Image-Based Fourier Reconstruction and Analysis of Free-Surface Wave Motion in a CNC-Excited Rectangular Reservoir

Free-surface sloshing in partially filled tanks and open reservoirs can produce spatially complex surface deformation under horizontal excitation. This study presents an image-based experimental and computational workflow for reconstructing measured free-surface profiles in a laboratory-scale rectangular reservoir mounted on a Haas VF-3/VF-3SS-family CNC machining center. The tank underwent a programmed horizontal back-and-forth motion. The retained CNC controller record shows commanded x positions spanning approximately $-19.976$ mm to $+19.851$ mm, corresponding to a programmed peak-to-peak stroke of approximately 39.8 mm. The realized forcing frequency was not independently measured; therefore, no experimental near-resonance condition is claimed. Videos were recorded approximately normal to the transparent tank wall, converted to still frames, calibrated directly in physical length units, and manually digitized with WebPlotDigitizer. A truncated spatial Fourier series was fitted independently to each measured time slice. The accompanying Python applications dynamically evaluate the experimentally fitted representation: the GUI tools animate reconstructed profiles over the recorded time sequence, while the final FourierN3 implementation builds time-dependent coefficient models, linearly interpolates coefficient histories, evaluates $y(x,t)$, and generates a three-dimensional space–time reconstructed surface. These outputs are described as dynamic Fourier reconstructions because they remain conditioned on measured coefficient histories and do not constitute independent CFD or potential-flow predictions. The Fourier harmonics are mathematical reconstruction functions and are not identified one-to-one with physical sloshing eigenmodes. For a 0.45 m tank length, linear theory gives first-mode reference frequencies of 0.687 Hz at a 4 cm water depth and 0.763 Hz at a 5 cm water depth. The workflow is demonstrated on three selected datasets that provided sufficiently clear free-surface records for consistent frame-by-frame digitization. Using a common three-harmonic benchmark, mean RMSE values were 0.8815 cm for Case A (4 cm), 1.419 cm for Case B (5 cm), and 0.5022 cm for Case C (4 cm), corresponding to mean NRMSE values of 22.0%, 28.4%, and 12.6%, respectively. For Case C, adding a fourth harmonic reduced the mean RMSE to 0.4427 cm (11.1% of water depth), an 11.8% reduction relative to the three-harmonic fit. All RMSE and NRMSE values are reconstruction errors relative to the manually digitized coordinates, not estimates of absolute experimental measurement accuracy. The results therefore demonstrate the reconstruction workflow for the three selected records rather than establish a general water-depth effect.