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The wave function $$ f(x,t) = A \sin (\vec{k}\cdot\vec{x} - \omega t+ \phi) $$

Can graphically describe the linear 2d wave propagation. Why this equations is written in this form: $$ f(x,t) = A [\cos (\vec{k}\cdot \vec{x} - \omega t+ \phi) + i \sin (\vec{k}\cdot \vec{x}- \omega t+ \phi)] $$ which is eventually converted to $$ f(x,t) = A e^{i \varphi(\vec{x},t)} $$

What does the addition of the imaginary part add to the graphical representation of the function?

Matteo
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Shadi Abdelhadi
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    Does this answer your question? What is the need of complex functions in wave analysis? – Michael Seifert Apr 21 '22 at 16:23
  • It's not about graphical representation, it's the fact that the exponential obeys the same linear equations the real solution does, and it is easier to manipulate (differentiate, group, etc.) terms like $e^{i(kz-\omega t)}$ than terms like $e^{-k''z}\cos(k'z-\omega t)$ (both can describe decaying wave due to absorption). – Ján Lalinský Apr 21 '22 at 16:35
  • My point as, why adding the term i sin (kx-ωt + ϕ) to the function when only the real part F(x,t) = A cos (kx-ωt + ϕ) would have perfectly sufficed the description of a linear sinusoidal wave function? Why must we add a redundant term i sin (kx-ωt + ϕ) just to fulfill the satisfaction of replacing this whole expression cos (kx-ωt + ϕ) + i sin (kx-ωt + ϕ) with the euler identity e^i(kx-ωt+ϕ) – Shadi Abdelhadi Apr 21 '22 at 17:50

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