  Vocal fold (VF) motion is fundamental to voice production and diagnosis in
speech and health sciences. The motion is a consequence of air flow interacting
with elastic vocal fold structures. Motivated by existing lumped mass models
and known flow properties, we propose to model the continuous shape of vocal
fold in motion by the two dimensional compressible Navier-Stokes equations
coupled with an elastic damped driven wave equation on the fold cover. In this
paper, instead of pursuing a direct two dimensional numerical simulation, we
derive reduced quasi-one-dimensional model equations by averaging two
dimensional solutions along the flow cross sections. We then analyze the
oscillation modes of the linearized system about a flat fold, and found that
the fold motion goes through a Hopf bifurcation into temporal oscillation if
the flow energy is sufficient to overcome the damping in the fold consistent
with the early models. We also analyze the further reduced system under the
quasi-steady approximation and compare the resulting vocal fold equation in the
small vibration regime with that of the Titze model. Our model shares several
qualitative features with the Titze model yet differs in the specific form of
energy input from the air flow to the fold. Numerical issues and results of the
quasi-one-dimensional model system will be presented in part II (view resulting
web VF animation at http://www.ma.utexas.edu/users/jxin).

Pattern Formation and Solitons