132 Lecture 8 
2 
w" + ww =0 (2) 
a 
and with the boundary conditions will lead to a wave velocity c=WNh/n for the 
first mode if WM is constant (i.e., c=1/m7[(Ap/p)gh]”, where Ap is the total density 
variation in a depth h. In the case of the deep thermocline (and sound channel) 
where large stability is confined to a limited part of the depth, one may expect 
intuitively that the Ap and the A appropriate to the problem are the change in 
density and the thickness of the thermocline itself. Thus, if there is a tempera- 
ture gradient of 8°C between, say, 400 and 1000 m, the velocity is approximately 
100 cm/sec, and for an 8°C change at a depth of 50 m (using the simple Stokes 
formula for two layers), we find thatc=90 cm/sec. There is rather little differ- 
ence between these two apparently dissimilar cases ofinternal waves, but on the 
other hand the velocity of surface waves in a total depth of, say, 3000 m is 
c=2-104 cm/sec, far greater. We shall return to this disparity later. Using this 
velocity (100 cm/sec), we find the wavelength equal to 31/, km for a wave of 
period of 1 hr. As the frequency approaches WN the velocity goes to zero. It is 
on such time and space scales that information is badly needed. The paper of 
Haurwitz et al. [6] has been referred to, and there is one further set of obser- 
vations by Ufford [7] (1947) which is of particular interest. He worked with a 
triangle of three temperature elements moored to the bottom up to 300 m apart. 
By correlation techniques the phase lag was computed between the records and, 
hence, the wavelength was deduced. The apparent period of the waves is about 
6.8 min, but this is affected by the mean current present. In most cases he was 
able to explain the o~k relation satisfactorily as the first mode of a simple Stokes 
two-layer theory or a simple extension of it. Recent progress reports from 
Scripps indicate that Cox has been getting satisfactory results also. It seems 
possible that work on acoustic propagation could yield much useful information 
about the waves (cf. Lecture 16). 
Alongside the development of experimental techniques there is room for some 
extension of the theory. The theory has been primarily concerned with the eigen- 
solutions of the equations. It would seem fruitful to apply statistical methods 
akin to those now being explored for surface waves in view of the "noisy" 
appearance of the spectrum and the apparent lack of discrete lines over most of 
it. There is the added complication that the o-k relation is not single-valued. 
A further problem is that in many cases, one of which will be illustrated shortly, 
the disturbance to the isotherms is too large to satisy a linear theory. As with 
long surface waves, this will lead to change of form and possible breaking. This 
could well increase the spectrum level at frequencies beyond the upper limit WV 
(i.e., the region of microstructure). The only other work of which the author 
is aware regarding the nature of the spectrum will also serve to introduce a 
discussion of the lower frequencies which are susceptible to observation by 
repeated dips of a standard bathythermograph. Two ships of the USHO occupied 
an anchor station to the north of Bermuda and made B/T observations every 
half hour for the whole period of 25 days. In generai, the spectrum of the fluctua- 
tions in level (Fig. 8.4) of anisothermdoes not contradict the idea that it may be 
regarded as noise. However, between 20- and 30-hr periods there is a resolved 
