  In this article, we study the ill-posedness of a quasilinear wave equation.
It was shown by Tataru and Smith in 2005 that for any $s>7/4$ (or $11/4$ in our
situation), the equation is well-posed in $H^{s}\times H^{s-1}$. We show a
sharpness result by exhibiting a quasilinear wave equation and an initial data
such that the Cauchy problem is ill-posed for in $H^{11/4} (\ln
H)^{-\beta}\times H^{7/4} (\ln H^{-\beta})$.
