Consider a spherically converging shock wave that was initiated by some means at a radial location  and directed towards the center. As the shock wave travels towards the origin, its strength increases since the shock wave compresses lesser and lesser amount of mass as it propagates. The shock wave location  thus varies with time. The self-similar solution to be described corresponds to the region , that is to say, the shock wave has travelled enough to forget about the initial condition.
Since the shock wave in the self-similar region is strong, the pressure behind the wave  is very large in comparison with the pressure ahead of the wave . According to Rankine–Hugoniot conditions, for strong waves, although , , where  represents gas density; in other words, the density jump across the shock wave is finite. For the analysis, one can thus assume  and , which in turn removes the velocity scale by setting  since .   
At this point, it is worth noting that the analogous problem in which a strong shock wave propagating outwards is known to be described by the Taylor–von Neumann–Sedov blast wave. The description for Taylor–von Neumann–Sedov blast wave utilizes  and the total energy content of the flow to develop a self-similar solution. Unlike this problem, the imploding shock wave is not self-similar throughout the entire region (the flow field near  depends on the manner in which the shock wave is generated) and thus the Guderley–Landau–Stanyukovich problem attempts to describe in a self-similar manner, the flow field only for ; in this self-similar region, energy is not constant and in fact, will be shown to decrease with time (the total energy of the entire region is still constant). Since the self-similar region is small in comparison with the initial size of the shock wave region, only a small fraction of the total energy is accumulated in the self-similar region. The problem thus contains no length scale to use dimensional arguments to find out the self-similar description i.e., the dependence of  on  cannot be determined by dimensional arguments alone. The problems of these kind are described by the self-similar solution of the second kind.
For convenience, measure the time  such that the converging shock wave reaches the origin at time . For , the converging shock approaches the origin and for , the reflected shock wave emerges from the origin. The location of shock wave  is assumed to be described by the function 
where  is the similarity index and  is a constant. The reflected shock emerges with the same similarity index. The value of  is determined from the condition that a self-similar solution exists, whereas the constant  cannot be described from the self-similar analysis; the constant  contains information from the region  and therefore can be determined only when the entire region of the flow is solved. The dimension of  will be found only after solving for . For Taylor–von Neumann–Sedov blast wave, dimensional arguments can be used to obtain 
The shock-wave velocity is given by
According to Rankine–Hugoniot conditions the gas velocity , pressure  and density  immediately behind the strong shock front, for an ideal gas are given by
These will serve as the boundary conditions for the flow behind the shock front.