However, we don't usually do it that way for open-channel flows.

More commonly, we use the Manning equation given here, which says that the velocity

is equal to K over N, R H to the two thirds, S to the one half power.

And this is the corresponding section from the reference handbook.

Where in this equation N is a constant,

known as the Manning coefficient or Manning's N.

Rh is the hydraulic radius, divided by the cross section area,

divided by the wetted perimeter.

In other words, a quarter of the hydraulic diameter.

S is the channel slope, and

K is a factor which is different depending on what units we're working in.

The Mannings equation is unfortunate in fluid mechanics,

in that it's not dimensionally homogeneous.

And it's a different equation, depending on what system of units you're,

you're working in.

In metric units or SI units, the value of K is 1.

But, in US customary system of units,

the value of the constant is one point four eight six.

Here is the equation again in terms of velocity.

And because the volume flow rate Q is equal to velocity times area,

we can also write the Manning equation in terms of volume flow rate as shown there.

And N in this equation, Manning's N, is a type of roughness parameter.

And the reason that Manning's equation is useful often

is that there is a lot of empirical evidence built up over the years

on what is a suitable value of N to use for different situations.

For example, for natural channels here a typical value of Manning's N for

clean and straight channels is 0.03.

For sluggish or major rivers with a lot of vegetation and

meandering, it increases to about 0.035.

For very, large roughness elements, for example heavy brush or

trees such as you would encounter in a flood plane,

Manning's N increases substantially to 0.075 or as much as 0.15.

On the other hand, for very smooth channels, for example,

artificial channels with glass lining, it's 0.01.

For steel, etc., it increases to 0.015.

So, N is a type of a roughness coefficient.

To illustrate that, let's do some examples.

So in this question we have a rectangular cross section channel four meters wide,

carrying a discharge of two cubic meters per second,

at a uniform depth of 1.5 meters.

Manning's N is 0.012.

And the question is the slope in percent of the channel bed

is most nearly which of these alternatives?