Augustine. Exposition on the Book of Psalms. Schaff, Philip (editor). New York: Christian Literature Publishing Co, 1886. Nicene and Post-Nicene Church Fathers: Series 1: Volume VIII. Re-published 2014, Catholic Way Publishing. Kindle electronic edition, ISBN-13: 978-1-78379-372-3.
Augustine's understanding of Psalm 137:1 is that "'The waters of Babylon' are all things which here are loved, and pass away" (Augustine Psalms, loc. 341539, par. 2). He expands the idea to emphasize the passing away. Our momentary pleasures, then, would be compared to Babylon's waters. The response of others, who are more aware of "the holy Jerusalem" is to sit and weep (Augustine Psalms, loc. 341545, par. 3). The emphasis to Augustine is that of sitting by the streams but remaining on shore, rather than being washed away or drowned in the waters.
Though loss of worldly things does cause weeping, Augustine encourages rather a weeping for Zion, knowing we are separated from God's city (Augustine Psalms, loc. 341554, par. 4). Verse two speaks of hanging instruments on willow trees, a species of tree which Augustine sees as essentially worthless. The worthless trees tain their sustenance from the waters of Babylon, joys which pass away. Augustine takes the instruments as a means by which the truth could be told, but they too are idled in the Psalm (Augustine Psalms, loc. 341560, par. 5). He sees this as a sign of captivity. As we are bound by earthly delights, we dare not sing the powerful hymns (v. 3) which might even release us (Augustine Psalms, loc. 341565, par. 6).
Verse four confesses how hard it would be to sing God's songs in a land like Babylon. Augustine understands this as an allegory for speaking the truth in a setting which surrounds us with falsehood (Augustine Psalms, loc. 341569, par. 7). This is no acceptable way to live. Verse five emphasizes the great importance of remembering the truth of God. It would be a tragedy to forget Jerusalem (Augustine Psalms, loc. 341574, par. 8). Augustine observes that the enemies of God never forget their desire against God and His people. Those who trust in God must also hold fast to him (Augustine Psalms, loc. 341584, par. 9).
Verse seven turns to God in prayer against God's enemies, whom Augustine classes as "carnal persons" (Augustine Psalms, loc. 341589, par. 10). We need to be rescued from those who would hate and persecute us. Those who would attempt to destroy God's people were a present danger (Augustine Psalms, loc. 341599, par. 1).
In verse eight, then, the daughter Babylon is a pitiable sight. It will be repaid for its evil through calamity (Augustine Psalms, loc. 341604, par. 12). In verse nine Babylon suffers repayment in kind for her faithless actions and persecutions. She has no defense (Augustine Psalms, loc. 341614, par .12). Augustine takes the Rock against which the little ones, the sinful attitudes of Babylon, are dashed, to be Christ.
Augustine's application of the concept in Psalm 137 is that the work of the godly is to keep from hanging up instruments in an idle manner, but to speak and sing the truth, longing for restoration to Christ (Augustine Psalms, loc. 341619, par. 13). This puts the attitudes which draw us away from Christ to death.