A theoretical study is made of steady, subcritical (Froude number F 1) two-dimensional free-surface flow due to a uniform stream flowing over smooth, locally confined bottom topography of large horizontal extent (L 1) and finite peak height (= O (1) ). In earlier work, this flow was analysed based on the nonlinear shallow-water equations which neglect the effects of dispersion altogether. This so-called hydraulic theory predicts a steady disturbance confined in the vicinity of the topography if is below a critical value ₂ₑ₈ₓ (F). The present asymptotic analysis of the full potential flow equations focuses on how dispersive effects (controlled by = 1/L 1) influence this steady state, particularly in regard to a steady short-scale radiating wave downstream that is ignored by hydraulic theory. Utilizing exponential asymptotics, it is shown that as is increased this dispersive wave, whose amplitude is formally exponentially small with respect to, grows sharply and ultimately it becomes comparable with the hydraulic wave disturbance when approaches ₂ₑ₈ₓ. Thus, the nonlinear shallow-water equations break down in the vicinity of ₂ₑ₈ₓ regardless of 1. The asymptotic results are supported by numerical solutions of the full potential flow theory, which also reveal a limiting, ₋₈₌ ₂ₑ₈ₓ, above which steady wave responses cannot be computed. For just below ₋₈₌, the downstream wave resembles a steep steady Stokes periodic wave, while for slightly above ₋₈₌ unsteady computations suggest that the downstream disturbance steepens and breaks.
Kataoka et al. (Mon,) studied this question.