The most efficient trapezoidal section of canal can be calculated as follows

Let,

Base width of the canal=B

Side slope=m (1V:mH)

Flow depth = h

Wetted perimeter= P

The cross sectional area, A=By+my^2

ie B= (A-m*h^2)/h

ie B=A/h-m*h

Wetted perimeter P= B+2*h *sqrt(m^2+1)

For most efficient channel, dP/dh=0

ie -A-m*y^2+2 y^2 sqrt(m^2+1)=0

ie B= y(2*sqrt(m2+1)-1-m)

Therfoer **B/y=2*sqrt(m2+1)-1-m i**s the required relationship