When the cone angle is very small, the flow is nearly parallel everywhere in which case, an exact solution can be found, as shown by Theodore von Kármán and Norton B. Moore in 1932. The solution is more apparent in the cylindrical coordinates  (the  here is the radial distance from the -axis, and not the density). If  is the speed of the incoming flow, then we write , where  is a small correction and satisfies
where  is the Mach number of the incoming flow. We expect the velocity components to depend only on , i.e.,  in cylindrical coordinates, which means that we must have , where  is a self-similar coordinate. The governing equation reduces to
On the surface of the cone , we must have  and conesequently .
In the small-angle approximation, the weak shock cone is given by . The trivial solution for  describes the uniform flow upstream of the shock cone, whereas the non-trivial solution satisfying the boundary condition on the solid surface behind the shock wave is given by
We therefore have
exhibiting a logarthmic singularity as  The velocity components are given by
The pressure on the surface of the cone  is found to be  (in this formula,  is the density of the incoming gas).