According to Wikipedia entry on Waltzing Matilda, the tune was

based on the Scottish Celtic folk tune "Thou Bonnie Wood of Craigielea",[8] written by Robert Tannahill and first published in 1806, with James Barr composing the music in 1818.[10] In the early 1890s it was arranged as the "The Craigielee" march music for brass band by Thomas Bulch.[8] This tune itself was possibly based on the old melody of "Go to the Devil and Shake Yourself", composed by John Field (1782–1837) sometime before 1812. It is sometimes also called "When Sick Is It Tea You Want?" (London 1798) or "The Penniless Traveller" (O'Neill's 1850 collection)

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