  To expand the narrow response bandwidth of linear point wave energy absorbers
(PWAs), a few research studies have recently proposed incorporating a bi-stable
restoring force in the design of the absorber. Such studies have relied on
numerical simulations to demonstrate the improved bandwidth of the bi-stable
absorbers. In this work, we aim to understand how the shape of the bi-stable
restoring force influences the effective bandwidth of the absorber. To this
end, we use perturbation methods to obtain an approximate analytical solution
of the nonlinear differential equations governing the complex motion of the
absorber under harmonic wave excitations. The approximate solution is validated
against a numerical solution obtained via direct integration of the equations
of motion. Using a local stability analysis of the equations governing the slow
modulation of the amplitude and phase of the response, the loci of the
different bifurcation points are determined as function of the wave frequency
and amplitude. Those bifurcation points are then used to define an effective
bandwidth of the absorber. The influence of the shape of the restoring force on
the effective bandwidth is also characterized by generating design maps that
can be used to predict the kind of response behavior (small amplitude periodic,
large amplitude periodic, or aperiodic) for any given combination of wave
amplitude and frequency. Such maps are critical towards designing efficient
bi-stable PWAs for known wave conditions.
