= Solution
For a flexible uniform string with <linear mass density> $\rho$ and <tension> $T$, <dimensional analysis> gives $[T/\rho]=L^2T^{-2}$ and hence $c\propto\sqrt{T/\rho}$. The small-slope <force> balance $\rho\psi_{tt}=T\psi_{xx}$ fixes the factor to one:
$$
\boxed{c=\sqrt{T/\rho},\qquad \omega=ck.}
$$
Here $\rho$ has units kg/m, not the volume-density units used for a bulk fluid. The <phase> moves along the string at $c$, while a material point moves transversely with $\psi_t=A\omega\sin(kx-\omega t)$. Its maximum <speed> is \b[$A\omega$], which is not the <phase velocity>. Small slope $Ak\ll1$ makes $A\omega\ll c$ in this model.
The <kinetic energy> and extra elastic <energy> of a segment are
$$
K=\frac12\int\rho\psi_t^2\,dx,\qquad U=\frac12\int T\psi_x^2\,dx.
$$
The second expression follows from extra <length> $\sqrt{1+\psi_x^2}dx-dx\simeq\psi_x^2dx/2$ against the pre-existing constant <tension>. Time averaging a harmonic <wave> gives $\langle\mathcal E_K\rangle=\rho A^2\omega^2/4$ and $\langle\mathcal E_U\rangle=TA^2k^2/4=\rho A^2\omega^2/4$. Therefore
$$
\boxed{\varepsilon_{\rm av}=\frac12\rho A^2\omega^2.}
$$
At the driven endpoint the string pulls along its local tangent into $x>0$, with vertical component $T\psi_x$ for small slope. The point exerts the opposite <force> on the string, so the <driven-string endpoint power> is
$$
\boxed{P(0,t)=-T\psi_x(0,t)\psi_t(0,t).}
$$
For the outgoing <wave> $\psi=A\cos(kx-\omega t)$, at $x=0$ the <derivatives> are $\psi_x=Ak\sin\omega t$ and $\psi_t=-A\omega\sin\omega t$. Thus $P=TA^2k\omega\sin^2\omega t$ and
$$
\boxed{P_{\rm av}=\frac12TA^2k\omega=c\varepsilon_{\rm av}.}
$$
The <energy flux of a transverse string wave> follows by combining the <energy density> $\mathcal E=(\rho\psi_t^2+T\psi_x^2)/2$ and flux $P=-T\psi_x\psi_t$, which satisfy $\partial_t\mathcal E+\partial_xP=0$. For a pure right-going nondispersive <wave>, $P=c\mathcal E$ even instantaneously. <energy> is transported with the <wave>; a material string element merely oscillates about its unperturbed position.
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