## 2 Governing Equations and Methods
### Rough Interfaces Model Generation  
In order to investigate the robustness of the wave phenomena and the practicalities of manufacturing specimens of the metamaterial in the future, accompanying studies of perturbed cross-shaped holes were performed. Surface roughness may be regarded as a random process that requires statistical techniques for characterisation [37]. Typically, a rough surface may be described by its deviation from a smooth reference surface. For example, the rough surface of a pipe would be described by height deviations from a smooth cylindrical surface, whereas the rough sea would be measured relative to a smooth plane. There are two important aspects of surface roughness to consider: the spread of heights above and below the mean reference surface and the lateral variation of these heights along the surface. Numerous statistical distributions and parameters have been used to describe these properties, including the root mean square (rms) height \(\delta\) and correlation functions [38], which are defined in two-dimensional space as follows:  
\[\delta=\sqrt{<h^{2}>},\ <h>\ =\ \int_{-\infty}^{\infty}h\ p(h)\,\mathrm{d}h\ =\ 0;\quad C(r)=\frac{<h(x_{1})h(x_{1}+r)>}{\delta^{2}}, \tag{8}\]  
where \(x_{1}\) is the coordinate direction along which the extent of the surface is defined, \(h\) is the variation in height from the mean line \(x_{2}=0\), \(p(h)\) is a probability density function, and \(<>\) denotes spatial averaging over the surface. The quantity \(r\) in (8) is the distance between any two points on the surface. A height correlation function \(C(r)\) describes the extent to which information about the height at one point on a surface determines, on average, the height at another point. A Gaussian correlation function is assumed here, following previous literature [38], but alternative distributions are also possible and investigated [37]. Note that with the definition in (8), \(C(0)=1\) and \(C(\infty)=0\) so that as distance \(r\) increases, \(C(r)\) decreases. The characteristic correlation length, \(\lambda_{0}\), is the distance over which \(C(r)\) falls to \(1/\mathrm{e}\). In additive manufacturing studies of surface roughness, an alternative measure of the variation of height about the reference surface is commonly used. For example, in the recent investigations [28; 29; 30], the arithmetic average of the absolute values of height profile deviations, \(R_{\mathrm{a}}\), was reported. The difference between \(R_{\mathrm{a}}\) and \(\delta\) is most easily understood by comparing their equations:  
\[R_{\mathrm{a}}=\frac{1}{L}\ \int_{0}^{L}\ |h(x)|\,\mathrm{d}x;\qquad\delta= \sqrt{\frac{1}{L}\ \int_{0}^{L}\ h(x)^{2}\,\mathrm{d}x}\,, \tag{9}\]  
where \(L\) is the length of the rough surface. Although the two measures are similar, in general, the rms value \(\delta\) will produce slightly higher values than \(R_{\mathrm{a}}\) - for example, in the event of a single large peak or deep valley. Recent investigations of the surface finish of additively manufactured parts made from alloys containing aluminium (for example, Ti6 Al4V, a titanium alloy stabilised with aluminium, and AlSi10 Mg) by [28; 29] indicate that the roughness parameter \(R_{\mathrm{a}}\) typically ranges from 2-25 \(\upmu\)m. For our study of the effect of roughness on the wave phenomena observed for the regular cross-shaped holes, randomly rough surfaces of length 0.3 mm (characterised by a value of \(\delta=5\) \(\upmu\)m, which is in the aforementioned range) were generated and used to define the perturbed crosses shown in Figure 4b. The rough surfaceswere generated by using a weighted moving average approach to correlate randomly generated height values, following the work of Ogilvy for ultrasonic nondestructive testing applications [38]. To ensure sufficient convergence and accuracy of the modelling results, the time-stepping of the FE simulation was adjusted to account for the addition of roughness in the models.