, have a continuous 
. Then 
. The kinetic
energy becomes: 
. So 
. This gives the wave equation directly.m
and 
. The lagrangian gives a solution
of the form 
. This gives a dispersion relation of the form: 
for
, you get 
. Then 
where 
= linear mass density, and 
= Young's
modulus.
Instead of having a discrete , have a continuous . Then . The kinetic
energy becomes: . So . This gives the wave equation directly.m