  In weakly nonlinear systems, the main effect of cubic nonlinearity on wave
propagation is an amplitude-dependent correction of the dispersion relation.
This phenomenon can manifest either as a frequency shift or as a wavenumber
shift depending on whether the excitation is prescribed as a initial condition
or as a boundary condition, respectively. Several models have been proposed to
capture the frequency shifts observed when the system is subjected to harmonic
initial excitations. However, these models are not compatible with harmonic
boundary excitations, which represent the conditions encountered in most
practical applications. To overcome this limitation, we present a multiple
scales framework to analytically capture the wavenumber shift experienced by
dispersion relation of nonlinear monatomic chains under harmonic boundary
excitations. We demonstrate that the wavenumber shifts result in an unusual
dispersion correction effect, which we term wavenumber-space band clipping. We
then extend the framework to locally-resonant periodic structures to explore
the implications of this phenomenon on bandgap tunability. We show that the
tuning capability is available if the cubic nonlinearity is deployed in the
internal springs supporting the resonators.
