1/a is the height of the capillary fringe above the water table, and d is the depth to the water table. Both of them decrease during rainfall period.
A root of the equation is intersection of y = tan(xad) and y=-2x . It says that first positive root is between pi/2 and pi. You may find it by Newton-Raphson's method.
I recommend Wolfram Alfa (https://www.wolframalpha.com/) to practical solve this problem
You should write the command similar to that
findroot (tan(1/3*x) +2* x=0)
and click icon compute. If you can’t see the results click sentence (“a math function”) which is below the task frame.
or you can drow tan(xad) and -2x. Confluence of two Graphs is root.
see attachment for example.
You can also use MATLAB to solve this problem (root-finder) by using the command 'fsolve'. After setting some values for 'a' and 'd', you may define a function 'f' and find the roots with 'fsolve'.
For example:
>> a=10;d=5;
>> f=@(x)[tan(a*x*d)+2*x]
If we give the initial condition 'x0' as 'x0=1', and use 'fsolve' in MATLAB, you get:
>> x0=1;
>> x=fsolve(f,x0)
Output:
x =
0.983302064820308
>> x0=2;
>> x=fsolve(f,x0)
x =
1.984140541948211
This goes on... With each initial guess as positive integer, you get a root
close to it..
The transcendental equation f(x) = 0 has infinitely many positive roots.
The 'OptimTool' in MATLAB is very handy for finding solutions (positive roots) for different values of the constants a and d.
Best wishes,
Sundar
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