On the Uniqueness of the Approximate Analytical Solution of the Surf-Riding Threshold in the IMO Second Generation Intact Stability Criteria

The paper derives the Melnikov balance in the quadratic form τ0 n^2 + τ1 E[u] n + τ2 E[u^2] = (1/2π)∫ R(u) dy (its Eq. (41)) with τ0>0, τ2<0 and E[u^2]>0 (Eqs. (18), (20), (40)) and proves that, if (1/2π)∫ R(u) dy > τ2 E[u^2], there are two distinct real roots with exactly one positive (its Eq. (42)) . The model’s solution reproduces the same algebraic argument explicitly via c<0 and Δ>0, concluding uniqueness of the positive root and noting that R enters only through its average R̄, hence independence from the approximation of R—matching the paper’s statement that the conclusion does not depend on how R(u) is approximated . The paper further provides a physical monotonicity rationale using Eq. (34) that also yields uniqueness of a positive solution (Sections 5–6, Figs. 2–3), which is an additional perspective rather than a contradiction . Overall, both are correct and essentially the same proof at the algebraic level, with the paper adding a physical interpretation.