Both '99 323i mid-range speakers cut out sometimes

Both mid-range speakers (the big ones at the bottom of each front door) on my 1999 323i have intermittently stopped working. From the times when they are working I can hear that the speakers themselves still work, but sometimes (increasingly frequently, now) they just cut out. At first I thought this might be a broken wire, but the same speaker on each front door goes out at the same time -- which seems unlikely if it is a set (L/R channels) of speaker wires. The battery died recently, but the radio is otherwise functioning properly, just not the mid-range speakers, so I don't *think* it's a security lock-out problem. Any ideas?