Adam Clarke on Psalms 79:11

The sighing of the prisoner - The poor captive Israelites in Babylon, who sigh and cry because of their bondage.

Those that are appointed to die - בני תמותה beney themuthah, "sons of death." Either those who were condemned to death because of their crimes, or condemned to be destroyed by their oppressors. Both these senses apply to the Israelites: they were sons of death, i.e., worthy of death because of their sins against God; they were condemned to death or utter destruction, by their Babylonish enemies.