Interior-fold and endpoint transitions in finite-depth capillary-gravity ship wakes

Binghai Lei 

We study how finite water depth changes the principal stern caustics and wave patterns of linear capillary-gravity waves generated by a steadily moving pressure source on an inviscid fluid of uniform depth. Using the dispersion relation and ray directions relative to the source, we determine the conditions for caustic formation. Let H denote the water depth divided by the capillary length. No regular stern caustics exist when H is positive and no greater than the square root of 3. For H greater than the square root of 3, analytical results and numerical scans support a sequence of zero, two and one regular principal caustics in each stern half-plane as the source speed increases, under a weak-contribution approximation. The pair emerges at a finite wavenumber. At the long-wave speed, the lower branch reaches the zero-wavenumber endpoint, removing the transverse-divergent boundary while leaving the divergent-fan boundary regular. Local expansions quantify the wavenumber and angular separation near onset and the approach to the long-wave endpoint. Near the critical depth, the squared-speed width of the two-caustic regime decreases quadratically in (H squared minus 3). Gaussian pressure-source calculations show that the transverse contribution stretches and weakens before its interval closes, while the surviving divergent and fan contributions continue to interfere. Source weighting and relative phases determine which caustics are prominent and where the largest response occurs. These results connect the creation and loss of principal caustics to changes in the visible wake and distinguish caustic directions from maximum-amplitude directions.