1193 
= ee 
3.2.1.2he Solutions of Equations (4) and (8) 
We consider various possibilities:- 
(a) p imaginary. This would only be possible if c, > ctl Imaginary). 
For L real Equation (4) implies that q must be real, and there Is 
then no solution of equation (8). 
(b) p real, q real. This corresponds to a mode which oscillates with y 
ir the top layer, and dies away exponentially in the bottom layer, 
so that most of the energy is carried by the top layer. We call 
this type of mode a "canalised" one, 
(c) p real, q imaginary. . This corresponds to a mode which oscillates 
with y In both layers. We call this type of mode "non—canalised". 
A rather troublesome problem is to prove that the canalised and 
Non=canalised modes together form a complete set, so that we need not consider 
complex values of p and q. Perhaps the best argument is the physical one. 
|f we allow | to approach zero, our problem goes over into that of uniform 
sound velocity and It can be shown that the modes of types (b) and (c) go 
over smoothly Into the Fourier Serics for the uniform medium, which is known 
to be a complete set, The introduction of complex values of p and q would 
lead to modes which would go over into something of the type sin [(a + i b) y]J 
which would be redundant in a Fourier expansion. 
There are only a finite nunber of canalised nodes, From equation 
(8), If p and q are both real, tan p h must be negative, which means that 
p h can only take certaln values between 7 and 7, between 27 and 27, and 
2 
so On. As p is less than | by equation (4), this Implies that | must be 
L L 
greater than 7 for a canallsed mode to exist. This implies a lower timit 
to the frequency for which such a mode can exist. As the frequency risss, 
two things happen. First, q, for any given mode, increases steadily from 
zero, which means that the exponential decay in the bottom layer becomes more 
and more rapid. Secondly, more and more modes of the canalised type appear 
as .~ passes the values a7 5ST, etc., so that we might reasonably Infer an 
Increasing concentration of energy in the upper layer, as the frequency rises, 
However, a rise in the frequency also increases the number of non-canalised 
modes ({p real, q imaginary) for which © > p and which therefore remain important 
c 
at great distances (say of the order of! 500 feet). We therefore Investigate 
the energy flow for a few frequencies in a typical case, 
3.3,Results of Calculations. 
We assume the following conditions:- 
Thickness of top layer !0 feet, velocjty In this layer 5,000 feet/ second 
Thickness of bottom layer 100 feet, velocity in this layer 5,200 feet/ 
second, 
Whence we have , ="5.49°x) 10) - > (11) 
the critical value of wis given by tan c =a , which, solves by successive 
approximatlons ...e0. 
