The subject of a monohull surface vessel rolling in longitudinal waves (head seas) is introduced. The discussion includes a treatment of the rigid-body dynamics behind coupled roll and pitch motions. Simplified non-linearly coupled roll-heave-pitch differential equations of motion including inertial, damping, and stiffness terms are developed. Direct and parametric excitation terms are also discussed. It is shown that inertial dynamic coupling, non-linear stiffness coupling, and parametric excitation can all contribute to rolling in longitudinal waves. The inertial dynamic coupling is shown to result from a product of inertia term appearing in the differential equations of motion. Results obtained from the analysis include frequency amplitude response plots for roll, heave, and pitch motions, as well as plots illustrating a saturation phenomenon and a transfer of energy between the pitch and roll modes that explains the presence of roll in certain purely head seas situations. Recommendations for naval architects and mariners are provided to guide design, weight distribution, and loading decisions in hopes of avoiding situations where head seas can drive significant and potentially dangerous roll motions.

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