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Abstract The radiation of steady surface gravity waves by a uniform stream$$U_{0}$$ $U_0$over locally confined (width$$L$$ $L$) smooth topography is analyzed based on potential flow theory. The linear solution to this classical problem is readily found by Fourier transforms, and the nonlinear response has been studied extensively by numerical methods. Here, an asymptotic analysis is made for subcritical flow$$D/\lambda > 1$$ $D/\lambda >1$in the low-Froude-number ($$F^{2} \equiv \lambda /L \ll 1$$ $F^2\equiv \lambda /L\ll 1$) limit, where$$\lambda = U_{0}^{2} /g$$ $\lambda =U_0^2/g$is the lengthscale of radiating gravity waves and$$D$$ $D$is the uniform water depth. In this regime, the downstream wave amplitude, although formally exponentially small with respect to$$F$$ $F$, is determined by a fully nonlinear mechanism even for small topography amplitude. It is argued that this mechanism controls the wave response for a broad range of flow conditions, in contrast to linear theory which has very limited validity. 

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $$\mu$$ , compares to the basic-state amplitude parameter, $$\epsilon \ll 1$$ . For $$\mu ={O}(1)$$ , the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $$\mu \ll 1$$ but $$\alpha =\mu /\epsilon \gg 1$$ , this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $$\mu$$ is further decreased holding $$\epsilon$$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $$\alpha ={O}(1)$$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $$\alpha ={O}(1)$$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.