energy density using Rowland's notation:
http://web.mit.edu/fluids-modules/www/basic_laws/1-1-LagEul.pdf
File:Transverse string wave longitudinal motion.svg
u = τ 0 ( ξ ′ + 1 2 η ′ 2 ) {\displaystyle u=\tau _{0}\left(\xi '+{\tfrac {1}{2}}\eta '^{2}\right)}
If ℓ = 1 {\displaystyle \ell =1} , I think I get:
u = κ T ( ξ ′ + 1 2 η ′ 2 ) {\displaystyle u=\kappa _{T}\left(\xi '+{\tfrac {1}{2}}\eta '^{2}\right)}
