Gravity waves over topographical bottoms: Comparison with experiment

The propagation of water surface waves over one-dimensional periodic and random bottoms is investigated by the transfer matrix method. For the periodic bottoms, the band structure is calculated, and the results are compared to the transmission results. When the bottoms are randomized, the Anderson localization phenomenon is observed. The theory has been applied to an existing experiment [M. Belzons et al., J. Fluid Mech. 186, 539 (1988)]. In general, the results are compared favorably with the experimental observation.

Figure 1

Conceptual layouts.

Figure 2

Situations considered in this paper, adopted from Fig. 2 of Ref. 7. (a) Bed P case: in this case, the steps are mounted periodically with lattice constant $2L_0$; the variation of the steps $\sigma H$ is fixed. (b) Bed RS case: in the case, the steps are allowed to move randomly from their initial periodic positions, as set in the bed P case—the allowed range is denoted as $\pm \Delta L$ and the variation of the steps $\sigma H$ is fixed. (c) Bed R: in this case, both the heights and the widths are allowed to vary randomly from their initial values in the bed P case within the ranges $[H_0-\Delta H,H_0+\Delta H]$ and $[L_0-\Delta L_0,L_0+\Delta L_0]$.

Figure 3

Band structure and transmission for the bed P case in two situations, referring to Fig. 2a: (a1) and (a2) The average water depth is $H_0=1.75\mathrm{cm}$, and the height variation is $\sigma H∕H_0=0.43$. (b1) and (b2) The average water depth is $H_0=3\mathrm{cm}$, and the height variation is $\sigma H∕H_0=0.25$. The left and right panes show the band structure and transmission results, respectively. The transmission is presented in the logarithmic scale for 100 steps.

Figure 4

Reflection versus frequency for the bed RS and P cases with three average water depths. Left panel: the results from the experiment [7]. Middle panel: the theoretical results with the average over five random configurations. Right panel: the theoretical results with the average over 10 000 random configurations, so to make sure the stability of the average.

Figure 5

Localization length versus frequency for the bed R case. The experimental results are shown in the inset, and the legends can be referred to in [7]; for example, the lines in the inset refer to the results from two other theoretical calculation [7]. The five black dots denote the results from an averaging over five random configurations.

Figure 6

Variation of the amplitude of wave elevation along the wave tank for the bed RS and bed R cases for different frequencies. The experimental [7] and numerical results are shown in the left and right panels, respectively.

Figure 7

The averaged variation of the amplitude of wave elevation along the wave tank for the bed RS from Fig. 6 with $f=1.6\mathrm{Hz}$. To show the behavior near the transmission site, the results are plotted in two length scales: (a1) up to $2\mathrm{m}$ and (a2) up to $20\mathrm{m}$.