I have to make a program (using finite difference method) to determine the lowest normalized eigenvalue, ka, for the TM mode defined by Helmholtz differential equation. I have to do this using Matlab (I have just started studying it!!) but the main question is: what does actually represent the eigenvalue of the TM mode (I understand niether eigenvalue nor the modes). What's the phisical meaning of eigenvalue? what about TM, TE, TEM? (if there are some free resources about the last topic, i would like to know) Thanks a lot!
It may be next to impossible to give you what you need in an online forum. Let me try to get you started. Ordinary Differential Equations generally have unique solutions. In contrast Partial Differential Equations have many solutions depending on the conditions at the geometrical boundry. An eigenvalue is a scaler quantity β which satisfies the following equation Code ( (Unknown Language)): Ax = β x for some matrix A and some vector x The existence and the interpretation of eigenvalues has to do with the geometry of the problem. To begin understanding "modes" it may be helpful to think of a vibrating string constrained at both ends vibrating at some fundamental frequency. Now imagine that the string begins vibrating at its second harmonic. This would be an example of a different mode. If you can get your arms around this explanation then we can go further. If not, I fear that you task will be difficult to complete in the available time.
I think I can (I have to!) Unfortunately it has to be clear to me in a fortnight (maybe if I would spend 20 hours a day with this!) I need to understand first... Question: For the example with the vibrating string: how/ why would it begin to vibrate on the harmonics of the fundamental frequncy? For example a LC circuit would resonate only on a single frequency (dependent on the values of L and C) and one needs to change C (or L) in order to oscillate on other frequency, right? What happens in microwaves that makes possible different modes? I don't know if my question is clear (it is not very clear in my mind either)
In the case of the vibrating string the initial condition might lead to an intial vibration at the second harmonic. As the amplitude decreased the mode might shift back to the fundamental. It was only an analogy to illustrate what a "mode" was. What happens in microwaves is the geometry of the problem. Some more definitions: TEM - Transverse Electromagnetic Field. The Electric Field Vector and the Magnetic Field Vector are perpendicular (transverse) to the direction of wave propagation. TM - Transverse Magnetic Field. The Magnetic Field Vector is perpendicular(transverse) to the direction of wave propagation. The Electric Field Vector may have a component in the direction of propagation. TE - Transverse Electric Field. The Electric Field Vector is perpendicular(transverse) to the direction of wave propagation. The Magnetic Field Vector may have a component in the direction of propagation. Hybrid Waves - These waves are a combination of TE & TM. Both the Electric Field Vector and the Magnetic Field Vector are allowed to have non zero components in the direction of propagation. Many Partial Differential Equations can be solved by a method called separation of variables. Doing this for the wave equation leads to time varying solutions multiplied by time invariant solutions. These correspond to standing waves and traveling waves. A Standing Wave is a time invariant solution that is a function of position only. A Traveling Wave is a position independent wave that is a function of time An example of such a function would be Code ( (Unknown Language)): exp(j ω t - β z) = exp(j ω t) * exp(- β z) where z - position along the z axis t - time β - a scalar constant ω - a scalar constant j - √ (-1) the imaginary unit Note how exp( -β z) is time invariant and only a function of the z coordinate? Note how exp(j ω t) is time varying and independent of position?
Thank you, Papabravo! I have started reading a book on microwaves. I have to refresh what the few things I once knew in mathematics and to learn much more. It is ok, excepting the time I have for this