  We construct an explicit model of gravity trapping domain wall potential
where for the first time we can study explicitly the graviton wave function
fluctuations for any thickness of the domain wall. A concrete form of the
potential depends on one parameter 0 <= x <= pi/2, which effectively
parameterizes the thickness of the domain wall with specific limits x -> 0 and
x -> pi/2 corresponding to the thin and the thick wall, respectively. The
analysis of the continuum Kaluza Klein fluctuations yields explicit expressions
for both the small and large Kaluza Klein energy. We also derive specific
explicit conditions in the regime x > 1, for which the fluctuation modes
exhibit a resonance behaviour, and which could sizably affect the modifications
of the four-dimensional Newton's law at distances, which typically are by four
orders of magnitude larger than those relevant for Newton's law modifications
of thin walls.
