Cubic speed-power law: where it breaks down
Where the cube law fails: the fitted exponent by segment and speed, the in-service versus calm-water dispute, and why merchant hulls never reach a hump.
The cubic speed-power law holds that delivered power varies with about the cube of ship speed, and it is the relation every slow-steaming saving is projected from. The exponent is a fitted quantity rather than a constant, and the published fits disagree in a way that matters to anyone forecasting a fuel saving at reduced speed.
Calm-water and design-condition fits cluster near 3 to 3.5 and rise above 4 for fast container ships operating over 20 knots. In-service regressions run much lower: Adland, Cariou and Wolff (2020), fitting noon reports from 16 crude oil tankers with a speed-dependent elasticity, reported values below 0.5 at low speed and near 2 at mid speed. Psaraftis and Lagouvardou (2023) argue those low figures are artefacts of treating speed and weather as independent regressors, and that calm-water resistance cannot carry an exponent below 2 on hydrodynamic grounds.
The full article will set out the resistance decomposition behind the exponent, the Froude number regime merchant hulls actually occupy (roughly 0.13 to 0.23, well clear of any residuary hump), the regression methods used on noon and high-frequency data, and what a practitioner should assume when the measured exponent is unknown. See slow steaming and ship resistance and powering .