340 Tides and Tidal Currents in the Proximity of Land 



and of continuity are the same as there: (VI. 5 and 47). The solution for 

 certain simple cases, in which both the width and the depth of the canal vary 

 proportionally to the distance from the closed end, has already been given 

 in Lamb's Hydrodynamics (2nd edition, 1931, para. 186). Defant has made 

 other applications, using the Chrystal theory of seiches. Other theories of 

 seiches can also be used ; however, the simplest one seems to be the method of 

 the step-wise numerical integration of the hydrodynamical equations (p. 165). 

 It can be applied also, when the cross-sections along the valley vary very 

 irregularly and gives at once the entire form of oscillation of the basin (po- 

 sition of the nodal lines, amplitude of the horizontal and vertical displacements 

 and the tidal currents respectively). 



To compute the co-oscillating tide one uses the equations (VI. 54 and 55) 

 and starts the computation at the closed end of the basin where must be 

 £ = and selects for r\ an arbitrary value, for instance 100. The period of 

 the co-oscillating tide is given by the selection of the component tide. There- 

 fore, £ and r\ can be computed step- wise from cross-section to cross-section. 

 Finally, one obtains at the end of the basin which opens into the ocean 

 (x = /) a certain value rj =v\ l . However, according to the boundary con- 

 dition indicated above, the amplitude of the tide for this point in the open 

 ocean is Z. In order to obtain the correct distribution of the amplitude of 

 this tide, it will suffice to multiply each value by the ratio Z/r] l . The equations 

 remain satisfied, inasmuch as the proportionality factor in £ and rj eliminates 

 itself from them. 



For the independent tide, which is generated directly by the tidal for- 

 ces, it is necessary to add to the equations a term for a periodic force 

 X = mcos{ot-\-e). The equations for the step-wise computation of £ 2 and r\ % 

 from the values £ x r} 1} of the preceding cross-section will then be 



l 2 = 



S 2 [l + (av 2 )l(4S 2 )] 





(XI. 17) 



fi + f. 



<? 2 = fc+^y^i 



in which AH = (m/g)Ax. The boundary conditions to be fulfilled are: at th 

 closed end (x = 0) £ = and at the open end (x = I) rj = 0. ?/ at the crosse 

 section (closed end) must be selected in such a way that the second con- 

 dition is fulfilled at the end of the computation. One varies the values of n- 

 which necessitate a repetition of the computation, until the correct value is 

 sufficiently narrowed down between two limits. This tedious calculation can 

 be considerably shortened by the following artifice. One selects for rj a value 

 resulting from the equation (XI. 15) for y = for a canal of equal length 



