Stationary Waves in Water. 99 



10 

 (a) It is not without interest to notice that G(w) = -^ 



gives a complete free surface with straight boundaries 

 inclined at +30° to the horizontal. 



{!>) G(w) = ic-i gives flow in a deep tank having an inlet 

 and outlet pipe in the bottom as in fig. 1. 



Fig. 1. 



o 



(c) Michell's examples in his paper on the Highest Wave 

 (Phil. Mag. 1893, xxxvi. p. 437) are given by 



G(w) = A( — i sin w e iw ) { 1 + (\e 2iw + c 2 e iiw + . . . } 3 . 



There is, however, no simplification in the calculation of 

 the constants by the use of (1). 



It appears that the method used by Michell is the most 

 direct when the form of the stream-bed is given. 



Equation (1) is useful when the shape of the free surface 

 is known approximately, and it is desired to obtain the general 

 characteristics of the motion. 



(d) Flow in a stream with a finite drop. 



Consider G(ia) = B — tanh xw B>1, a<l. 



— ■'-- = — r-rr : pr { 1 — a" SOCll 4 $w\$ — ia SCcll 2 OtW . 



dw /x{B-tanhau'}i L l } J 



H 2 



