# Potential Energy of wave in a string (mechanism wave)

In physics textbook the potential energy of a segment is often given by

$$\Delta U=F(dl-dx).$$

I know this is the work done to stretch the string. However, shouldn't we consider the work done that move this segment up and down under the action of wave?

Explicitly,

$$\Delta U=\int F_y ~dy$$

where $F_y$ is net force exerted on the segment.

• Comment to the post (v2): Is $x$ and $y$ supposed to be perpendicular coordinates? – Qmechanic Dec 23 '16 at 2:54
• Yes in the usual sense – Math The Novice Dec 23 '16 at 3:37
• Combined transversal and longitudinal displacements of the non-relativistic string are worked out in my Phys.SE answer here. – Qmechanic Dec 25 '16 at 21:57

• When we calculate gravitational potential we use $F={GMm \over r^2}$ Can we use same reasoning that \$F_y=T{d^2y \over dx^2}dx ? So the total potential would be this plus the potential of elongation – Math The Novice Dec 23 '16 at 3:44