= Solution
For $h=\eta\sin(qx)$, the linearized mean curvature is
$$
K=(\partial_x^2+\partial_y^2)h=-q^2\eta\sin(qx).
$$
The boundary value in the upper phase is therefore
$$
g(x,0)=\frac{\sigma q^2\eta}{2\alpha\phi_B}\sin(qx).
$$
The decaying harmonic extension into $z>0$ is
$$
g(x,z)=\frac{\sigma q^2\eta}{2\alpha\phi_B}e^{-|q|z}\sin(qx).
$$
With the normal directed into the upper half-space,
$$
\boxed{J_n(x,0^+)=-M\alpha\partial_zg(x,0^+)=\frac{M\sigma|q|^3}{2\phi_B}\eta\sin(qx)}.
$$
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