In this work, we carried out a theoretical analysis of the reflection, transmission and surface deformation of long linear water waves, propagating on a submerged breakwater whose cross-section obeys a cycloidal geometric transition. We use the well-known one-dimensional governing equations for the propagation of linear shallow-water waves, which are presented in their dimensionless version. Considering harmonic wave propagation, the governing equations can be reduced to a dimensionless second-order differential equation with variable coefficients for predicting the elevation of water waves. This equation is solved using a matrix method based on Taylor polynomials. In particular, we evaluate the reflection and transmission coefficients for three breakwaters, namely, cycloidal, semi-cycloidal and quarter-cycloidal. The first case exhibits the smallest transmission coefficient. The present mathematical model is compared with a simple numerical solution and with another analytical solution expressed in terms of the Legendre functions. The present mathematical model can be used as a practical reference for design of the geometrical configurations of submerged cycloidal breakwaters under shallow-flow conditions.