  This article is devoted to investigate the singular profile of the free
boundary of two-dimensional incompressible inviscid fluid with external force
near the stagnation point. More precisely, given an external force with some
polynomial type decay close to the stagnation point, the singular profile of
the free boundary at stagnation point possible are corner wave, flat and cusp
singularity. Through excluding the cusp and flat singularity, we know the only
singular profile is corner wave singularity, and the corner depends on the
decay rate of the solution near the stagnation point. The analysis depends on
the geometric method to a class of Bernoulli-type free boundary problem with
given degenerate gradient function on free boundary. This work is motivated by
the significant work [E. V$\breve{a}$rv$\breve{a}$ruc$\breve{a}$ and G. Weiss,
Acta Math, 206, 363-403, (2011)] on Stokes conjecture to the incompressible
inviscid fluid acted on by gravity.
