Integrating wave energy converters into coastal structures such as breakwaters, seawalls or jetties not only offers benefits in terms of construction costs but also improves wave energy extraction. In this paper a novel theoretical model based on linear potential flow theory is developed to study the performance of an oscillating water column (OWC) integrated into a vertical structure in water of finite water depth. The model has three fundamental advantages relative to previous works: no thin-wall restriction (the thickness of the OWC chamber wall is considered), no singularities, and far fewer truncating terms in the eigen-function expansions. The OWC chamber is a vertical cylinder semi-embedded in the structure with a submerged, open bottom. As water waves impinge on the structure, the water column in the chamber oscillates and drives an air turbine installed at the chamber top to extract wave power. Using linear wave theory, the velocity potential in the water domain is decomposed into scattering and radiation potentials whose unknown coefficients are determined by the eigen-function matching method. Upon successful validation, the model is used to investigate the influence of the thickness of the chamber wall and the radius and submergence of the chamber on wave power absorption.