J. Crease 131 
remarks. For a temperature gradient of 1°C/m, v =4.4-10-? sec”! for 1°C/10m, 
N =1.4-1072 sec7', The first figure is more appropriate to the diurnal thermo- 
cline, and the latter to the seasonal thermocline at a depth of 150 ft or so. 
Together with the boundary conditions, Eq. (1) constitutes an eigenvalue problem. 
The boundary condition at a flat bottom is W =0 and the same condition may be 
used at the free surface, bearing in mind that the solutions of most interest in 
the present context have maximum amplitude in the interior with little deforma- 
tion of the free surface. Bythis artifice the surface wave solutions are, of course, 
lost. 
With the equation in this form it isclear that there is fundamentally different 
behavior of the solution in different ranges of o. First, if o is greater than the 
maximum value of WN (i.e., Vmax), all solutions will be of exponential type through- 
out the depth range; and in view of the boundary conditions this means that in 
fact no solutions at all are possible. If Nmax >a>f there is at least a range of Z 
in which the solution is oscillatory, and there is then the possibility of a solution 
satisfying the boundary conditions. Finally, for o<f thereare again no solutions 
(other than the unallowable exponential ones). I have tacitly assumed, as is 
generally true in the ocean, that V >f. Eckart [4] (1960) has givena full treat- 
ment of this eigenvalue problem for a compressible fluid. 
From these remarks we may conclude that Wand f are upper and lower bounds 
to the frequencies of free internal waves in the sea, although there are further 
free oscillatory motions which have periods greater thana week or so which are 
quasi-geostrophic in character and will not be considered here. Groen [5] in 1948 
first drew explicit attention to the upper limit N on the frequency, but there has 
been little in the way of observations (in the open literature) to confirm it. Only 
now are the techniques becoming available to make the required observations in 
deep water at sufficiently close time intervals. An appropriate reference is the 
paper of Haurwitz, Munk, and Stommel [6] (1958). A long series of temperature 
measurements were made with thermistors on the sea floor at depths of 50 and 
500 m offshore from Bermuda. From Fig. 8.2, a section of the record, it is im- 
mediately clear that the deep thermistor shows the presence of higher frequen- 
cies. This is confirmed bythe respective spectra. They both approach a low level 
rapidly with increasing frequency, and the shallower record differs from the 
deeper record by a factor of ten. Figure 8.3, showing the variation of W with 
depth, indicates Mmax at 5th cycles/hr; at the upper thermistor, V equals Yo, 
while at the lower it is 15/4. It would seem likely that herein lies the explanation 
of the observed difference in frequency response. Thetwo peaks in W correspond 
to the seasonal and main thermoclines—often one may also expect a diurnal 
thermocline responding to meteorological conditions with possibly large values of 
wv. The development of chains of thermistors at some laboratories, notably at 
Woods Hole Oceanographic Institution, for rapid sampling of the vertical profile 
of temperature in the top 400 ft of the sea, would seem to be of the greatest 
importance in investigating this part of the spectrum. 
It is possible to make some statements about the velocity of propagation 
without getting too involved in equations. Fjeldstadt [3] in 1933 and Groen [5] in 
1948 treated the analytical details in special cases. At frequencies away from 
either limit, Eq. (1) becomes 
