Chta-Shun Yth 
With v given by (43), (40) becomes (remembering the factor 
exp i(kz - ct) ) 
2 d 
Aa" ¢ 2 2 3 
bomen ie Sap we Fy Sgt (44) 
ie 0 oy 5 ee 
0 0 
The boundary condition at the solid boundary is 
po ea ONG (45) 
where u and v are given by (34) and (43), and nn and n, (=0) 
are the direction cosines of the normal to the solid Soeee The 
Bernoulli equation is still valid in the free surface (if there is one). 
In fact, the Bernoulli equation obtained by integrating (29) is simply 
(39). But we must recall that p is the pressure perturbation only. If 
we require the total pressure p+ po to be zero on the free surface, 
and use (28), we have (with the new definition for ¢ ) 
ails 2 + gn = constant (at the free surface) , (46) 
PO 
which gives, upon idfferentiation with respect to t, 
p 
BPO %y 
2 
+ ! 
a BPO 
2 2 2 
V. VARIATION OF. 5° AND c WITH k 
For the purpose of establishing the comparison theorems, we 
shall write (44) as 
1408 
