208 Long Waves in Canals and Standing Waves in Closed Basins 



so that, besides a constant factor, 



xZ = e-«y (VI. 108) 



(VI. 106) becomes 



+a 



I 6 -Shje-^dy = . (VI. 109) 



There is only one real value of x, which fulfils this equation. 



Proudman deals especially with a canal with a parabolic cross-section 

 (see equation (VI. 20) which was explained on p. 148) for a non-rotating 

 earth. From (VI. 109) we obtain the conditional equation for x in the form 



*f2*a= ' (VI .n0) 



2xa 1 + 2a 2 a 2 /gh v ' 



For a = 100 km, h Q = 100 m and a = 1 4x 10 -4 sec -1 (semi-diurnal tide) 

 a 2 a 2 /gh = 0-2, and one obtains, according to (VI. 110), for the ratio of x 2 

 and the value lo 2 /gh , which applies in the case of a narrow canal or when 

 the earth's rotation is neglected (see equations (VI. 24) and (VI. 25)), the 

 following values: 



a 2 a 2 /gh : 01 2 03 4 05 6 07 8 09 10 

 "-:104 109 113 118 1 22 1 27 1-32 1 38 1 43 1 49 



§(*W*o) 



c and c are the velocities of propagation respectively of a canal at rest 

 and rotating. It is obvious that if the canal is rotating the velocity of progress 

 decreases when the width of the canal increases. 



Moreover, we can derive values for U and Kin a simple manner from 

 (VI.108) with the aid of (VI. 106). If we select o 2 a 2 /gh = 01, 0-5 and 10, 

 we obtain the distribution of the velocities in the longitudinal and transverse 

 direction of the canal as represented in Fig. 89. We see that the transverse 

 current is distributed very unsymmetrically in respect to the centre of the 

 canal. This transverse current vanishes the closer to the left side of the canal, 

 in the direction of the wave propagation, the greater a 2 a 2 /gh is. In the narrow 

 strip to the left, the transverse velocities also are smaller, the greater this 

 value is. Only on the right side of the canal the transverse velocity is note- 

 worthy in proportion to the strength of the current in the longitudinal direc- 

 tion of the canal. 



Poincare Waves. Besides the Kelvin waves, there can be in a canal other 

 waves which are called Poincare waves (1910, p. 126) after the name of their 

 discoverer. If we assume again a solution of the equations of motion in the 

 form (VI. 15) and the depth of the rectangular canal to be constant, U, V 

 and Z must fulfil the equation (VI. 104), of which the last one takes the form 



g + l-^C-^z-o 



