= Solution
The <tension> term $Tw_{xx}$ is the transverse restoring force from a nearly straight, tensioned rod. The bending term $-EIw_{xxxx}$ is the force from Euler-Bernoulli flexure, where $E$ is <Young's modulus> and $I$ the <second moment of area>. The right side is transverse <inertia>. In the displayed one-dimensional equation $\rho$ denotes <linear mass density>; if it denotes volumetric <density> instead, the inertial coefficient must include cross-sectional area.
For a <Fourier mode>, $w_{xx}=-k^2w$, $w_{xxxx}=k^4w$ and $w_{tt}=-\omega^2w$, giving
$$
\boxed{\rho\omega^2=Tk^2+EIk^4.}
$$
For $k>0$ the <phase speed> is $c_p=\omega/k=\sqrt{T/\rho+EIk^2/\rho}$, strictly increasing with $k$ when $EI>0$ and $T\geq0$. Thus shorter <wavelengths> propagate faster in <phase>. The <group velocity> $c_g=(T+2EIk^2)/\sqrt{\rho(T+EIk^2)}$ is also increasing, so the statement is consistent with wave-packet transport.
At a simple support the <displacement> and <bending moment> vanish: $w=0$ and $EIw_{xx}=0$ at $x=0,L$. The <eigenmodes> are $\sin(n\pi x/L)$ for $n=1,2,\ldots$. Setting $T=0$ in the <dispersion relation> therefore yields
$$
\boxed{\omega_n=\sqrt{\frac{EI}\rho}\frac{n^2\pi^2}{L^2}.}
$$
The integer is positive for a nonzero mode; $n=0$ gives the identically zero sine function, and negative integers repeat the same modes. <Bending stiffness> makes the frequencies quadratic in mode number, unlike a purely tensioned string.
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