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DYNAMICAL OCEANOGRAPHY
7.5. Exercises on chapter 7
(7.1) Standing waves: closed basin
Consider one-dimensional gravity waves with a sea surface amplitude η(x, t)
and velocity u(x, t) in a basin with length L and in a water layer of depth H;
assume rotation is absent.
a. Show that the equations (7.4) reduce to
∂η
∂t
+ H
∂u
∂x
=0;
∂u
∂t
+ g
∂η
∂x
=0
b. Show that both u and η satisfy wave equations with wavespeed C =
√ gH.
c. What are the boundary conditions at x =0and x = L?
d. Show that the solutions of these equations can be written as
η(x, t)=
∞
n=1
A n cos k n x cos σ n t
u(x, t)=
C
H
∞
n=1
A n sin k n x sin σ n t
where the A n are constants and
k n =
nπ
L
; σ n =
nπC
L
(7.2) Standing waves: half open basin
Consider the same situation as in the previous exercise, but now with an open
boundary at x = L.
a. What is the boundary condition at x = L in this case?