22 MB. G. I. TAYLOR ON TIDAL FRICTION IN THE IRISH SEA. 



Let us assume that the tidal phenomena in the South Channel can be represented 

 by the superposition of two waves, one of amplitude a going in, and the other of 

 amplitude b, going out. These may be represented mathematically by the formula 



2v /. X\ j 2ir /. , X\ / .v 



h = aoo8-isr[t 6cos7=- U+- ) ...... (34) 



1 \ c/ I \ cl 



where the first term represents the wave entering the channel, and the second 

 represents the reflected wave leaving the channel. 2a and 2b are the ranges of the 

 tides due to the two waves separately ; x is the distance measured along the channel 

 in the direction of the Irish Sea ; and c is the velocity of a long wave in water of the 

 depth, D, of the channel so that < vgD.* 



Our problem is to analyse the observed tidal phenomena so as to find the values of 

 a, and b, and to show that the various characteristic features of the tidal phenomena 

 of the South Channel can be accounted for by considering them as being due to these 

 two waves. 



The current due to the entering tidal wave is A/ ^ cos -= I t - ] I" The current 



due to the out-going tidal wave is b A/ ^ cos ~( t+'~}- 



They are both positive at x = if n, and b are both positive, because the original 

 formula (34) assumed for //, gives h as the difference of two terms and not as the 

 sum. 



Hence the tidal current, r, is 



* It has been suggested that the velocity of the waves into and out of the Irish Sea are not equal to 

 JgD, because they are forced waves due to the moon. The moon's attraction, however, does not appear 

 to be capable of exerting sufficient force to alter appreciably the velocity of a free wave of the amplitude 

 with which we are concerned travelling down a channel of a depth of about 37 fathoms. 



If / be the horizontal component of the moon's attraction, the maximum possible value of / is 

 8-57 x lO~ B g (see LAMB'S ' Hydrodynamics,' 4th edition, p. 256). 



The maximum value of the horizontal force F, due to the pressure gradient in a free wave of height 2a 



2ir 



from trough to crest, is F - /Tvp. where T is the tidal period of 12 -4h. 



s gD i 



It will be seen later that the semi-amplitude of the smaller of the two waves with which we are 

 concerned, i.e., the out-going wave, is 145 cm. Taking a = 145, D = 37 fathoms = 6800 cm., it will be 

 found that F = 8 x I0~ f g. 



It appears therefore that /, the horizontal force due to the moon's attraction, is only ^^ of F, the 

 force due to the horizontal pressure gradient in a free wave of the height with which we are concerned. 

 The velocity of these waves cannot therefore differ appreciably from that of free waves in the channel. 



t This is the well-known formula connecting the current velocity and tidal range in a tidal wave. 



