Field Patterns in Waveguides
Except for some very special cases, you actually can't solve the field equations analytically. For most cases, you can only approximate the solution by a numerical analysis. Typically you use something like the finite element method or finite difference time domain method to obtain these approximate solutions. 

Even for the special, solvable cases, there's no real "method" to solving them. Like with many partial differential equation problems, you basically guess at an answer, and then demonstrate that your guess is correct. For example, if the problem has a radial symmetry, you might guess that the solution has a Bessel function component in the radial direction. 


  What is the practical use of these patterns?


Knowing the solutions can help you guess how some change to the problem might affect the solution. 

For example, if you have a waveguide mode with a null in the E field along the y-axis of the waveguide, then you can guess that putting a wire in that location won't cause much reflection, or that a feed at that location won't efficiently excite that mode.