SECT. 5] 



76^ 



2. The Hydrodynamic Equations and Their Application to Tidal 



Problems 



The elevation of the tide above the mean sea surface, ^, and the components 

 of tidal currents, u, v, are also set up by harmonic constituents: 



^ = ^0 cos{at — K) = ^1 cos at + ^2 sin at ~ ^e"*"' 

 u = Uq cos {at-Ku) = ui cos at + U2 sin at ~ we-'''^ 

 V = Vq cos {at — Kv) — vi cos at + V2 sin at ~ ?;e~*'"^ 

 Q-iat = cos o-^ + ^ sin at; i = -y/ — 1 . 



Amplitudes and phases, for instance f and /<:, are called harmonic constants ; 

 normally they are derived from long-term records of sea-level. These records 

 are also used in the derivation of tidal currents. The introduction of these kinds 

 of harmonic constituents simplifies the mathematical treatment of tidal prob- 

 lems by means of the hydrodynamical differential equations. These are 



8u 8u 8u . „, , 8(^ — i) 

 dv dv 8v . „, ^ 8U-i) 



(3) 



in cartesian, and 



8u 



,^^ - 2co sin m-v + B(^) +1 cosm-— U-i) = 

 8t ^ a ^ 8\ 



T^ + 2a} sin q)-u + R^f) +-■ — (^ - 1) = 

 Ct ^ a 8(p ^ ' 



1 



^i 



bt a cos 



-[(A + ^)w]+— [(/t + |)vcos99] - 



(4) 



in spherical co-ordinates without convective terms. In these equations it is 

 assumed that the components of velocity, u, v, are mean values from the sea 

 floor to the surface, and furthermore that the density is constant in the ocean. 

 Neglecting the convective term and the variation of ^ compared with mean 

 depth h and using the relations 



R(x) = r-u. 



Riy) ^ r-v. 



(5) 



the equations become linear. 



Equations (2), when introduced into the system of differential equations, 



