On Friday, October 11, 1839, Francis Allyn Olmstead sailed on the whaler "North America" from the port of New London, returning in early February of 1841. This voyage took him to Hawaii and to Tahiti. He arrived in Tahiti on Thursday, September 10, 1840 (p. 271). This was about thirteen years before the "Julia Ann" put in there sometime between April and June of 1853.

http://books.google.com/books?id=sqxy9F9a9ggC&dq=Francis+Allyn+Olmstead&printsec=frontcover&source=bl&ots=azcwa7ziV8&sig=cEFPLm9

In a journal entry for February 11,1840, Olmstead mentions two sea shanties, and gives a verse and the music for each of them. One is a version of "What shall we do we a drunken sailor?" And the other is perhaps a version of "Heave 'er Away" (p. 115-116).

The fact that these songs were being sung on board a whaler rather than a merchant ship doesn't mean they couldn't have been sung on board of the "Julia Ann". It is important to remember that her first mate on the third and fourth voyages, Peter Coffin, who was "an old whaler of fifteen years experience on the Pacific Ocean." (Pond's memoirs). This would put him all the way back to the time of Olmstead's voyage.

Since we know that these two songs were being sung in the South Pacific prior to the voyages of the "Julia Ann" I would suggest that they "could" have been sung thirteen to fifteen years later on her trips from San Francisco to Sydney between 1853 and 1855.