= Solution
The symbol $\rho$ here is <linear mass density>, with units $\mathrm{kg\,m^{-1}}$. A short element $[x,x+\delta x]$ has transverse <force>
$$
F_y=T\sin\theta(x+\delta x)-T\sin\theta(x)
\simeq T[\Psi_x(x+\delta x,t)-\Psi_x(x,t)],
$$
because small slopes give $\sin\theta\simeq\tan\theta=\Psi_x$. Its <mass> is $\rho\delta x$. Applying <Newton's second law>, dividing by $\delta x$, and taking the limit gives the <wave equation on a string>,
$$
\rho\Psi_{tt}=T\Psi_{xx},\qquad
\boxed{\Psi_{tt}=v^2\Psi_{xx},\quad v=\sqrt{T/\rho}}.
$$
The derivation assumes constant positive <tension>, small transverse displacements and slopes, a flexible string with negligible bending stiffness, and negligible <damping>. Variations of <tension> due to the motion enter beyond the linear approximation.
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