= Solution
An element with unperturbed coordinate length $dx$ has mass $\rho\,dx$ and transverse speed squared $\dot\psi_y^2+\dot\psi_z^2$. Its actual length is
$$
ds=\sqrt{1+(\psi_y')^2+(\psi_z')^2}\,dx
\simeq\left[1+\frac12((\psi_y')^2+(\psi_z')^2)\right]dx.
$$
At fixed <tension> the extra <elastic energy> to this order is $F(ds-dx)$. Subtracting it from the <kinetic energy> gives
$$
\boxed{L=\int dx\left[\frac\rho2(\dot\psi_y^2+\dot\psi_z^2)
-\frac F2((\psi_y')^2+(\psi_z')^2)\right].}
$$
Here prime means $\partial_x$ and dot means $\partial_t$. The approximation retains quadratic transverse energy and neglects higher-order slope effects and longitudinal motion. The resulting field equations are $\rho\ddot\psi_i=F\psi_i''$ for $i=y,z$, with <wave speed> $\sqrt{F/\rho}$ when $F>0$.
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