About energy equations of unidimensional transverse waves Studying the unidimensional waves, at least 3 books, and a lot of address on the web used the following idea (at least i think that it happened):
In order to describe the rate of change of potential energy per width on the string, is assumed the tension and density to be constant as the vibration goes on, also that the amplitude are small, the medium is continuous, and possibility another things, until here everthing is ok, the problem is that to describe this they use a small amount of the wire saying it to have a mass $dm$,..., but when use the mechanical relations, like work, they use the tension on only one side, why not use the tension on both sides? like in the way "to get" the wave equation?

 A: The tension T is the force force (per unit area) acting along the string across all cross sections, you can think of it to correspond to the pull on the string on one end compensated by the equal and opposite pull at the other end. This also holds for the unperturbed string element, which therefore doesn't experience a net force. When you extend a string element from length dx to length ds at constant T, you perform a work (per unit area) of T(ds-dx). You can think of this being accomplished by pulling on one end of the string element with force T and holding the other fixed or by pulling on both ends of the string element with opposing T. The work done is the same as long as the sum of extensions at both ends is ds-dx.  This work corresponds to the gain in potential energy of the string element when deformed from the unperturbed state. For the derivation of the wave equation, you have to consider the forces acting on a perturbed small string element of length dx and mass dm  in order to be able to apply Newton's 2nd law F = dm·a to it. To obtain the total force on the dm element you have to add vectorially both forces (of equal absolute value T but different directions) at both ends of the element with the appropriate angles. Therefore the forces on both ends of the string element are needed in the derivation of the wave equation.
