43 
ordinates. As this is very rarely done in papers, a minor digression may be justified. 
Let us consider a wave of angular frequency co progressing in water of constant depth 
H. As derived in standard literature, its wave number k is subject to the dispersion 
equation co 2 = gk tanhfA/f). Our representation in spherical co-ordinates is a little more 
involved than that known from Cartesian representation. We need a 2D- function 
a=a{)i,cp) providing the phase of the wave. It should be harmonic and of constant 
slope, i.e. 
1 
’ 5 
( 1 5a 
R cos cp 
dA 
K R cos cp dA ^ 
( 1 da'I 
2 
' 1 da'' 
K R cos cp dA ^ 
[r d< p) 
5 ( cos cp da' 
\- — —— 
5ç?^ R d(p; 
= k 2 
= 0 
The wave formula reads 
a 5(sin(a - cot)) 
P = pga 
co dt 
cosh (k(H + z)) 
= a cos (a - cot) 
cos(a - cot) 
u = aco 
v = aco 
w = aco 
cosh (kH) 
cosh(k(H + z)) 1 d(sin(g - cot)) 
s\vds\(kH) kRcoscp dX 
cosh(Æ(W + z)) 1 ^(sin(o: - cot)) 
sinh(AW) kR dcp 
sinh(A(// + z)) . 
aco 
sinh(^W) 
-sin(a -cot) 
cosh{k(H + z)) . . 1 da 
= aco costa -cot) 
sinh(kH) kR cos tp dA 
cosh (k(H + z)) . . 1 da 
sinh(A/i) kR dcp 
This construction was made in order to fulfil the continuity equation 
f *- J aw' _ 
+— = o 
l 
R cos cp 
du ô(cos (pv) 
dA dcp 
dz 
It also makes vorticity vanish, i.e. 
1 
dv' 
~dA 
d(coscpu') 
1 dw' 
R cos cp 
du' 
dz R cos cp dA 
1 dw' dv' _ 
R dcp dz 
dcp 
-0 
= 0 
The local direction of propagation is aligned with the gradient of the phase function, and 
we may characterize it by an angle as a function 6 = 9{A,cp) of spherical co-ordinates, 
defined to be 
1 da 
cos 6 =■ 
kRcoscp dA 
I da 
sm 6 
kR dcp