Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 93

For waves with l ≪ h, termed deep water waves, kh ≫ 1 and cosh kh ! e kh /2 so

e kz . At the same time, tanh kh that the attenuation factor simplifies to H a (z) ¼ ! 1so

that the phase velocity in Equation (4.4 ) becomes:

o 2
k 2 ¼

g
k ¼

gl
2p :

c 2 ¼ ð 4 : 7 Þ

Since we must also have c ¼ l/T p , we can substitute for l and obtain the useful relations:

gT p
2p

gT p
2p ;

c ¼ l ¼ ð 4 : 8 Þ

giving the phase velocity and wavelength of waves in deep water in terms of the wave
period T p .
In many situations where the depth is 100 metres or greater, locally generated

wind waves in the shelf seas satisfy the condition l

∼

h, which means that they are not
substantially influenced by the presence of the bottom. They can reasonably be
treated as deep water waves conforming to Equation (4.7) and decaying with depth
according to H a (z)

<

e kz . We shall next look at the particle motions for these waves
and the way in which they are modified when the waves encounter shallow water.

¼

4.1.2 Orbital motions

From Equation (4.5) , the particle velocities for deep water waves simplify to:

¼ @

f

oA 0 e kz sin u

x ¼

ð kx ot Þ

@

ð 4 : 9 Þ

¼ @

f

oA 0 e kz cos w

z ¼

ð kx ot Þ:

@

This means that the particles describe circular orbits with a radius A 0 at the surface and
decreasing with depth as A 0 e kz . These circular particle motions, which are in-phase at
all depths, are illustrated in Fig 4.2a ; the particles move forward (in the direction of
wave propagation) under the wave crests and backwards under the troughs. As l/h
increases, the flow extends to the seabed where the condition that flow cannot pene-
trate the seabed becomes important. With only horizontal flow allowed along the
bottom boundary, the circular orbits give way to ellipses which flatten towards the bed
as shown in Fig. 4.2b . For very long waves (l

>>

h), like those discussed in Section 3.6 ,
the ellipses become extremely flattened (see Fig. 4.2c ) ; the amplitude ratio of w to u is
k(z þ h) which is zero at the bed (z ¼ h) and takes a value of kh ¼ 2ph/l at the surface

(z 0). The motion is, therefore, practically rectilinear at all depths; an obvious

example of this in the ocean is the particle motion in tidal flow.

¼

4.1.3 Waves of finite amplitude

Of course real waves are not infinitesimally small and, if we want to be rigorous,
the above first order theory has to be refined when we are dealing with steep waves. In