Stokes Waves in a Fluid of a Finite Depth

Ali Aghayev (McMaster)

Monday, March 02, 11:30 (HH403)

This work explores the formulation of free surface Euler equations using conformal variables to derive Babenko’s equation for a fluid of finite depth h0. By considering two-dimensional 2π-periodic waves, we establish the relationship between physical depth and conformal depth through the mean value of the surface elevation η. The study utilizes a Lagrangian functional with Lagrange multipliers to preserve constraints and derive the system's equations of motion. Key highlights include:

• • The derivation of travelling wave reductions where steady solutions with constant speed c lead to the Babenko formulation.

  • The construction of small-amplitude (Stokes) expansions for the wave profile η and the dispersion relation c2.

  • A comparison between expansions in conformal depth versus physical depth, identifying specific discrepancies in existing literature regarding high-order asymptotic analysis.