Complex notation in telegrapher's equations

[solanojedi]

Hi everyone,
I'm reading about the solution of the telegrapher's equations (e.g. the generalities are here https://en.wikipedia.org/wiki/Telegrapher's_equations ). Supposing we are treating only time [itex]t[/itex] and space [itex]z[/itex], this is a second order partial differential equation on an infinite domain of space and infinite domain of time. If we use the variable separation technique, we get two differential equations of the second order in [itex]t[/itex] and [itex]z[/itex]. They are called singular (since the domain is infinite) Sturm-Liouville problems and their solution is a continuous superposition of complex exponentials.
The complete solution should be something like this [tex]V(z,t)=\int_{- \infty}^{+\infty} \left ( V_{1}e^{ikz}+V_{2}e^{-ikz}\right ) e^{i \omega t} d\omega[/tex] Then, usually, when we want to discuss this kind of solution, we take the real part of it and get a cosinusoidal function that is the usual traveling wave.

My question is: what do we lose when we do this operation? What the imaginary part of the partial differential equation solution represents?
(in many books they do a different procedure: first they introduce the Fourier transform, then the concept of "phasors" that is basically a compact writing for a cosinusoidal function using only complex exponential and finally they modify the differential equation eliminating the time variable. In this way they have only one variable, they resolve the equation and finally they attach the time solution and take the real part of the entire solution. However, there is something that doesn't convince me with this real part selection, since if I look inside a partial differential equation book I see that the solutions are already made of complex exponentials, without introducing those phasors)

Thank you in advance for your help.
Nicola

 

[jasonRF]

These equations describe the behavior of physical quantities that are manifestly real. Various mathematical techniques used to solve the equations may yield complex quantities as intermediate results, but a final result that describes the physical voltage in space and time must be real. Your pde book may allow the final answer to be complex, but nature doesn't.

The phasor analysis technique is used all the time by engineers for quick steady state analysis of time invariant linear systems. The fact that the final result must be real is explicitly taken into account. Other techniques, such as explicit Fourier analysis, must also take it into account.Edit: perhaps a better way to say the above is that mathematically the solutions can be complex valued. A physical solution, though, must be real valued. This is an often unwritten constraint that scientists and engineers require.

 

[solanojedi]

Thank you very much Jason!
This is a very reasonable fact.

I actually never saw (or I don't remember) this type of explicit justification in books: in EM books (like Harrington, Balanis...) I find the usual use of the phasor approach that seems reasonable since it's a compact view for real signals and all the theory is developed with this construction.
But in an opposite approach (without using the Fourier harmonic decomposition at the very beginning of the discussion), when I get to the solution of the PDE that involves complex exponentials, I find more strange considering only the real part of the solution without any particular notice.

Eventually, starting from a real PDE, finding the very general mathematical solution (i.e. with complex numbers) and keeping only the real part because it is related to the physical world since the starting quantities involved are real, it seems a reasonable justification.

Thank you again!