= Simply supported beam normal modes
{title2=$\omega_n=(n\pi/L)^2\sqrt{EI/\rho_\ell}$}
A uniform <elastic beam> with endpoint conditions $y=y_{xx}=0$ has <normal modes> $\sin(n\pi x/L)$ for positive integers $n$. Substitute $y=Xe^{-i\omega t}$ into $EIy_{xxxx}+\rho_\ell y_{tt}=0$ and impose the four endpoint conditions to obtain the displayed spectrum. Here $\rho_\ell$ is <mass> per unit <length>, $I$ is a <second moment of area>, and $EI$ is <bending stiffness>. <frequencies> proportional to $n^2$ distinguish <flexural waves> from the nondispersive ideal stretched string.
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