B737-300 CREW GOT ACFT BUFFET; RUMBLE; AND STICK SHAKER AT FL350; AFTER THE CAPT'S AIRSPD INDICATOR STUCK; IN ZSE CLASS A.

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FLT TO SEA. APCHING BTG VOR. VECTORED OFF RTE BY ZSE. HDG 260 DEGS. ALT FL350. DUE TO STACKING TFC; ATC ASKED US TO SLOW AS MUCH AS POSSIBLE. FO CHKED THE OPC AND DETERMINED THE LOW BUFFET SPD WAS 213; WT 110.3; TEMP -18. I SAID I DIDN'T WANT TO FLY THAT SLOW; BUT WOULD MAINTAIN AT LEAST 10-15 KTS ABOVE THAT SPD. I SLOWED THE ACFT TO 235-231 KTS. MY AIRSPD INDICATOR SHOWED THE ACFT WAS MAINTAINING THAT SPD WITH THE THROTTLE SETTING I HAD. THERE WERE NO OFF FLAGS OR ANY OTHER INDICATION THAT THE AIRSPD INST WAS NOT FUNCTIONING PROPERLY. AS SEA WAS 1600 FT OVCST; THE FO WAS READYING HIMSELF WITH HIS CHARTS AND PREPARING FOR THE APCH. THE ACFT STARTED TO RUMBLE. I PUSHED THE THROTTLES UP; AND THE RUMBLE CONTINUED. I INITIATED STALL RECOVERY BY DISCONNECTING THE AUTOPLT AND USING MAX THRUST ON THE THROTTLES. SECONDS LATER; ATC ASKED US TO TURN R DUE TO TFC AND MAINTAIN FL350. THE STICK SHAKER STARTED MOMENTARILY; AND I SAID WE WERE DSNDING TO FL310 AS THE ACFT WOULD NOT MAINTAIN FL350. THE FO IMMEDIATELY DECLARED AN EMER; AND TFC WAS MOVED TO ACCOMMODATE US. I SAID WE WERE DOING 235 KTS; AND THE FO SAID HE HAD 180 KTS. THAT'S WHEN WE REALIZED THE DISCREPANCY BTWN THE 2 AIRSPD INDICATORS. RECOVERY WAS COMPLETE BY FL345-FL343. ATC DIRECTED US TO DSND TO AND MAINTAIN FL310. MY AIRSPD INDICATOR WAS STUCK AT 235 KTS; BUT THE MACH WAS CORRECT VERSUS THE FO AIRSPD/MACH INDICATOR. WE CROSS-REFED THE STANDBY; FO AIRSPD AND THE HUD TO DETERMINE THE CORRECT AIRSPD INDICATOR. ONCE THE ACFT WAS UNDER CTL; WE TERMINATED THE EMER AND RAN THE 'UNRELIABLE AIRSPD' CHKLIST. I ELECTED TO HAVE THE FO FLY THE APCH AND LNDG. AT 4000 FT; MY AIRSPD INDICATOR UNWOUND AND PROCEEDED TO WORK PROPERLY. I NOTIFIED THE FLT ATTENDANTS OF WHAT OCCURRED; AND ASKED IF ANY PAX WERE UNDULY UPSET. THE FLT ATTENDANTS INDICATED THAT THE PAX APPEARED TO BE OK; AND THAT NO ONE COMPLAINED. I DECIDED AS LONG AS THEY WERE NOT UPSET; THAT I WOULD DEAL WITH THEM ON THE GND IF ANYONE DECIDED TO ASK QUESTIONS. LNDG WAS UNEVENTFUL. MAINT CAME AND RAN THEIR TESTS; AND FOUND THEY COULD NOT DUPLICATE THE PROB. MAINT CLRED THE LOGBOOK; AND WE PROCEEDED. SUPPLEMENTAL INFO FROM ACN 583923: I NOTICED A LOW RUMBLE THAT TURNED OUT TO BE THE LOW SPD BUFFET. I LOOKED AT THE THROTTLES AND NOTICED THAT THE CAPT HAD ADVANCED THEM TO FULL PWR. THE AIRSPD HAD DECREASED TO AROUND 180 KTS; AND WAS NOT INCREASING EVEN WITH FULL PWR. I THEN TOLD ZSE THAT WE HAD A PROB AND NEEDED AN IMMEDIATE DSCNT. AT THE SAME TIME; THE CAPT EXECUTED A STALL RECOVERY AND STARTED TO DSND. DURING THE DSCNT; THE STICK SHAKER ACTIVATED A COUPLE OF TIMES FOR A BRIEF PERIOD OF TIME. AFTER THE RECOVERY; WE NOTICED THAT THE CAPT'S AIRSPD INDICATOR WAS STUCK AT 235 KTS. THERE WERE NO FLAGS OR ANYTHING ELSE TO INDICATE THE UNRELIABILITY OF THE INDICATOR. THE MACH METER CONTINUED TO FUNCTION THROUGHOUT THE FLT.