Flume measurements of a one-dimensional sliding hump starting from rest in quiescence fresh water indicate that when the hump travels at speed less than the shallow-water wave celerity, three waves emerge, travelling in two directions. One wave travels in the opposite direction to the sliding hump at approximately the shallow-water wave celerity (backward free wave). Another wave travels approximately in step with the hump (forced wave), and the remaining wave travels in the direction of the hump at approximately the shallow-water wave celerity (forward free wave). These experiments were completed for a range of sliding hump speed relative to the shallow-water wave celerity, up to unity of this ratio, to investigate possible derivation from solutions of the Euler equation with non-linear and non-hydrostatic terms being included or excluded. For the experimental arrangements tested, the forced waves were negative (depression or reduced water surface elevation) waves while the free waves were positive (bulges or increased water surface elevation). For experiments where the sliding hump travelled at less than 80% of the shallow-water wave celerity did not include transient behaviour measurements (i.e. when the three waves still overlapped). The three wave framework was partially supported by these measurements in that the separated forward and forced waves were compared to measurements. For the laboratory scale experiments, the forward free wave height was predicted reasonably by the long-wave equation (ignoring non-linear and non-hydrostatic terms) when the sliding hump speed was less than 80% of the shallow-water wave celerity. The forced wave depression magnitude required the Euler equations for all hump speed tested. The long-wave solution, while being valid in a limited parameter range, does predict the existence of the three waves as found in these experiments (forward travelling waves measured quantitatively while the backward travelling waves visually by video). Nevertheless, the forced wave transient development required non-linear and non-hydrostatic terms for higher sliding hump speeds. The forward free wave, controversially, does not need non-linear and non-hydrostatic terms until much higher hump speeds, suggesting that the forward free wave falls in the parameter space where long-wave equations apply whereas the forced wave more often falls into the parameter space requiring non-linear and non-hydrostatic terms. It does raise the question of why the forced wave transient dynamics does not impact on the initial transient dynamics where the forward free wave is in the long-wave theory, escaping the forced wave (at least for speeds less than 80% of the shallow-water wave celerity).
Experiments in Fluids – Springer Journals
Published: Aug 19, 2011
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