We can find a the eigenfunction which satisfy
[math]\displaystyle{ \partial_x^4 w_n = \lambda_n^4 w_n }[/math]
plus the edge conditions.
[math]\displaystyle{ \begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L,
\end{matrix} }[/math]
[math]\displaystyle{ \begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm L.
\end{matrix} }[/math]
This solution is discussed further in Eigenfunctions for a Free Beam.
[math]\displaystyle{
\frac{1}{2}
xx
}[/math]
