# Confused over the complex term in the simple harmonic wave equation

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. $$\cos(kx-\omega t)$$?

• 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". – pasaba por aqui Jan 27 '19 at 19:48
• – Emilio Pisanty Jan 27 '19 at 20:54

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.