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Assume $\kappa ,\omega >0$ and impose upward group velocity in $z>0$. The two positive-frequency components then have wave vectors ${\mathbf{k}}_1=(\kappa ,m)$ and ${\mathbf{k}}_2=(-\kappa ,m)$ with $m<0$. Equal lengths are automatic, and orthogonality gives $m^2={\kappa }^2$. HenceThe given orthogonal-wave condition is therefore possible only at this forcing frequency. The upward radiation condition excludes the $m=+\kappa$ branch.

Use superposition with vertical amplitude $W$ in each component and internal-wave polarization:The two sets of crests have slopes $+1$ and $-1$; their energy propagates upward and outward at angles of $45^{\circ }$, while their phases travel downward. Their interference makes fixed horizontal bands of vertical-velocity nodes, at $\cos \kappa x=0$, and horizontal-velocity nodes, at $\sin \kappa x=0$.

Put $A=\kappa x$, $B=\kappa z+\omega t$. The first-order displacement relative to mean parcel labels isThe horizontal displacement correction vanishes identically, whileConsequently the formal cross-wave Stokes drift isIt has alternating downward and upward columns, no depth decay in this ideal infinite plane-wave field, and zero horizontal-period-averaged vertical transport. Each wave alone has zero drift; the interference terms create this result.

This calculation is a kinematic second-order consequence of the linear field, not a complete prediction of mean transport. In particular, an impermeable oscillating lower boundary cannot have that nonzero mean normal parcel velocity. Its first-order displacement is ${\zeta }_b=(2W/\omega )\cos (\kappa x)\sin \omega t$. Expanding the exact kinematic boundary condition on $z={\zeta }_b$ givesThis Eulerian mean velocity cancels the computed normal Stokes drift at the material boundary. Away from it, the second-order mean-flow and buoyancy equations must also be satisfied. Thus the alternating drift is a valid formal cross-wave effect, but treating it alone as persistent transport with zero Eulerian mean would be physically incomplete. A prescribed permeable boundary would require its own second-order flux specification instead.