Long Waves in Canals and Standing Waves in Closed Basins 213 



canal, however, a disturbance has to be added, through which the boundary 

 condition is fulfilled. Taylor's condition for the total reflection is 



°*-P<gh% (VI. 11 5) 



in which b represents the width of the canal. 



2n r 2n%\x\(b , 2b . 



As o- == — , / = — and as — — = T a the natural period of the 



T 12 hours y a n 



canal in the transverse direction, this relation can also be given the form 



'Tq\ 2 ^ , , / T q sin $ \ 2 

 T 



® 



" + (§££)' < VI| "> 



in which the periods are to be taken in hours. 



If we take for the wave the semi-diurnal tidal period (12 hours) we 

 obtain from (VI. 116) 



5<-^r, (VI. 117) 



T cos cf> ' 



for 45° of latitude = 1-42, i.e. if total reflection is to occur at the closed 

 end of the canal (bay), the natural period of the transverse oscillations in 

 tide waves must not, for average conditions, exceed \\ times the tidal period. 

 Although Taylor's results apply strictly only for canals with a rectangular 

 cross-section, they nevertheless can be used to fairly correctly evaluating the 

 possibility of a reflection of Kelvin waves in the case of a more complicated 

 configuration of the basin. 



The solution by Taylor is mathematically very elegant, but difficult. A simpler one, in connec- 

 tion with Taylor's solution, has been given by Defant (1925, p. 25). Let the origin of the co-ordi- 

 nate system be situated in the center of the canal, its width be n and extend along the j-axis from 

 — \n to +\n. The superposition of two Kelvin waves progressing in opposite directions gives 

 for the horizontal velocities in the x- and y- direction at the point x = Xx of the canal, 



/ ax x ax t \ 



w„ = S\ coshajsin - — cosa/+ sinhcrycos — sina/ 

 \ c c j 



(VI. 11 8) 



0, 



where a =//c and c = \/gh. A second solution of the equations of motion (VI. 102) is being 

 sought, which will give also always v = for y = ± £tt, but for x = x x the value u of the solu- 

 tion (VI. 118). The difference between the two solutions then always is that at the longitudinal 

 walls the transverse current is zero, but at the cross-section x = x x the longitudinal current is zero, 

 so that a barrier-like partition can be erected here, without disturbing the wave motion in the canal. 

 Taylor derives the second solution from the differential equation in v, (VI. 102), by putting 



v = v 1 cosat+ v 2 s'mat , (VI. 119) 



v x and v 2 must then fulfil this equation. A solution satisfying the boundary condition for y = ± \n, 

 v = and extending practically over a small area in the x- direction, has the form: 



v t = 2j Cn e ~ "*sin«y and v 2 = 2j C'n e ~ "^cos/ry . (VI. 120) 



even n odd n 



