On the solution of the initial-boundary problem in a half-strip for a hyperbolic equation with a mixed derivative

An initial-boundary problem for an inhomogeneous second-order hyperbolic equation in a half-strip of a plane with constant coefficients and a mixed derivative is studied. This problem describes transverse oscillations of a finite string with fixed ends. We introduce the notion of a classical solution of the initial-boundary problem, prove a uniqueness theorem for the classical solution, and obtain a formula for the solution in the form of a series whose terms are contour integrals containing the initial data of the problem. A definition of a generalized solution is given and finite formulas for this generalized solution are found.