6.31

The Manning equation can be written for a rectangular open channel as

Q = ((sqrt(S) * (B*H)^(5/3))/(n*(B +2*H)^(2/3)));

where Q = flow [m^3/s], S = slope [m/m], H = depth[m],
and n = the manning roughness coefficient. Develop a fixed-point iteration
scheme to solve this equation for H given Q = 5, S = 0.0002, B = 20 , and
n = 0.03. Prove that your scheme converges for all initial guesses greater
than or equal to zero.


Matlab code:

H = 0:.0001:4;

Q = 5;

S = 0.0002;

B = 20;

n = 0.03;

hold on

%Q = @(H) ((sqrt(S) * (B*H) .^(5/3))/(n*(B +2*H) .^(2/3)));  %Original Q

f = @(H) (sqrt(S)*(B*H).^(5/3) ./ (n*(B+2*H).^(2/3)))-Q;        %Rearranged

plot(H,f(H))

g = @(H) (((Q * (n*(B +2*H) .^(2/3))) ./ (sqrt(S))) .^(3/5))/B ;  %arranged to solve for H

H = 1;

for ii = 1:10,

H = g(H);

plot(H,f(H),'*g')

end

EDU>> H

H =

    0.7023

EDU>> f(H)

ans =

     0