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We show how the zeroth-order Hankel transform diagonalises the radial bi-Laplacian, clarify the sign convention \(\nabla^4 \leftrightarrow k^4\), and then apply the same transform machinery to an axisymmetric linear free-surface problem to obtain explicit representations for \(\eta(r,t)\) and \(\hat w(s,z,t)\).

We collect a few basic identities for the gamma and polygamma functions, their link to the Hurwitz zeta function, standard asymptotics for Bessel functions, and a useful Bessel product expansion and inequality for \(J_n(z)\).

We review the Eulerian continuity and Euler momentum equations, clarify the relationship between Eulerian and Lagrangian descriptions via the material derivative, and briefly connect these fundamentals to the physics of hydraulic jumps as transitions between supercritical and subcritical flow.