  In this paper we propose and analyze a class of simple Nystr\"om
discretizations of the hypersingular integral equation for the Helmholtz
problem on domains of the plane with smooth parametrizable boundary. The method
depends on a parameter (related to the staggering of two underlying grids) and
we show that two choices of this parameter produce convergent methods of order
two, while all other stable methods provide methods of order one. Convergence
is shown for the density (in uniform norm) and for the potential postprocessing
of the solution. Some numerical experiments are given to illustrate the
performance of the method.
