stream. In particular, I attempted to clarify the different circumstances under which 

 on the one hand, the development of a drag on the obstacle is accompanied by the 

 formation of a train of periodic waves on the downstream side and, on the other hand, 

 the receding stream converges towards a uniform state. As Professor Lighthill ex« 

 plained this morning, a great many different flow phenomena possible in a straight 

 channel can be summarized very neatly in the energy/momentum diagram * which 

 was put on the blackboard this morning. It is still there, and we may conveniently reier 

 to it now. 



The diagram shows that whenever a slightly subcritical stream (i.e., one for 

 which the Froude number is slightly less than unity) is obstructed, the reduction of 

 momentum associated with the drag on the obstacle gives rise to a train of waves on 

 the downstream side. Increasing the drag, by deeper immersion of the obstacle, causes 

 the amplitude and length of the waves to become larger; and eventually the wave train 

 resembles a succession of 'solitary waves.' 



However, a different state of affairs arises when the undisturbed stream has a 

 Froude number less than about 0.8, the value at which the solitary wave of maximum 

 height (i.e. the sharp-crested form of the wave) is formed by reducing the momentum 

 of the stream as much as is possible. In this lower range of Froude numbers, the re- 

 sistance experienced by an obstacle cannot, except for very small values, be ascribed 

 to wave formation alone. The effect of gradually increasing the resistance is first to 

 produce periodic waves of maximum height, and then to produce breaking waves. In 

 other words, the reduction in the momentum of the stream must eventually be ac- 

 companied by a loss of energy. Nevertheless, by increasing the resistance sufficiently it 

 is possible to bring about once again a steady lossless flow on the downstream side. In 

 this extreme case, the receding stream converges steadily towards a uniform super- 

 critical state, and waves are entirely absent; this is exemplified by the flow under a 

 vertical sluice-gate, although such a flow is possible under an obstacle of any shape 

 which is adequately immersed. 



These effects were clearly demonstrated in some experiments in a horizontal 

 water channel with glass sides. A metal plate was supported from a horizontal axle, and 

 spanned the whole width of the channel. The plate could be swung down broadside 

 to the stream, and was balanced in such a way that the drag due to contact with the 

 water could be fixed at any desired value. The observed effects of gradually lowering 

 the plate into a subcritical stream with Froude number less than 0.8 was as follows. 

 First, a smooth train of periodic waves was formed, whose amplitude and length in- 

 creased as the drag was increased. Then the leading wave broke at its crest, accom- 

 panied by a reduction in the amplitude of the waves in the rear, so that the front of 

 the wave train resembled a turbulent 'hydraulic jump.' On a further increase of the 

 drag, the wave train was swept downstream out of the channel, and the flow at a 

 distance behind the plate became approximately uniform; thus, the maximum drag oc- 

 curred for a uniform supercritical flow downstream, as indicated by the theoretical 

 considerations summarized above. 



To account theoretically for the case of maximum drag, where waves are absent, 

 it is convenient to invert the problem; that is, the depth and Froude number far down- 

 stream are specified, so that the drag is determined by a simple momentum relation 

 between the uniform flows up and downstream, and the position of the obstacle remains 

 to be found. In my paper I showed that the profile of the converging flow in the rear 

 of the obstacle is determined uniquely by the ultimate conditions reached far down- 

 stream; that is to say, the flow pattern is independent of the shape of the obstacle. How- 

 ever, this conclusion does not apply to the region where the free surface first springs 

 clear of the solid boundary, and where its curvature is large. This general wake profile 

 has a simple approximate analytic expression in which the Froude number is a param- 

 eter, and it may be seen that this expression is the same as a well-known equation for 



Figure 9 of Prof. Lighthill's paper in this volume. 



134 



