 String sigma model in an isotropic plane wave

In conformal gauge and light-cone gauge in string theory $X^{+}=\kappa \tau$, the transverse part of the string sigma model in a spacetime metric reduces toThe omitted longitudinal term is a total time derivative. Each transverse coordinate obeys a massive wave equation. If the spatial period is $L$, the longitudinal momentum is $p^{+}=T\kappa L$ in this normalization. Keeping the gauge slope explicit prevents missing a factor of $p^{+}$ when changing conventions.