Details of the proof of equivalence : the “ cosine ” versus “ vertical eigenfunction ” representations

The boundary-value problem for the linear horizontally-forced sloshing problem can be solved using two different classes of eigenfunction expansions. The first will be referred to as the ”cosine” expansion since the organizing centre is a cosine series in the x−direction (the horizontal direction), and the second is called the “vertical eigenfunction expansion” since the organizing centre is a class of y−direction (the vertical direction) eigenfunctions. These two methods lead to very different forms of solution, and their equivalence is not obvious. This technical report gives the details of a proof of equivalence. The strategy of the proof was suggested to the authors by McIver [7].