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,
$$
\boxed{J_n(x,0^+)=-M\alpha\partial_zg(x,0^+)=\frac{M\sigma|q|^3}{2\phi_B}\eta\sin(qx)}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 2 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-344/2/d/solution