Fig. 2: Dispersion relations of the mass-and-spring model.

a Surface plot of real component of frequency \(\omega\) versus the real and the imaginary components of the wavenumber \(k\) following Eq. (5). For the conditions discussed in this paper, the angular frequency \(\omega\) is purely real. The wavenumber \(k\) is also purely real for a Bloch-periodic solution of an infinite periodic model. For a finite-thickness slab (see Fig. 1b), evanescent modes can play a role and the imaginary part of \(k\) is generally not zero. The imaginary part of the complex-valued angular frequency \(\omega\) is shown by the false-color scale. Only the positive parts of the real and imaginary components of the wavenumber are shown here as the corresponding negative parts can be obtained by mirror symmetry. The four highlighted black lines on the surface lead to purely real angular frequency \(\omega\). Among them, one corresponds to purely real wavenumber and the other three correspond to complex wavenumbers in the range of \({{{{{\rm{Re}}}}}}(k) > 0\). For a normalized frequency of \(\omega /{\omega }_{0}=1.0\), in between the local maximum and roton minimum, three real wavenumbers (see the three gray dots) can be obtained from the dispersion relation. For \(\omega /{\omega }_{0}=0.2\) below the roton minimum, a real wavenumber and a pair of complex conjugate wavenumber are obtained (see two yellow dots). Parameters are \({K}_{3}/{K}_{1}=1.0\) and the normalization frequency is \({\omega }_{0}=\sqrt{4{K}_{1}/m}\). b Parameters corresponding to the critical case without roton minimum in the dispersion relation, i.e., \(m=1\) and \({K}_{3}/{K}_{1}=1/3\). Note that still three solutions for the complex-valued \(k\) in the range of \({{{{{\rm{Re}}}}}}(k) > 0\) occur at a given angular frequency \(\omega\).