# One dimensional lattice dispersion relation

I'm doing an experiment, in which I have a one dimensional lattice held up by strings. That is I have a series of n masses each of mass M each connected to each other by springs with spring constant C and unstretched length a. Each mass is suspended from the ceiling by a string of length L. I'm trying to find a relation that relates the angular frequency to the wave number for transverse oscillations. I found one for longitudinal oscillations, w^2=(g/L)+(4C/M)(sin(ka/2)^2, where w is the angular frequency and k is the wave number. For transverse oscillations things become more difficult. I know the relation has the form w^2=(g/L)+(4D/M)(sin(ka/2)^2, where D is a constant that involves C and is smaller than C. I'm guessing D has the form D=B(a/L)C, where B is a constant. You have to setup the equation of motion for transverse oscillations and assume a solution of the form y(x,t)=exp(i(k(na+x)-wt), where y is the transverse displacement. From there you make a bunch of approximations to get an equation of the above form. I've spent a lot of time working on this. I'll post more of the work I've done, when I have more time. I attached a pdf file with more info to the post.

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