Radiation analysis of sound waves from semi-infinite coated pipe

An analytical solution is presented for the problem of radiation of sound waves from a semi-infinite circular cylindrical coated pipe which is partially lined from inside. By stating the total field in duct region in terms of normal waveguide modes (Dini's series) and using the Fourier transform technique elsewhere, we obtain a Wiener-Hopf equation whose solution involving three sets of infinitely many unknown expansion coefficients satisfying three systems of linear algebraic equations. This system is solved numerically and the influence of some parameters (pipe radius, impedances, extension, etc.) on the radiation phenomenon is displayed graphically.