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Figure 3: Waveguides with hard boundary walls, operated near the higher-order (2,0) mode cut-off of the intermediate channel. We can see that the higher-order mode matches the phase pattern of a soft-hard (1,0) mode (from Figure 2) in the center of the channel. This presents an “effective soft boundary” along the planes where the phase flips by π. This flip results from a change in sign of the pressure, which is a purely real-valued standing wave. (a) Numerical results for transmission phase and amplitude. The design parameters are similar to the experimental setup, with only the length being modified (a = 0.450 m, b = 0.382 m, and d = 12.6 mm). (b) Phase distribution through the CNZ channel near the (2,0) cut-off frequency for hard-hard configurations of different lengths. For bottom figure: ftunneling (i) = 762.9 Hz, ftunneling (ii) = 763.4 Hz, and ftunneling (iii) = 764.8 Hz. (c) Spatial variation of phase at the CNZ tunneling frequency under 90° and 180° bends. Here, a = 0.450 m, b = 0.382 m, ftunneling_90 (i) = 764.40 Hz, and ftunneling_180 (ii) = 757.38 Hz.