pm 1$, and
two $0$'s. To identify the polarizations of gravitational waves, it is
explicitly demonstrated how the gauge-invariant approach reproduces the earlier
results.
