<p>The static and dynamic analysis of beams and plates resting on or embedded within an elastic half-space has attracted increasing attention due to their widespread applications in civil, mechanical, and geotechnical engineering. Achieving reliable and accurate designs requires either improving existing calculation methods or developing new approaches capable of more effectively capturing their static and dynamic behavior. This study investigates the dynamic behavior of rectangular plates resting on the surface of an elastic half-space with inertial properties according to Lamb’s model. A semi-analytical approach based on the Zhemochkin method is employed. This method combines the Ritz technique, used to evaluate plate deflections, and Green’s function for computing the surface displacements of the elastic half-space. The plate–foundation system is discretized into identical rectangular elements, whereby the continuous contact is replaced by partial contacts established at the centers of the elements while ensuring continuity of contact between them. The canonical equations incorporate all relevant parameters, including the mechanical and geometric properties of the plate and the foundation, reactive forces, inertial effects, plate deflections, and vertical surface displacements of the elastic half-space. After incorporating the relevant formulas, the mathematical transformations yield a matrix formulation enabling the determination of the reactive forces in the contact zone. Using the principles of elasticity theory, further quantities such as eigenfrequencies, natural modes, and dynamic responses under various external excitations are obtained. The accuracy and applicability of the proposed approach are demonstrated through validation against the modal superposition method and comparison with results obtained from Boussinesq’s model and the finite element method. The proposed methodology provides a versatile and efficient framework for the dynamic analysis of plate–foundation systems.</p>