A B757'S UPDATED FMC INCORPORATING ACARS FUNCTIONS HAS NO SECONDARY MSG ALERTING FUNCTION WHEN OPERATING IN A NAV MODE. A NORDO ACFT CANNOT BE ALERTED ABOUT AN ATC CONTACT FREQUENCY VIA AN ACARS MSG IN THE NAV MODE.

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AT CRUISE; FL370; WE SENT AN ACARS MESSAGE FOR ZZZ2 ATIS AND DISCOVERED THAT THERE HAD BEEN SEVERAL ACARS MESSAGES TO CONTACT ATC ON VARIOUS FREQS. BOTH PLTS HAD BEEN IN THE FMS MODE AND THERE HAD BEEN NO CHIME TO BRING OUR ATTENTION TO ACARS. WE ALSO DID NOT GET ANY SELCAL ALERTING US TO CALL ANYONE. NO CONTACT WAS HEARD ON GUARD 121.5. WE FINALLY GOT IN CONTACT WITH ZZZ1 AFTER TRYING SEVERAL FREQS AND BEING TOLD WE WERE ON THE INCORRECT FREQ AND TO RETURN TO OUR LAST ASSIGNED. WE THEN GOT AN ACARS MESSAGE ASKING FOR MY SPOUSE'S NAME AND ADDRESS FOR PLT AUTHENTICATION. AFTER LOOKING UP PLT AUTHENTICATION IN THE MANUAL TO SEE WHAT IT WAS; I SENT THE REQUESTED INFO. ZZZ1 CTR THEN RELAYED A PHONE NUMBER FOR US TO CONTACT ZZZ WHEN WE GOT DOWN AND WE GOT ANOTHER ACARS MESSAGE TO CONTACT FLT CTL AFTER THE FLT. THE REMAINDER OF THE FLT WAS UNEVENTFUL. I MADE A LOGBOOK WRITE UP FOR THE CHIME AS BEING INOP. IN SUMMARY; I FOUND OUT LATER THAT IF NEITHER PLT IS IN THE ACARS MODE AND A MESSAGE IS RECEIVED; THERE WILL BE NO CHIME. THIS IS A FLAW IN THE EQUIP. SOME OTHER MEANS SHOULD HAVE BEEN EMPLOYED TO CONTACT US; IE; SELCAL; ACFT RELAY ON LAST KNOWN FREQ. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE REPORTER STATED THAT HIS FLT WAS NOT GIVEN A FREQUENCY CHANGE AND DID NOT REALIZE THAT THEY WERE NORDO. WHEN ACARS WAS A STAND ALONE TECHNOLOGY A CREW WAS ALERTED IN ONE OF SEVERAL WAYS WHEN AN INCOMING MSG WAS AVAILABLE FOR VIEWING. THE NEW COMBINED FUNCTIONALITY IN THIS FMC REQUIRES SPECIFIC MODE SELECTIONS IN ORDER TO BECOME AWARE OF A RECEIVED MSG. IN THE REPORTER'S ACFT NO SECONDARY ALERTING METHOD IS PRESENT. THE REPORTER WAS TOLD THAT THE ACR COULD ISSUE A SPECIAL OVERRIDING MSG TO ACTIVATE A COCKPIT CHIME BUT THAT DID NOT OCCUR EVEN THOUGH THE FLT'S DISPATCHER SAID THAT WAS DONE. THE REPORTER BELIEVES THAT WITH THE NEW MULTIFUNCTION FMC NO SECONDARY ALERTING METHOD EXISTS AND SO AT LEAST ONE CREWMEMBER MUST REMAIN IN THE ACARS MODE IN ORDER TO RELIABLY KNOW THAT MESSAGES HAVE BEEN SENT TO THEIR ACFT.