In the paper, a two-degrees-of-freedom ship model with quadratic coupled pitch and roll modes under sinusoidal harmonic excitation is considered. The Multiple Scales perturbation technique is applied to yield the first-order expansions for the special internal resonant case where the pitch frequency is twice the roll frequency. Increasing the wave frequency from zero to infinity, five resonant situations are detected. For each case, the governing equations for the transition towards the steady-state solutions, the first-order approximations for these solutions and the frequency-amplitude relationships are presented. A detailed analysis is performed only for the case where the excitation frequency is half of the roll frequency. The reliability of the analytical results derived in the paper is checked in a companion contribution by comparison with numerical solutions.