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amplitudes of surface elevation and particle velocity of the wave both increase
exponentially in the y direction. Waves of this kind are referred to as Kelvin waves
and are widely used to represent tidal motions in shelf seas. The amplitude of the
wave has its maximum value A f at the boundary (y
¼
0) and decreases in the negative
¼
p
y direction to A f =
e
'
0
:
37A f at y
¼
Ro
gh
=
f, i.e. at a distance from the coast
which is
250 km for a typical shelf depth (h
¼
100 m) in midlatitudes. In the deep
ocean, where h
4000 m, the scale Ro is increased to
1600 km.
3.6.3
Amplification and reflection of the tide
Tidal energy, input by the tide-generating forces acting on large basins of the deep ocean,
travels around the ocean basins in the form of very long Kelvin waves (l
8000 km).
At the shelf edge, it is transmitted on to the shelf, still in the form of Kelvin waves
but travelling at a slower speed as h decreases. We shall see in Chapter 4 that the
energy of these waves moves at the wave speed c, and that for a wave of amplitude A 0
the energy flux onto the shelf is given by
2 0 gA 0 is the energy
density of the wave. Consider what happens to a wave travelling in the deep ocean
with amplitude A d , changing to amplitude A s after crossing onto the shelf. As the
waves moves into shallower water, c decreases, so, in order to maintain a steady
energy flux,
1
E w c, where
E w ¼
E w must increase and with it the wave amplitude. The amplitude on the
shelf will then be related to its value in deep water by:
1 = 4
A s
A d ¼
h d
h s
:
ð
3
:
69
Þ
Taking h s
4000 metres to be the corresponding depths, the
amplitude of the tidal elevation therefore increases by a factor of
100 metres and h d
2.5 as it crosses
A 0
p (Equation
onto the shelf. The particle velocity amplitude is
j
u
A 0 g
=
c
¼
g
=
3.68) so the ratio of shelf to deep water velocities is just
1 = 2 A s
A d ¼
3 = 4
u s
u d ¼
h d
h s
h d
h s
:
ð
3
:
70
Þ
This implies a much larger amplification factor of
16.
The incoming waves are also reflected at the coastline, and the combination of
incident and reflected waves generates a standing wave component as in Equation
(3.62) and Fig. 3.11 but with the added complication that we are dealing with Kelvin
waves. Let us think about what happens to a Kelvin wave when it enters a rectangu-
lar gulf with width B G and aligned in the x direction with y
0 along the central axis.
We can write a complete description of the motion by combining two Kelvin waves
travelling in opposite directions:
¼
A f e y = Ro sin
A b e þ y = Ro sin
¼
ð
Þþ
ð
þ
Þ
kx
ot
kx
ot
n
o
:
ð
3
:
71
Þ
g
c
A f e y = Ro sin
A b e þ y = Ro sin
u
¼
ð
kx
ot
Þ
ð
kx
þ
ot
Þ
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