78 A THEORY OF FLUID FRICTION. 
head, the final change being an increase, it would appear that the whole decrease in 
velocity will occur on the smooth surface. A more rapid decrease in velocity at the 
surface of the water will serve to prevent the occurrence of a decrease in the head in 
the early stages of the change in velocity. The change back to the greater velocity 
will occur partly on the rough surface and partly on the smooth, as in the previous 
case (Fig. 6, Plate 59). | 
Consider an open trough of some width, with a strip of rough bottom intro- 
duced at the middle of the width. The tendency will be for the depth of the water 
over the rough strip to increase as its velocity decreases. This increase in depth 
will cause a side flow, and the depth over the smooth bottom adjacent to the rough 
strip will increase. Shortly, the depth will become uniform over the trough, the 
new depth being such as will give the same volume of flow. The velocity over the 
smooth bottom will be unchanged, and the velocity over the rough strip will be that 
suited to its coefficient of friction and the slope of the trough. 
FLOW IN THE OPEN. 
Consider a long vertical plane, of considerable width but no thickness, partly 
immersed in water flowing past the plane with a velocity Vo where not affected by 
the friction of the plane. The conditions affecting the flow are composite. If the 
velocity is great, the retardation adjacent to the plane will be progressive. The 
width of the wake will increase as the velocity at the plane decreases, and the cross- 
section of velocities will be of some parabolic form following the laws applicable in 
the case of the pipe. The total volume of fluid passing will remain constant, but 
not the volume of individual laminz, as the height of the water at the plane will 
not increase in the same proportion as the velocity decreases. The elevation of the 
surface of the water will extend beyond the outer limit of the wake, and the sur- 
face will be only slightly raised at the plane. 
If we neglect the effect of eddies, we may consider that the reduction in veloc- 
ity which has occurred at a given distance from the front of the plane will be great- 
est adjacent to the plane and decrease as the distance from the plane increases, 
until the original velocity of flow is reached. The width of this frictional wake 
will depend upon the reduction of velocity adjacent to the plane, and upon the value 
of tana corresponding to the roughness of the surface of the plane, and the reduced 
velocity. It will depend also upon the nature of the curve of velocities across the 
section of the wake, as in the case of the pipe. This curve of velocities at any cross- 
section can be determined by considering that the loss of head outboard of any 
section parallel to the plane must be balanced by the friction occurring in the 
wake at the longitudinal section in front of the cross-section selected. 
If the surface of the plane is such that the distance from the cutwater is con- 
siderable before a great reduction in velocity occurs, so that the elevation of the 
surface of the water can be neglected, it seems probable that the curve of velocities 
across the wake resembles a cubic parabola. 
