  The determination of the polarization modes of gravitational waves (GWs) and
their dispersion relations is a crucial task for scrutinizing the viability of
extended theories of gravity. A tool to investigate the polarization states of
GWs is the well-known formalism developed by Eardley, Lee, and Lightman (ELL)
[Phys. Rev. D 8, 3308 (1973)] which uses the Newman-Penrose (NP) coefficients
to determine the polarization content of GWs in metric theories of gravity.
However, if the speed of GWs is smaller than the speed of light, the number of
NP coefficients is greater than the number of polarizations. To overcome this
inconvenience we use the Bardeen formalism to describe the six possible
polarization modes of GWs considering general dispersion relations for the
modes. The definition of a new gauge-invariant quantity enables an unambiguous
description of the scalar longitudinal polarization mode. We apply the
formalism to General Relativity, scalar-tensor theories, $f(R)$-gravity, and a
wide class of quadratic gravity. We derive an explicit relation between a
physical observable (the derivative of the frequency shift of an
electromagnetic signal), and the gauge-invariant variables. Then we find an
analytical formula for the pulsar timing rms response to each polarization
mode. To estimate the sensitivity of a single pulsar timing we focus on the
case of a dispersion relation of a massive particle. The sensitivity curves of
the scalar longitudinal and vector polarization modes change significantly
depending on the value of the effective mass. The detection (or absence of
detection) of the polarization modes using the pulsar timing technique has
decisive implications for alternative theories of gravity. Finally,
investigating a cutoff frequency in the pulsar timing band can lead to a more
stringent bound on the graviton mass than that presented by ground-based
interferometers.
