  In this work, we investigate the shape identification and coefficient
determination associated with two time-dependent partial differential equations
in two dimensions. We consider the inverse problems of determining a convex
polygonal obstacle and the coefficient appearing in the wave and Schr\"odinger
equations from a single dynamical data along with the time. With the far field
data, we first prove that the sound speed of the wave equation together with
its contrast support of convex-polygon type can be uniquely determined, then
establish a uniqueness result for recovering an electric potential as well as
its support appearing in the Schr\"odinger equation. As a consequence of these
results, we demonstrate a uniqueness result for recovering the refractive index
of a medium from a single far field pattern at a fixed frequency in the
time-harmonic regime.
