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I've read that plane wave equations can be represented in various forms, like sine or cosine curves, etc. What is the part of the imaginary unit $i$ when plane waves are represented in the form $$f(x) = Ae^{i (kx - \omega t)},$$ using complex exponentials?

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It doesn't really play a role (in a way), or at least not as far as physical results go. Whenever someone says

we consider a plane wave of the form $f(x) = Ae^{i(kx-\omega t)}$,

what they are really saying is something like

we consider an oscillatory function of the form $f_\mathrm{re}(x) = |A|\cos(kx-\omega t +\varphi)$, but:

  • we can represent that in the form $f_\mathrm{re}(x) = \mathrm{Re}(A e^{i(kx-\omega t)})=\frac12(A e^{i(kx-\omega t)}+A^* e^{-i(kx-\omega t)})$, because of Euler's formula;
  • everything that follows in our analysis works equally well for the two components $A e^{i(kx-\omega t)}$ and $A^* e^{-i(kx-\omega t)}$;
  • everything in our analysis is linear, so it will automatically work for sums like the sum of $A e^{i(kx-\omega t)}$ and its conjugate in $f_\mathrm{re}(x)$;
  • plus, everything is just really, really damn convenient if we use complex exponentials, compared to the trigonometric hoop-jumping we'd need to do if we kept the explicit cosines;
  • so, in fact, we're just going to pretend that the real quantity of interest is $f(x) = Ae^{i(kx-\omega t)}$, in the understanding that you obtain the physical results by taking the real part (i.e. adding the conjugate and dividing by two) once everything is done;
  • and, actually, we might even forget to take the real part at the end, because it's boring, but we'll trust you to keep it in the back of your mind that it's only the real part that physically matters.

This looks a bit like the authors are trying to cheat you, or at least like they are abusing the notation, but in practice it works really well, and using exponentials really does save you a lot of pain.

That said, if you are careful with your writing it's plenty possible to avoid implying that $f(x) = Ae^{i(kx-\omega t)}$ is a physical quantity, but many authors are pretty lazy and they are not as careful with those distinctions as they might.


(As an important caveat, though: this answer applies to quantities which must be real to make physical sense. It does not apply to quantum-mechanical wavefunctions, which must be complex-valued, and where saying $\Psi(x,t) = e^{i(kx-\omega t)}$ really does specify a complex-valued wavefuntion.)

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From Euler's formula, we have:

$$Ae^{i\theta} = A(\cos \theta + i\sin \theta)$$

This is a common way to express $sine$ or $cosine$ waves as it makes the math easier.

As you are asking, many get startled by that $i$ in the exponent. The $i$ does not mean that the quantity or the wave is imaginary. It does not have anything to do with it. While solving problems, we consider the real or the imaginary part of that expression only (in most cases we consider either one of them but not both).

For example, $A\cos (\omega t -kx)$ can be expressed as $Real(Ae^{i(\omega t - kx)})$ where $Real(x)$ gives the real part of the complex number $x$.

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Actually the complex expression for a plane wave is extremely useful in signal processing and areas of electronic engineering like communications systems and signals. It is called the analytical signal. Wikipedia has a pretty good explanation of what it is useful for, the mathematics of how and why. See https://en.m.wikipedia.org/wiki/Analytic_signal

From that article "The analytic representation of a real-valued function is an analytic signal, comprising the original function and its Hilbert transform. This representation facilitates many mathematical manipulations. The basic idea is that the negative frequency components of the Fourier transform (or spectrum) of a real-valued function are superfluous, due to the Hermitian symmetry of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complex-valued function instead. That makes certain attributes of the function more accessible and facilitates the derivation of modulation and demodulation techniques, such as single-sideband. As long as the manipulated function has no negative frequency components (that is, it is still analytic), the conversion from complex back to real is just a matter of discarding the imaginary part. The analytic representation is a generalization of the phasor concept:[2] while the phasor is restricted to time-invariant amplitude, phase, and frequency, the analytic signal allows for time-variable parameters."

And uyes the analytic function is used by physicists to analyze physics related signals - keep in mind that signals are what much of the electronic engineering community uses to refer to waveformsl. An example is the way it is used in analyzing and preparing waveform templates for detection of gravitational waveforms and later processing for the correlation against the received signal to detect and then obtain the waveform parameters. An example is in the paper in Arxiv at https://arxiv.org/pdf/1606.03952.pdf, where it is used for that purpose.

Complex exponentials are much easier to analyze and process than using sines and cosines, and easily processed nowadays (and for over 20 years) using signal processing techniques and Fourier transform, correlation , parameter estimation (eg, for spectral analysis and modulations, distortions, and so on) and other techniques. The real and imaginary parts are related by a Hilbert transform, which is expensively used in the processing of the signals. The real and imaginary parts are sometimes simply called the in-phase and quadrature components, and they carry information since they are mutually orthogonal. For physics comex exponentials are used to describe and best analyze any wave motion and effects and for perturbations it is the best way to do it also.

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