How to create a sparse disk image for Time Machine backups

First create the sparse image:

sudo hdiutil create -size 500g -type SPARSEBUNDLE -fs HFS+ -volname TigerCat /Volumes/Passport/TigerCat.sparsebundle

Next tell Time Machine to add the disk to its list of backups:

sudo tmutil setdestination -p /Volumes/TigerCat