Hankel–Bessel Ladder Identities Claim. For suitable \(f_m(r)\) and \(k>0\), \[ \mathscr{H}_{m+1}\!\left[\left(\partial_r-\frac{m}{r}\right) f_m\right](k) =-\,k\,\hat{f}_m(k). \tag{1} \] Here \(\hat f_m(k):=\mathscr H_m[f_m](k)=\displaystyle\int_0^\infty f_m(r)\,J_m(kr)\,r\,dr\).

We summarise Lamb’s treatment of a pressure disturbance on a free surface in a uniform stream, emphasising the contour‐integral evaluation of the far-field wavetrain, the underlying simple-harmonic free-surface condition, a Bessel–delta normalisation, and a geometric construction for envelopes of straight lines.

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 \(J_n(z)\).