Curved-Stern Internal Waves with Applied Pressure in a Finite-Depth Two-Layer Fluid

The wave train generated by a moving vessel remains a persistent challenge in ship hydrodynamics, particularly in stratified oceans where internal waves contribute significantly to drag. This research investigates steady interfacial wave generation by flow past a semi-infinite curved plate in a two-layer fluid where both layers are of finite depth. Previous studies were limited by flat-plate geometries and infinite-depth approximations, neglecting the critical role of applied pressure as a design variable. We address these limitations to facilitate the engineering goal of wave drag reduction through optimal hull and pressure configurations. Using a scalar Wiener–Hopf formulation, we construct an analytical solution where the kernel is factorized via Cauchy-type integrals, bypassing the technical constraints of traditional infinite-product expansions. We derive a closed-form far-field amplitude and a wave-free condition, M(μ)=0. This condition is a single complex equation, i.e., two independent real constraints. A one-parameter hull design can therefore generally only minimise the downstream wave amplitude; exact cancellation at linear order generally requires a second, independent real design freedom, such as a jointly optimised applied pressure. We quantify both regimes explicitly. Our findings demonstrate that physical stern flow patterns exhibit spatial relaxation and advection scales essential for energy transfer. This result provides a mathematically sharp framework for wave drag minimisation, quantifying the explicit performance trade-off between hull curvature and dynamic pressure control in stratified environments, and we are careful throughout to distinguish the formal linear theory cancellation from what is achievable with a single physically realisable (real-valued) design parameter.