A method of solution of the boundary-value problem is proposed for the linearized Navier-Stokes equations in the half-plane. The Navier-Stokes system is linearized about the flow velocity vector with an exponential profile. The adhesion boundary condition is imposed at the infinite elastic beam whose deflection from the zero state is subject to the elasticity equation. The flow is supposed to be steady. After the Fourier-Laplace transform, a system with shifted argument is solved. The inverse Laplace transform leads to the dispersion equation and to the explicit solution of the initial system.