And Now For the Good Stuff !



sinplot
[8]
A one dimensional wave equation can describe a vibrating string with tension T



wave-eqn
where v is the wave velocity along the string


given certain, different bounary conditions this mehtod can explore  open strings and closed strings

For open strings there are two main boundary conditions
for Neumann boundary conditions the endpoint is free to move about but no momentum flows out
if the Dirichlet boundary conditions are considered -where the ends of the strings are held at specific locations on a manifold
the classical solution can be
fourier

this is the non-relativistic solution


for the relativstic case  a similar solution is produced
relative_fourier


modesum

with string mode action from Euler-Lagrange equations for a world sheet

actionb
[8]
Fixed string tensions represent the elementatry particles vibrating with thier own specific frequencies

The string tension in string theory is denoted by the quantity 1/(2 p a'), this is equal to the square of the string length scale.

This sets the length of the string at around the Planck lenth 10^-33 cm

These strings span out a 2-d space in time called world sheets ,(this is similar to 0-d  particles -world line)

sheets
[9]




it get's more exiting