1191 
; abla en = fut (an - fan= 2 \2 |, 
Pn = sin ets cman pas = oe at 
Oe Tah ep 
4 
(1) 
We need a sine function of this aoa in order to satisfy oe O at the free 
surface, and yn = 9 at the rigid surface. (The function J, could be 
replaced by iy. We solve the problem of a point source first by adding 
together such solutions fn order to make the potential near y = y, agree 
with that of a point source. We could then add up these solutions for 
various frequencies to make a pulse of the form we wish to study. For the 
present we confine ourselves to a single frequency source. The appropriate 
solution: for this case can be shown to be:- (3) fe 
Ch) 
pee ene fees @ sin (yy) sin (1 yo) Ho at - A, \tr| elt (a) 
; aC anes Tame) 2 
ot 4 Hea sin2 (y%) dy c 
where y, = {20 = lI ang eee is the first Hankel function. At large 
Ninn ath) chet er 
7 ix 
distances this function may be written Buia (x)~ 
(lb ss 
2 
and therefore represents, if x is real, a wholly outgoing wave, the factor 
being the famillar one associated with cylindrical spreading. If x is 
imaginary, this function represents a decreasing exponential, and will dle 
away to a very small value in a distance r of the order of h + |. Thus, at 
distances great conpared with the depth, the contributions of those modes 
for which y, > 2 will be negligible so that the pressure will fall off as 
_| the sound being continually reflected between surfaces and bottom. As 
= 
we approach to distances of the order h + | the contributions of these 
"exponential-like" modes become appreciable until finally very near the source, 
we find that the pressure is falling off according to the familiar inverse 
distance law en For brevity, we shall describe those modes for 
u y2)2 
which y, <= which alone are important at great distances, as the "carrier" 
modes. 
Th2 mathematical problem is now quite clear: we have to repeat the 
work we have just summarised, using nornal modes appropriate to conditions 
when the velocity in the two layers is different. 
The New Normal Yodes. 
Let c, be the velocity in the upper layer, and c that in the lower 
layer. By continuity considerations, we should expect our normal modes to be 
similar to those that occur in the und form case. We therefore try solutions 
of the wave eauation of the following types. 
iat 2 
one © sin py Beta es - p? 2 in top layer 
c 
4 | 
b \ p er (3) 
=the (= cd + coe) Hea (4 + 4 ) in lower layer, 
c 
where for the present we leav2 open the question of whether p_and q are real 
or imaginary, We use a sine in the top layer in order that = may vanish 
at cecvee 
