When discussing strings, it is useful to talk about tension. Consider a small piece of string from to , with
In the case of transverse oscillations, the longitudinal force is negligible and so it is only neccessary to compute the net vertical force. If we let be the tension, we find that for a small amount of vertical force
Since the length of the string is and if we denote the mass per unit length by , the total mass is . According to Newton’s second law,
This is a just wave equation describing a wave with transverse velocity .
To compute the action for a nonrelativistic string, we must find the Lagrangian, . In this case, the kinetic energy, is equal to the sum of the kinetic energies of all the infinitesimal pieces of the string:
The potential energy, can be thought of as the spring potential when a piece of string with ends at and is stretched by in the x-direction and in the y-direction. The change in length can be computed as
so our potential becomes
Our Lagrangian is therefore
and so our action becomes
By varying by and using the Euler-Lagrange equation,
we are able to get the equation of motion:
Close examination reveals this to be the wave equation.