While solving for the electric and magnetic wave components using Maxwell's equations in the free space, I have read that it was assumed that propagation of the wave is in z direction and that all the field components in this direction may be expressed in the form e^(-r*z) where r is a complex number. It is stated that it is a reasonable assumption for any uniform transmission guide. Why is it so?
And what does the term 'propagating in z direction' mathematically mean, with respect to the solution obtained for each field component through variable separable method where the solution is in the form of the product of a separate functions each of which is a function (constant * exponential) of just one independent variable?
The solution you are describing is the plane solution for a plane wave with harmonic time dependency, e.i. a wave that propagates in the z-direction and has uniform electrical and magnetic fields in the x and y directions, normal to the z-direction. This solution can certainly only exist in free-space. It is not the only free-space solution. Another example is the spherical symmetric solution. In fact the existence of a plane or spherical wave, depends on the way the wave has been generated (initial and boundary conditions). Using these two solutions, as building blocs, one can generate an infinite number of other solutions by superposition based on a spatial Fourier analysis for plane waves. By spatial differentiation of the spherical wave, one can generate other elementary solutions such as the dipole, quadrupole, octuple...solutions. These can be used to generate a Taylor expansion of sound generated by sources in a region small compared to the wavelength of the radiation. The radio antenna radiates in first approximation as dipole.
In pipes or cavities the boundary conditions are completely different than in free space. Plane wave solutions do not necessarily describe the actual solutions. In fact when considering an electromagnetic wave in a metal pipe (zero electrical resistance) of uniform cross-section, it will not propage until its wavelength is short enough (high enough frequency). It can then be described as a plane wave, that propagate along a zig-zag path (bouncing up and down on the walls of the pipe). Its effective propagation speed along the pipe axis (for the energy) will be lower than the speed of light.
There are a lot of excellent textbooks discussing this. I would look for radar technology.
Regards
Mico
Hello,
In free space, the electric field and magnetic field, E and B also satisfy the wave equation. For example, since B=∇×A, you can get a differential equation for B by taking the curl of given equation. Similarly, the order of the operations curl and ∂/∂t can be interchanged.
You can refer any book of electromagnetic, you can easily get there.
Thanks,
Instantaneous solution of Maxwell's equation that lead to a standard second order differential equation solved by Laplace equation.
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