Separating the frequency and amplitude in the strengths of colliding plane gravitational waves

The usual wave equation, being a linear second-order ordinary differential equation, typically involves two arbitrary constants. Physically, these constants correspond to the amplitude and frequency of the waves. However, it is not immediately clear whether “nonlinear wave equations” must also have two constants. Interestingly, the exact solutions of the nonlinear Einstein Field Equations for plane and cylindrical gravitational waves do indeed involve two constants, which are naturally assumed to have the same physical significance. In the original solutions for colliding plane gravitational waves — specifically, the “impulsive” solutions by Khan and Penrose and the “sandwich” solutions by Szekeres — two constants are also present. Although we might have expected four constants (two for each wave), only two are involved. Recently, a solution for “colliding plane gravitational waves of arbitrary strengths” was derived and was thought to have two constants. However, upon closer examination following up a comment by Roger Penrose, this solution actually involves three constants. We demonstrate how the frequency and amplitude of the two colliding plane gravitational waves are coded into the three constants, by considering the momentum imparted to test particles in this spacetime.