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 an 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.

在弱非线性系统中，三次非线性对波传播的主要影响是色散关系的幅度相关校正。这种现象可以表现为频率偏移或波数偏移，具体取决于激励是分别规定为初始条件还是边界条件。已经提出了几种模型来捕获当系统受到谐波初始激励时观察到的频移。然而，这些模型与代表大多数实际应用中遇到的条件的谐波边界激励不兼容。为了克服这一限制，我们提出了一个多尺度框架来分析捕获谐波边界激发下非线性单原子链的色散关系所经历的波数位移。我们证明了波数偏移会导致异常的色散校正效应，我们将其称为波数空间带限幅。然后我们将框架扩展到局部共振周期结构，以探索这种现象对带隙可调性的影响。我们表明，如果在支持谐振器的内部弹簧中部署三次非线性，则可以使用调谐能力。