Free waves
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b. Show (again with separation of variables) that the solutions can be written
as
η(x, t)=
∞
n=1
cos k n x(A n sin σ n t + B n cos σ n t)
u(x, t)=
C
H
∞
n=1
sin k n x(−A n cos σ n t + B n sin σ n t)
where A n and B n are constants and determine the k n and σ n .
Suppose that the flow is forced at the open boundary by a tide with amplitude
η(L, t)=η 0 cos ωt
Assume that the sea-surface is at rest for t =0.
c. Determine the total solution of this forced problem for t>0.
(7.3) Phase versus group velocity
For Poincar´ e waves, the phase speed is given by (7.14). The group velocity is
defined as:
C g =
∂σ
∂k
∂σ
∂l
a. Determine the group velocity for Poincar´ ewaves.
b. Are these waves dispersive?
(7.4) Kelvin wave
At midlatitudes, Kelvin waves move along coasts.
a. In which direction (north or south) propagate these waves if they move
along the European Atlantic coast?
Suppose that a Kelvin wave propagates in a layer of water with a thickness
H 0 =10 3 m.
b. Calculate the Rossby deformation radius of such a wave at 45 ◦ N.