makes water of the particular depth 0.5 cm. specially suitable for producing flows 

 closely similar to gas flows. 



5. Viscous attenuation 



Viscous effects modify the flow near the free surface and near the bottom in 

 boundary layers of thickness about 



(6) 



V" 



(where v = kinematic viscosity) and cause damping by a factor of e~ x in amplitude 

 in a time 



£rf (wavelength in cm.) 2 (7) 



Hence the capillary waves with A < 1.7 cm. travel only a short distance before being 

 damped out by viscosity. The longer waves are not seriously affected, but of course 

 in a river turbulent friction is a much more important source of damping for these, 

 as we will be discussing later. 



6. Wave-making resistance to two-dimensional obstacles 



The most striking evidence of frequency dispersion in river waves is to be found 

 by observing the wave pattern due to obstacles in the stream or steps in the bed. We 

 may consider first two-dimensional obstacles or steps (see Fig. 2). In steady flow these 

 can produce only waves whose wave-velocity c equals the velocity V of the stream, so 

 that their crests remain stationary. The energy in the waves is propagated upstream 

 at the group velocity U and swept downstream with the stream velocity V, which 

 equals c. Thus the energy input due to the resistance of the obstacle travels upstream 

 for capillary waves with U > c, which therefore are found in front. The energy travels 

 downstream, however, for the gravity waves with U < c, which therefore are found 

 behind, and for larger obstacles the bulk of the wave-making resistance of the obstacle 

 leaks back through these at the speed c — U. On the other hand, for supercritical 

 streams, with V > \/ (gh), no gravity wave with c=V exists, and so the wave- 

 making resistance of a two-dimensional obstacle is lowered. Similarly, the drag of a 

 barge on a narrow canal is reduced when its speed rises above \/(gh), and Scott 

 Russell [28] describes how this was first discovered due to a horse taking fright, and 

 galloping off pulling its barge behind it, when it was observed "to the proprietor's 

 astonishment, that the foaming stern surge which used to devastate the banks had 

 ceased, and the vessel was carried on through water comparatively smooth, with a 

 resistance very greatly diminished." The proprietor went on to achieve "a large 

 increase in revenue" as a result of this scientific observation. 



7. Stationary waves on non-uniform streams 



I might remark that Mr. J. C. Burns and I investigated [4] what relation corre- 

 sponding to c = V a stationary wave must satisfy in a stream whose velocity varies 

 with depth according to a one-seventh power distribution typical of turbulent flow over 

 a smooth bottom. For smali X/h the waves are on the surface only, so that for them to 

 be stationary we must have c — surface velocity = 1.143 V m (V m = mean velocity). 

 As X/h increases, c drops (Fig. 2) till it equals V m for X/h = 10 and is 0.966F m 

 when Xjh—> oo ; this lower value is because the oscillatory motion for very long waves 

 tends to be slightly stronger in the low-speed region near the bottom. 



20 



