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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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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.

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