Leonhard Euler’s derivation of the water-hammer equations in 1775

The present paper describes the first known appearance of the water-hammer equations in the archival history. The equations were derived in an article by Leonhard Euler that may have been submitted for a prize competition in 1742, was presented to the Petersburg Academy in 1775, was partially published in 1862, and was finally completed with a newly discovered missing fragment in 1979. Study of this historical work reveals a significant scientific and mathematical breakthrough that underpins much of late 19th and early 20th century developments in transient hydraulics. Here, the history of this remarkable achievement is traced, and the work is placed in historical context as it relates to the theory of fluid transients today.