External USB drive unmounts spontaneously

Hi, all,

My wife just got a brandy new MacBook Air (running Yosemite, of course), and she finds that her trusty old USB Time-Machine backup drive keeps unmounting itself in the middle of the backup. She has yet to complete a single successful backup. We've tried reformatting and repartitioning the drive (reluctantly, since we would rather have preserved the backups from her previous machine) , but that made no difference. I've plugged the drive into my ancient MacBook Pro (also running Yosemite), and so far it seems to be working, although it still has about 16 hours to go before it completes a backup, so I can't tell for sure. Another forum that I checked mentioned that there were several similar posts here in Apple Support Communities, but I can't seem to find them. Anyone got a suggestion about what we might do?