  We consider the following degenerate half wave equation on the one
dimensional torus $$\quad i\partial_t u-|D|u=|u|^2u, \; u(0,\cdot)=u_0. $$ We
show that, on a large time interval, the solution may be approximated by the
solution of a completely integrable system-- the cubic Szeg\"o equation. As a
consequence, we prove an instability result for large $H^s$ norms of solutions
of this wave equation.
