Therearesomecurvedinterfacesinoceanacousticwaveguides.Tocomputewavepropagationalongtherangewithsomemarchingmethods,aflatteningoftheinternalinterfacesandatransformingequationareneeded.Inthispaperalocalor-thogonalcoordinatetransformandanequationtransformationareconstructedtoflatteninterfacesandchangetheHelmholtzequationasasolvableform.Forawaveguidewithaflattop,aflatbottomandncurvedinterfaces,thecoefficientsofthetrans-formedHelmholtzequationaregiveninaclosedformulationwhichcanbethoughtofasanextensionoftheformalworkrelatedtotheequationtransformationwithtwocurvedinternalinterfaces.Inthetransformedhorizontallystratifiedwaveguide,theone-wayreformulationbasedontheDirichlet-to-Neumann(DtN)mapisthenusedtoreducetheboundaryvalueproblemtoaninitialvalueproblem.NumericalimplementationoftheresultingoperatorRiccatiequationusesalargerangestepmethodtodiscretizetherangevariableandatruncatedlocaleigenfunctionexpansiontoapproximatetheoperators.Thismethodisparticularlyusefulforsolvinglongrangewavepropagationproblemsinslowlyvaryingwaveguides.Furthermore,themethodcanalsobeappliedtowavepropagationproblemsinacousticwaveguidesassociatedwithvarieddensity.
