130 Lecture 8 
under Nansen's leadership, on approaching the ice edge in northern waters, en- 
countered the phenomenon of Dead Water, well known to seamen, but not pre- 
viously investigated scientifically. Ships encountering it found themselves able 
to make little or no headway. Ekman was able to show with an excellent series 
of model experiments that Dead Water arose when a ship running into a layer of 
fresh water overlying salt creates ship waves on the internal interface between 
the layers; the extra power required to generate the internal waves was sufficient 
to reduce the headway drastically. It was at about the same time that the repeated 
use of the more accurate deep-sea thermometers then becoming available indi- 
cated naturally occurring oscillations within the sea unaccompanied by visible 
surface displacement. 
Rather than catalog here the historical record of the subject, it is now pro- 
posed to give a qualitative account ofthe theory and the most significant observa- 
tions. Stokes (cf. Lamb [2], p. 370) was the first to consider waves on the inter- 
face between two liquids of different densities. We need only note here that in a 
homogeneous fluid there is a unique frequency-wave number relationship 
o?/k? = (6/k) tanh kh, Which for short waves becomes o*/k? = g/k, and for long waves 
is o?/k?=gh, where h is the depth of water, o is the frequency, and & is the 
wavenumber. For a two-layer system on the other hand, there are essentially 
two degrees of freedom (analogous in some ways to the motion of a double pen- 
dulum) and the o-k relation is no longer single-valued. In fact, for liquids of 
almost equal densities the relation factors into o?/k?=g/k, and into o7/k? = 
gh(Ap/p) for the special case of an upper layer of depth h overlying a deep lower 
layer whose density is greater by Ap - where h is small compared to the internal 
wavelength. The first relation is appropriate to ordinary deep-water surface 
waves and the secondtoa wave with maximum amplitude at the interface. The lat- 
ter has the appearance of a long surface wave with gravity reduced by a factor 
Ap/p. This may be seen to arise from the reduced potential energy required to 
deform the interface between liquids of almost the same density, as opposed to 
that required for the deformation of the free surface. By introducing successive- 
ly more interfaces, one may approximate to a continuous distribution of density; 
and it should be clearthat this will be associated with an infinite sequence of o-k 
relations. 
The following equation for the vertical velocity (and displacement) was de- 
rived by Fjeldstadt [3] in 1933 for an ideal, incompressible, rotating fluid with a 
continuous distribution of density; the notation is somewhat different from his: 
2 4 4 
we = fi N2/o? -1 a - Mw -0 (1) 
where W = p”; we!“*-7 is the vertical velocity (or displacement); the derivatives 
are with respect to the vertical coordinate z; k and o are the horizontal wave- 
number and frequency; f is the Coriolis parameter equal to 2Qsin@, where 
Q is the earth's angular velocity and 6 is the latitude. Finally, N is the Brunt- 
Vaisala frequency equal to [(g/p)(dp/dz)]”. This is clearly a stability parameter and 
is the frequency of oscillation ofa parcel of fluid displaced vertically from its 
position of equilibrium in a gravitational field. The last two terms in the equa- 
tion are generally small in the ocean and will be neglected in the subsequent 
