6.4 Internal Waves when the Density Varies Continuously with Depth
193
az
Po
Or---------r---------~
b z
N(z)
Fig. 6.7: Vertical profile of: a water density, b the Brunt-Viiisiilii frequency for the
case when N is constant
and
( 6.23)
in which n = 1,2,3,... Therefore, internal wave motion is only possible for
selected wave numbers, kz , or selected lengths of the vertical component of the
motion, Lz = 27r/kz .
Substituting Eq. (6.23) into Eq. (6.21) yields:
[ n7r(z + h)]
w(x,z,t) = an sin
h
sin(kx-wt).
(6.24)
The wave number kx should be found from the dispersion relation which takes
the form (Roberts, 1975):
(6.25)
If w 2 » f2 (waves are of short period), the Earth's rotation may be neglected
and Eq. (6.25) becomes:
(6.26)