Hi, I have this question:

The longitudinal displacement u(x,t) for longitudinal vibrations of a beam of uniform cross-section satisfies the wave equation:

v$\displaystyle u_{xx}$ - p$\displaystyle u_{tt}$ = 0

where v is the modulus of elasticity of the beam and p is the mass density. For a beam of length L with the end x = 0 fixed and the end x = L free, the boundary conditions are:

u(0,t) = 0 , $\displaystyle u_x$(L,t) = 0.

Determine the natural angular frequencies of vibration of such a beam.

Using separation of variables u(x,t) = F(x)G(t) I get two decoupled first order equations:

F"(x) - (c/v)F(x) = 0 and G"(t) - (c/p)G(t) = 0.

I'm not sure where to go from here though. In all my examples from lectures, both ends have been fixed so the form of F(x) is known to be a function of sine and cosine. But I have no idea how to deal with the free end.

Please help,

Katy