I am trying to derive the general equation of lamb wave. My book says that y = Aexp(i(kx−ωt) is the general equation of simple harmonic wave propagating in +ve x direction. but I am confused with its imaginary term. What is the purpose and its physical interpretation. Is it fine to derive the the equation by considering its real part only i.e. cos(kx-wt)???
Complex notation is largely a mathematical convenience, but the usefulness becomes apparent when you start talking about relationships between waves (phase and magnitude) and when dealing with (complex) impedance, Z.
When comparing the phase and magnitude of two waves, you could use A cos(omega t + phi) and just talk about the difference in A or phi for the two waves, but how do you visualize it? If I were to use polar coordinates to plot each value as a vector with a radius A and direction phi then it's easy to see if they have the same phase (point in the same direction) or magnitude (same radius), as well as what their other relationships may be (delta between them, etc.) Once you start from this visual relationship in polar coordinates, then you can translate that back to Cartesian, but using X and Y would have been confusing. Real and imaginary are good axis labels and happen to have mathematical relationships that match the sinusoidal oscillations we see. (Don't ask me whether the math existed before the need to use it for this application or vice versa, but there it is!)
As far as wave impedance, reactance (inductance or capacitance) is represented as an imaginary component since the energy in the wave is stored and re-released, while resistance is always a real (literally) loss term where the wave energy is lost as heat.
Communications engineers use complex vector notation heavily for representing complex modulations such as quadrature amplitude modulation (QAM) used in most wireless devices. Confusing the issue, the axes are labeled as I and Q instead of real and imaginary, but it's easy to create "constellations" of evenly spaced points in I and Q that have the same vector delta between each point in the IQ plane. If those points are the only valid (relative) magnitude and phase combinations for a given protocol, it's easy to resolve the symbols of the communication even with a significant amount of noise added to the signal. Visually a constellation is simply a rectangular grid of points. Noise causes those points to become "fuzzy", but as long as they're close enough to where they belong, the correct value can be extracted.
This links may help you
www.sjsu.edu/.../ch%201-complex%20representation%20of%20EM...
Use the book https://www.abebooks.com/book-search/title/rayleigh-lamb-waves-physical-theory-applications/author/viktorov/
and substitute everywhere cos( ωt) for exp(-i ωt) .
Dear Haider Rizvi ,
You can not considering the real part only as some info will be lost.
There are some formulations to convert the complex to real. However, this increases the complexity of system in term of bandwidth, components required etc... Maybe you can refer to below link for more info about this transformation.
Wish you all the best
https://learning.oreilly.com/library/view/practical-applications-in/9780133038446/ch07.html#
Ayad
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