The potential energy density in transverse string waves depends critically on longitudinal motion

The question of the correct formula for the potential energy density in transverse waves on a taut string continues to attract attention (e.g. Burko 2010 Eur. J. Phys. 31 L71), and at least three different formulae can be found in the literature, with the classic text by Morse and Feshbach (Methods of Theoretical Physics pp 126-127) stating that the formula is inherently ambiguous. The purpose of this paper is to demonstrate that neither the standard expression nor the alternative proposed by Burko can be considered to be physically consistent, and that to obtain a formula free of physical inconsistencies and which also removes the ambiguity of Morse and Feshbach, the longitudinal motion of elements of the string needs to be taken into account, even though such motion can be neglected when deriving the linear transverse wave equation. Two derivations of the correct formula are sketched, one proceeding from a consideration of the amount of energy required to stretch a small segment of string when longitudinal displacements are considered, and the other from the full wave equation. The limits of the validity of the derived formulae are also discussed in detail.