B737 CREW INBOUND FOR DCA ARE CLRED FOR AN LDA RWY 19 APCH. WHEN UNABLE TO PICK UP COURSE SIGNAL THEY ARE ADVISED THEY WERE CLRED ON THE ROSSLYN LDA.

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ON THE PUBLISHED ARR INTO DCA RWY 19; APCH CTL ADVISED US TO EXPECT A VISUAL TO RWY 19. ON THE DSCNT; WE WERE STABLE AND CONFIGURED EARLY AS WE EXPECTED TO BE HIGH ON THE APCH; WHICH IS NORMAL. IN THIS CASE THOUGH; OUR VECTORS WERE FURTHER OUT THAN NORMAL SO OUR APCH WOULD BE NORMAL. AS WE GOT CLOSER TO THE LDA COURSE; APCH CTL ADVISED US THAT WE WERE CLRED FOR THE LDA RWY 19 APCH. THIS WAS THE APCH THAT WE HAD ANTICIPATED. AT THIS TIME WE WERE JUST BECOMING VISUAL ON OUR DSCNT. THE CAPT TRIED TO IDENT THE LDA; BUT IT WOULDN'T IDENT AND; WHEN WE SAW WE WERE GOING TO GO THROUGH THE COURSE; I DISCONNECTED THE AUTOPLT AND MANEUVERED THE ACFT BACK TO THE LDA COURSE WHICH STILL WASN'T ACTIVE. AT THIS POINT WE HAD THE FIELD IN SIGHT AND WERE ON PROFILE. THE CAPT ASKED APCH CTL ABOUT THE LDA AND WAS INFORMED FOR THE FIRST TIME THAT WE WERE NOW CLRED FOR THE ROSSLYN LDA WHICH IS A DIFFERENT APCH AND LDA FREQ. WE WERE VISUAL AND SET THE NEW FREQ AND WERE SWITCHED TO THE TWR. AT THIS POINT AT ABOUT 4 MI OUT; THE TWR ASKED WHY WE WERE ON THE LDA COURSE AND SAID WE WERE CLRED FOR THE PUBLISHED VISUAL APCH TO RWY 19. WOW; TALK ABOUT A LOT OF CONFUSION ON THE PART OF ATC/TWR/US. WE PROCEEDED TO JOIN THE RIVER VISUAL AND HAD AN UNEVENTFUL LNDG. AFTER LNDG; WE DISCUSSED WHAT HAD TRANSPIRED AND CAME TO THE CONCLUSION THAT ATC HAD PROBABLY THOUGHT THEY HAD CLRED US FOR THE ROSSLYN LDA; WHEN WHAT THEY SAID WAS JUST CLRED FOR THE LDA; WHICH IS A DIFFERENT APCH. ALSO; APCH CTL FAILED TO INFORM THE TWR THAT WE WERE ON THE LDA APCH AND NOT A VISUAL. ALSO; ON THE FLT DECK WAS AN FAA INSPECTOR; WHO SAID WE DID A GREAT JOB AND THAT THE APCH CTLR WAS PROBABLY GETTING REPRIMANDED AS WE SPOKE. THIS SIT COULD HAVE BEEN AVOIDED IF WE HAD JUST BEEN GIVEN THE VISUAL LIKE WE WERE ADVISED. THAT SAID; THERE WAS DEFINITELY A COM BREAKDOWN WHICH I AM SURE WE ALL LEARNED SOMETHING FROM.