23. TIDES 



W. Hansen 



1. Introduction 



Newton was the first to give a physical explanation of oceanic tides. Later 

 Bernoulli, Laplace, Hough, Airy, Kelvin, Darwin and Poincare set up a 

 classical theory of tides, the aim of which was to understand qualitatively and 

 quantitatively this natural phenomenon. To some extent this theory was 

 completed by the work of Proudman and Doodson. 



The theory is of special importance since observations are only available 

 from coastal areas. Thus the theory may be helpful in gathering information 

 about the tides in the open sea. This theory is based on the hydrodynamical 

 differential equations, consisting of two equations of motion and the continuity 

 equation. The task is to develop mathematical solutions of this system, if 

 possible in the shape of analytical terms. The problem becomes more simple if 

 the system is a linear one, and the tide generating forces are developed in series 

 consisting of harmonic terms. The frequency a as well as amplitudes and 

 phases are known from astronomical data. The potential V of the tide -producing 

 forces or the equilibrium elevation | = - Vjg with gravitational constant g is 

 given by : 



1 = 21" cos {a^t-K^) = 2 I"'! cos (7,^ + 1., 2 sin a^t. (1) 



V 



There are three groups of periods determined by the frequency ct. They are 

 long-period, diurnal and semi-diurnal tides. 

 The principal harmonic terms are : 

 M2: principal lunar semi-diurnal constituent, frequency 28.98 degrees per 



mean solar hour. 

 S2: principal solar semi-diurnal constituent, frequency 30.00 degrees per 



mean solar hour. 

 K2 : luni-solar declinational semi-diurnal constituent, frequency 30.08 



degrees per mean solar hour. 

 Ki : luni-solar declinational diurnal constituent, frequency 15.04 degrees per 



mean solar hour. 

 Oi: lunar declinational diurnal constituent, frequency 13.94 degrees per 



mean solar hour. 



As a result of tidal observations it has been established that, normally, the 

 most important harmonic term is the lunar semi-diurnal constituent, M2. 

 In order to avoid mathematical difficulties, theoretical investigations prefer 

 the K2 — or, in the case of diurnal tides, the Ki — constituent. Doodson (1921) 

 has given the complete tidal potential (see also Bartels, 1957). A more detailed 

 representation of tides has been given by the following authors : Defant (1957, 

 1961), Doodson (1958), Lamb (1932), Proudman (1952), Sverdrup et al. (1955) 

 and Thorade (1931). 



[MS receivedJuly, I960] 764 



