The early stage of two-dimensional jet flow generated by an impulsive start of a wedge, initially floating on a free surface otherwise flat, is investigated taking into account liquid compressibility. During this short initial stage the flow close to the intersection points is self-similar. Velocity of the body is assumed much smaller than the sound speed in the liquid at rest. The acoustic solution of this problem reveals that the flow velocity is singular about the intersection points. In order to obtain a uniformly valid description of the flow, an inner solution is derived by using stretched variables, which are dependent on the Mach number of the problem. It is shown that, for small Mach numbers, the inner flow is approximately potential and is governed by the Laplace equation. The solution of the boundary-value problem is achieved numerically through an iterative procedure. A modified velocity potential, which significantly simplifies the boundary conditions on the free surface, is introduced. Accurate solutions are presented in terms of free surface shapes and jet lengths for different dead-rise angles of the wedge. The analogy between this problem and that of jetting flow caused by shock wave impact on a wedge-shaped cavity is discussed as well.