1192 
a flies 
at the free surface y = 0, From the conditions that the pressure and normal 
velocity are continuous at the boundary y = h, we obtain:— 
2 2 
p2 + q2 =e = Oe ee RSA) (L is always real since c,<c) (4) 
A RT 
| Ee 
sin p h = 8, ef" + . e7ah (Pressure continuous at boundary) (5) 
= gh -gh A . 
p cosp,.h=qi(B,e* - C, e )(Normal velocity continuous at (6) 
boundary) 
B, and C, are. related by the fact that the normal velocity at the bottom 
must be zero. This makes: 
s inestona lines -g (h + 1) 
Bneeed =UGAwe id (7) 
Eliminating B, and C, we obtain the following relation between p and q:- 
tanp nh +P coth g | =0 (8) 
; : 
and the following form for the typical normal mode:- 
: 2 
¢, = elt cin p y Hee +5 -p*\2 - In top layer 
c 
| 
(9) 
ea = e!4t sin p h cosh q (y - h-— 1) Healt hexe _ p*\2 rl in bottom layer 
cosh q | ( re 
Jt can te shown without difficulty that these functions are orthogonal if 
two distinct ones are multiplied together, and integrated fron y = 0 to 
Sen tral They can therefore be treated exactly IIke Fourier series in 
every respect but one, In equation (2) the normalisation factor is 
independent of y, and is simply h + lt. For normal modes of type (9) the 
2 
expression is:- 
ir sin@ LYE OV) iat 4 sin? p_h cosh? qly-h- 1) a 
cosh“ q | v 
Sifierstacip sh) Al emcos= pul “id= hee Sy (10) 
-_- re Soe a 
5 a2 p q2 q2 2 2 
which has to be computed separately for each mode. Repeating the work 
that led to equation (2), we have to replace sitin (mh Yo! 
fine sin? (ym y) dy 
ty sin (p y) or by cosh q (yo - h = 1) sin ph according to whether the 
Ky Ky cosh q 1 
source is in the top layer or the lower layer. Our problem is now solved 
if we can determine p and q for each mode, 
3.2. lasceee 
