1195 
Sf = 
C stands for a canalised mode, N for a non-canalised mode, 
K, would take the value 55 for all modes in 110 feet of uniform water, 
The values of "q for uniform water" are calculated on the assumption 
that the speed of sound is the same as in the lower layer, in 
we took It equad to the speed in the upper layer, extra modes, 
corresponding to the canallsed modes, would appear. p is always 
real, q ls real for a canalised mode, Imaginary for a non-canalised 
‘mode, 
The K, (the normalising factors) are all calculated on the basis of 
the amplitude of the sine wave in the top layer being unity. We 
therefore have three possible cases:— 
(a) A very large value of K, indicates that the amplitude of the sine- 
wave in the lower layer Is large compared with that in the top layer. 
(b) A value of K, of the order of 5 indicates that ths mode Is 
practically confined to the top layer (a canalised mode), 
(ec) A value of K, of the order of 55 indicates that the amplitude of 
the mode In the two tayers Is practically equal. 
By generalising equation (2), we find that the extent to which any 
given mode is stinulated is proportional to the expression obtained by putting 
y = Yq (where y, Is the depth of our source) In equation (9), and then 
dividing by Ky. We thus see that, for a source in the top layer, modes of 
type (b) (canalised modes) will be strongly stimulated, modes of type (c) 
moderately stimulated, amd modes of type (a) will be weak, For a source in 
the bottom layer, the stimulation of the canalised modes will be weak (due 
to the exponential decay), while modes of types (a) and (c) will be moderately 
stimulated, In other words, we should expect a canalisation effect if the 
source is in the top layer dus to reflection at the boundary tending to 
confine the sound, tut, If the source is In the bottom layer, the sound will 
be transmitted very much as if the top layer were not present at all. We 
therefore confine our attention to sources in the top layer. The stimulation 
factor of every mode will then te of the order of magnitude. Strictly 
speaking it is S'7 P Yo, but we are not concerned with cshautaciore of pressure 
flelds due to sourcas at particular depths, but we want rather a rough average 
Over sources at all depths within the top layer. In the same way, it is 
permissible to average the Hanke! functions (provided that thelr arguments 
are real) over a distance of the order of a wave-length, so that we neglect 
the oscillating property of the function and simply retain the factor cit 
vr 
Of course, we may not do this If the argument is Imaginary. 
Expressed more precisely, what we are really doing is to estimate 
the energy residing in a hollow cylinder of large radius due to point sources 
distributed along the r axis at all depths in the top layer. To do this, we 
take an expression of the type (2) for the pressure, substitute in it normal 
modes given by equation (9), square and then integrate over the whole depth 
of water. All the cross-terms then vanish owing to the orthogonality, and 
the contribution of all except the carrier modes fs nagtigible owing to the 
exponential decay of the others with rf. We have therefore simply to estimate 
how the energy of each mode divides itself between the upper and lower layer, 
and how much 2ach contributes to the energy. The energy residing in the 
upper layer is. proportional to h and that in the lower layer 
ie dh dy 
K 2 
n 
TO weesee 
