24 
MR. G. I. TAYLOR ON TIDAL FRICTION IN THE IRISH SEA. 
in some other way. With this end in view, we shall discuss the movement of the 
co-tidal lines in the South Channel. 
First, let us consider the theoretical aspects of the case. A co-tidal line is a line at 
all points of which it is H.W. at the same time. In a progressive wave the co-tidal 
lines are the positions of the crest of the wave at a series of successive times. The 
distance apart of the co-tidal lines corresponding witli a series of times, separated by 
intervals of 111., will be a measure of the velocity of the wave. In drawing a 
map of co-tidal lines, therefore, one is apt to think that they I’epreseiit the successive 
stages of advancement of a progressive tidal wave. This idea is incorrect. In the 
case of two superposed waves moving in opposite directions, for instance, it will be 
found that the co-tidal line moves in the same direction as that one of the two waves 
which has the greater amplitude ; but that it does not move at the same speed. 
In certain places the line moves faster than the wave, while in others it moves more 
slowly, and a knowledge of the relationship between the velocity of the co-tidal line 
a]id the velocity of the wave will enable us to determine the ratio of the amplitudes 
of the two waves. 
The height of tlie tide above mean sea-level at any time is 
h = a cos^(^—-)—6 cos + 
I \ c/ I \ cj 
This may be written in the form 
where 
and 
h = Acos 
A = 
/yX a^ + V^—'2ab cos 
dvra' 
a — h 
a + h 
cot 
27ra' 
cT' 
■ (39) 
■ ( 40 ) 
.( 41 ) 
d'lie co-tidal line, therefore, moves in time t,, through the distance ,r, from the 
place where the phases of current and tide are the same, x and are related by the 
equation (41). 
The velocity, V^, of the co-tidal line is therefore obtained by differentiating (4l) 
_ djJ' 
a- 
= c 
cot 
2 27rX 
cT 
+ 1 
coF 
^ttX 
+ 1 
a + b\ 
\a — hj 
(42) 
At the point x — 0 where the tidal heights of the two waves <.)ppose, and the tidal 
streams conciu-, the velocity of the co-tidal line is therefore a fraction ^ ^ of the 
a-\-b 
