  The wave equation $u_{tt} = c^2 u_{xx}$ is generally regarded as a linear
approximation to the equation describing the amplitude of a transversely
vibrating elastic string in the plane. But, as is shown in \cite{BC96}, the
assumption of transverse vibration in fact implies that the wave equation
describes the vibration precisely, with no need for approximation. We give a
simplified proof of this result, and we generalize to the case of an elastic
string vibrating (transversely or not) in a Riemannian surface $M$. In the more
general setting, the assumption of transverse vibration is replaced by the
assumption of "perfect elasticity," and we show that the wave map equation
$\nabla_{\bu_t} \bu_t = c^2 \nabla_{\bu_x} \bu_x$ gives a precise description
of the vibration of a perfectly elastic string in $M$, with no need for
approximation. Finally, we give examples describing the motion of various
vibrating strings in $\R^2$, $S^2$, and $\mathbb{H}^2$.
