Show that the function

describes a (plane) gravity wave propagating in the x̂ direction in a uniform ocean of depth h. In particular, show that ψ(x, t) satisfies eq. (31.7)

and the boundary condition that s_z = 0 at z = −h. (See Appendix B.4.2 for properties of the hyperbolic functions cosh x and sinh x, and in particular, for their expansions near x = 0.) Show that for the limiting cases kh ≪ 1 and kh ≫ 1 this correctly reproduces eq. (31.22)

 and eq. (31.9)

For waves of fixed period, graph H/H₀, λ/λ₀ as functions of depth, where H₀, λ₀ are the wave height and wavelength in the limit of infinite depth.