I am trying to derive the general equation of Lamb wave. My book says that $$y = A\exp(i(kx−\omega t))$$ 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. $\cos(kx-\omega t)$?
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$\begingroup$ Note in $cos(kx-wt)$ you have not a term for a phase displacement. Euler's formula simplifies calculus (derivation, ...) and includes the phase in the complex constant (from x and t point of view) "A". $\endgroup$– pasaba por aquiJan 27, 2019 at 19:48
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1$\begingroup$ Related: What is the physical significance of the imaginary part when plane waves are represented as $e^{i(kx-\omega t)}$? $\endgroup$– Emilio PisantyJan 27, 2019 at 20:54
1 Answer
Yes, the real part is what you use in the end.
The main advantage of writing the complex form is when you want to do interference related calculations. Consider two waves $y_1 = A\cos(kx - \omega t)$ and $y_2 = A\cos(kx - \omega t + \phi)$. The complex representation is $y_1 = A_1e^{i(kx - \omega t)}$ and $y_2 = A_2e^{i(kx - \omega t)}$, where $A_1 = A$ and $A_2 = Ae^{i\phi}$. The combined wave can be written as $y = \text{Re}\left[(A_1 + A_2)e^{i(kx - \omega t)}\right]$, which is easier to deal with than doing it without complex numbers.
tl;dr: It's a mathematical convenience and it is the real part that contains the physics.