Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 16C / c

Table of contents
- Solution c

Solution

At

z = 0

, the two free-surface conditions for a normal mode give

- i σ_{n} ϕ + g ζ = 0, - i σ_{n} ζ - \partial_{z} ϕ = 0.

Eliminating

ζ

yields

σ_{n}^{2} ϕ = g \partial_{z} ϕ .

For the vertical factor found in part (b),

\frac{\partial _{z} ϕ}{ϕ}_{z = 0} = k_{n} tanh (k_{n} h) .

Thus the gravity-wave dispersion relation is

σ_{n}^{2} = g k_{n} tanh (k_{n} h) = \frac{g}{h} Ψ (k_{n} h),

where

Ψ (x) = x tanh x .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version