Hydrostatic lower bound for a travelling-pressure response ($a\ge \frac{p_1}{\rho g}$)

For the subcritical travelling-pressure response of finite-depth surface gravity waves, put $c_{\ast }^2=g\tanh (kh)/k$. When $0\le c<c_{\ast }$, the amplitude isThe inequality follows immediately because the denominator belongs to $(0,1]$. Equality holds precisely at $c=0$, where hydrostatic equilibrium gives $\rho g\eta =-p_1\cos (kx)$. If only strictly positive $c$ is allowed, $p_1/(\rho g)$ is the infimum, approached as $c\downarrow 0$, and is not an attained minimum. This bound applies to the travelling forced component, rather than an arbitrary superposition with free surface gravity waves.