Summary Elastic lateral dynamic impedance functions are defined as the ratio of the lateral dynamic force/moment to the corresponding lateral displacement/rotation at the top ending of a foundation at very small strains. Elastic lateral dynamic impedance functions have a defining influence on the natural frequencies of offshore wind turbines supported on cylindrical shell type foundations, such as suction caissons, bucket foundations, and monopiles. This paper considers the coupled horizontal and rocking vibration of a cylindrical shell type foundation embedded in a fully saturated poroelastic seabed in contact with a seawater half‐space. The formulation of the coupled seawater–shell–seabed vibration problem is simplified by treating the shell as a rigid one. The rigid shell vibration problem is approached by the integral equation method using ring‐load Green's functions for a layered seawater‐seabed half‐space. By considering the boundary conditions at the shell–soil interface, the shell vibration problem is reduced to Fredholm integral equations. Through an analysis of the corresponding Cauchy singular equations, the intrinsic singular characteristics of the problem are rendered explicit. With the singularities incorporated into the solution representation, an effective numerical method involving Gauss–Chebyshev method is developed for the governing Fredholm equations. Selected numerical results for the dynamic contact load distributions, displacements of the shell, and lateral dynamic impedance functions are examined for different shell length–radius ratio, poroelastic materials, and frequencies of excitation. Copyright © 2016 John Wiley & Sons, Ltd.