How are the field patterns for magnetic and electric field obtained using the field equations? What is the intuitive way of understanding and drawing them? I searched a lot regarding the method for drawing field patterns but at most places there were just figures saying - "The field patterns obtained from equations are as follows!" Plus, what is the practical use of these patterns? How are they useful?
One starts with Maxwell's equations and some boundary conditions involving derivatives and vector components. In general, this is hard to deal with but we begin making some approximations and assumptions.
We are interested only in fields oscillating at some frequency f. Therefore, all fields will be varying with time as \$exp(j\omega t)\$ where \$\omega=2\pi f\$ Then all time derivatives become multiplications by phase factors: calculus becomes algebra, at least along the time dimension. Oh, I just used complex numbers. That's another trick - taking the physical fields to be the real part of some mythical complex-valued field lets us use simpler algebra.
The waveguide has the same shape along some length. Just as we assume simple sinusoidal (or complex exponential) oscillation along time, we can do so also along the 'z' axis. (We take x,y as the plane perpendicular to the waveguide, where we define it cross-sectional shape, and z along the waveguide, as a common convention.) So Curl, Div and all that get simpler, with derivatives with respect to z becoming simple coefficients involving wavelength (or wavenumber, its reciprocal) and complex phase factors.
We're dealing with radiation only, not static electric or magnetic fields, not motor or generator dynamics, not high energy particle beams. We can say that the electric and magnetic fields are perpendicular for a wave of any given wave direction. When multiple waves are superimposed, maybe we can't say so any more, but not a problem because we solve for field patterns for pure waves first, then add together what we like. Anyway, the point is we can eliminate all magnetic field terms from the equations, replacing them with terms involving the electric field only.
Finally, we luck out in that the most easily manufactured waveguide shapes are rectangles or circles. We have a differential equation in two dimensions (x and y) for an electric field (a vector) and some boundary conditions applying to the parts of the field parallel or perpendicular to the wall of the waveguide. For a rectangle, we can treat the x part and the y part separately. For a circle, the radial and angular parts can be separated. The technique is called "separation of variables" and is very popular in electromagnetics, quantum mechanics, acoustics and basically almost everything else in physics. Result: a pair of simple ordinary differential equations, each easily solved.
For more complex shapes, or waveguides with ridges, holes, dents along the way, then it gets messy and we lose one or two of these nice simplifications. Time to turn it over to a computer, then.
Once we have the field pattern, the usefulness is in knowing where the field is strongest - you want to minimize obstructions, and use the best materials. Where the field is zero or fairly weak, you can put screws, welding seams, holes, or whatever without affecting the waves. Examination of the field patterns also helps with determining how well polarization is preserved within the structure, which is important to radio astronomers and for doubling information capacity in telecommunications. Finally, field patterns make for great illustrations to impress the public, congresscritters and investors .
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.