Somewhat taming the bump function's derivatives

Often I need to initialize some smooth pulse in a simulation. Today, for example, I'm working with acoustic and entropy pulses to test the effectiveness of nonreflecting boundary conditions for the Euler equations (see, e.g., Baum et al). Bump functions are nice for this purpose as they're infinitely smooth and have compact support.

Awhile back I realized that using just a classical bump function produced large gradients which had to be resolved. Years ago I used Mathematica to see if taking some power of the classical bump function, which remains infinitely smooth, could produce smaller gradients better suited for numerical use.

The punchline is that using the fourth power of the bump function, BumpPower[4] in the following parlance, is Goodness (TM). The following Mathematica notebook demonstrates the problem and discusses how one arrives at this empirical result: