Two waveguide trifurcation problems

We consider the diffraction of the dominant acoustic wave mode which propagates out of the mouth of a semi-infinite waveguide made of a soft and hard half plane. This semi-infinite waveguide is symmetrically located inside an infinite waveguide whose infinite plates are soft and hard. The whole system constitutes a trifurcated waveguide. Another trifurcated waveguide is obtained by interchanging the infinite plates. A closed form solution of the resulting matrix Wiener-Kopf equation is obtained for each configuration. Thus we present exact closed-form solutions to two new waveguide trifurcation problems.