Hydraulic exponents for rectangular and triangular

Definitions

• First hydraulic exponent $M$ (for critical flow): $Z^2=\dfrac{A^3}{T}\propto y^{M}$. It is used in the critical flow condition $\dfrac{Q^2}{g}=\dfrac{A^3}{T}$.
• Second hydraulic exponent $N$ (for uniform flow): the conveyance $K=\dfrac1nAR^{2/3}\propto y^{N/2}$, i.e. $K^2\propto y^N$.

Wide rectangular channel (width $b$, depth $y$)

• $A=by$, $T=b$, and $R\approx y$ since $b\gg y$.

$$\frac{A^3}{T}=\frac{b^3y^3}{b}=b^2y^3\;\Rightarrow\;\boxed{M=3}$$ $$K=\frac{1}{n}(by)\,y^{2/3}=\frac bn\,y^{5/3}\;\Rightarrow\;K^2\propto y^{10/3}\;\Rightarrow\;\boxed{N=\frac{10}{3}=3.33}$$

Triangular channel (side slope $z$)

• $A=zy^2$, $T=2zy$, $P=2y\sqrt{1+z^2}$, so $R=\dfrac{zy}{2\sqrt{1+z^2}}=c\,y$, where $c=\dfrac{z}{2\sqrt{1+z^2}}$ is a constant.

$$\frac{A^3}{T}=\frac{z^3y^6}{2zy}=\frac{z^2}{2}y^5\;\Rightarrow\;\boxed{M=5}$$ $$K=\frac1n(zy^2)\,(c\,y)^{2/3}\propto y^{2+2/3}=y^{8/3}\;\Rightarrow\;K^2\propto y^{16/3}\;\Rightarrow\;\boxed{N=\frac{16}{3}=5.33}$$