  In this study a new approach to the problem of transverse vibrations of an
ideal string is presented. Unlike previous studies, assumptions such as
constant tension, inextensibility, constant crosssectional area, small
deformations and slopes are all removed. The main result is that, despite such
relaxations in the model, not only does the final equation remain linear, but,
it is exactly the same equation obtained in classical treatments. First, an
"infinitesimals" based analysis, similar to historical methods, is given.
However, an alternative and much stronger approach, solely based on finite
quantities, is also presented. Furthermore, it is shown that the same result
can also be obtained by Lagrangian mechanics, which indicates the compatibility
of the original method with those based on energy and variational principles.
Another interesting result is the relation between the force distribution and
string displacement in static cases, which states that the force distribution
per length is proportional to the second spatial derivative of the
displacement. Finally, an equation of motion pertaining to variable initial
density and area is presented.
