118 BENDS, TRANSITIONS, AND OBSTRUCTIONS 



best approximated by the angle ^o. or ACF. From this point down- 

 stream the influence of the negative wave which started at point B 

 begins to aff^ect the height of the water surface, which had been rising 

 along the outer wall from A to D' , so that at point D the water-surface 

 profile along the outside wall will have a maximum. At some point 

 farther downstream the greatest intensity of the negative wave will 

 reach the outside wall and the water-surface profile will have a minimum. 



In a similar manner the positive wave which started at A will reach 

 the inside wall at point E' , and cause the profile along the inside wall, 

 which drops continuously from B to E , to turn upward. 



The positive and negative waves continue to be reflected back and 

 forth across the channel, causing the profiles along both inner and outer 

 walls to have a series of maximums and minimums at angles Oq, ZBq, Sdo, 

 from the beginning of the curve. Moreover, it is likely that the effect 

 will continue past the end of the curve into the tangent below, giving 

 rise to the possibility of aggravated conditions at the next curve down- 

 stream. 



The phenomenon is much like that which would occur if a railway car 

 with undamped springs ran suddenly onto a circular curve without super- 

 elevation or spiral. The equilibrium position of the carlDody would be 

 changed instantaneously from horizontal to a slant corresponding to 

 the speed of the car and the radius of the curve. The car body, however, 

 could not assume the required slant instantly. By the time it did reach 

 the equilibrium position it would have considerable angular momentum 

 about its longitudinal axis, which would carry it well beyond the equi- 

 librium position before the springs could start it back, thus giving rise 

 to a series of oscillations. At the end of the curve, the equilibrium 

 position would suddenly change back to the horizontal. This would 

 start another series of oscillations, which might, if the curve were of 

 just the right length, cancel or diminish the series started at the begin- 

 ning of the curve, but which would be just as likely, in general, to 

 reinforce partially or even double the effect of the first series. 



Ippen and Knapp give equations for determining the phase length of 

 the disturbance and the depth of flow along the outside wall up to the 

 first maximum. These may be summarized as follows. (The notation 

 has been changed to conform to the authors', except as shown on Fig. 

 1002.) The central angle to the first maximum is given by 



W 

 do = arctan ,,,,,^,, ^ [1004] 



{Re + Pr/2)tani3o 



This follows from Fig. 1002, by geometry. The depths along the walls 



