A residual logarithmic term in low-frequency Chebyshev approximations of the finite-depth free-surface Green function: Closed form and finite-term correction

The frequency-domain free-surface Green function in finite water depth and its derivatives underpin boundary-element solvers for wave radiation and diffraction. In the formulation of Xie et al. (2022), P P = H G ̃ P and P M = H G ̃ M are approximated by Chebyshev polynomials for H = k 0 h ∈ [ 0,1 / 2 ] . For fixed R ≥ 0 and 0 < Z < 1 , we derive the exact ln H multiplier in P P using Mackay’s (2019) frequency reduction. As H → 0 + , the leading logarithmic contributions are − ( Z / 4 ) H ln H in P P and − ( 1 / 4 ) H ln H in ( P P ) Z , while P M has zero H ln H coefficient. These two quantities have published orders 18 and 15; no causal link is claimed. We subtract one or two leading logarithmic terms before interpolation and restore them after evaluation. In a 20-case evaluation-grid comparison at the sampled 1 0 − 6 criterion, our raw degree-18 reconstruction passed 0 / 20 cases for each quantity; degrees 18 and 10 were the first tested common degrees to pass all cases after the one- and two-term corrections, respectively. Against an independent high-precision John-series reference, the two-term variant also met the 1 0 − 6 mixed-error criterion at all eight noncoincident states of a representative upward-normal derivative kernel. The comparison was limited to the kernel level.