= Solution
The interface is material: fluid particles on $F=z-\eta(x,t)=0$ remain there. Its <material derivative> therefore vanishes, giving $w_j=\eta_t+u_j\eta_x=D_j\eta/Dt$ on the interface. Since $w_j=\partial_z\phi_j$, this is the <kinematic boundary condition for a free-surface graph> on each side. The material derivatives use the respective tangential fluid velocities; they enforce the same moving normal boundary.
The <Unsteady Bernoulli equation> gives $p_j+\rho_j[\partial_t\phi_j+|\nabla\phi_j|^2/2+gz]=C_j(t)$. Absorb the spatially constant functions $C_j$ into the gauges of the <velocity potentials>. Evaluating at $z=\eta$ and subtracting the equations yields the stated <dynamic boundary condition for an inviscid interface>. <Surface tension> supplies the pressure jump, rather than pressure continuity. Thus \b[the first condition enforces material motion, and the second enforces the Bernoulli pressure balance].
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