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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

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# A Complex notation in telegrapher's equations

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