  Pulsar timing experiments are currently searching for gravitational waves,
and this dissertation focuses on the development and study of the pulsar timing
residual models used for continuous wave searches. The first goal of this work
is to re-present much of the fundamental physics and mathematics concepts
behind the calculations and theory used in pulsar timing. While there exist
many reference sources in the literature, I try to offer a fully self-contained
explanation of the fundamentals of this research which I hope the reader will
find helpful. The next goal broadly speaking has been to further develop the
mathematics behind the currently used pulsar timing models for detecting
gravitational waves with pulsar timing experiments. I classify four regimes of
interest, governed by frequency evolution and wavefront curvature effects
incorporated into the timing residual models. Of these four regimes the
plane-wave models are well established in previous literature. I add a new
regime which I label "Fresnel," as I show it becomes important for significant
Fresnel numbers describing the curvature of the gravitational wavefront. Then I
give two in-depth studies. The first forecasts the ability of future pulsar
timing experiments to probe and measure these Fresnel effects. The second
further generalizes the models to a cosmologically expanding universe, and I
show how the Hubble constant can be measured directly in the most generalized
pulsar timing residual model. This offers future pulsar timing experiments the
possibility of being able to procure a purely gravitational wave-based
measurement of the Hubble constant. The final chapter shows the initial steps
taken to extend this work in the future toward Doppler tracking experiments.
