Question

is it possible to understand a simple wave without using complex numbers

Answer 1

\[ \sin(kx - \omega t)\]
describes a traveling wave, is that what you mean?

Answer 2

f(x-vt), for any function f, is a wave, as is g(x+vt) for any function g.

I think the issue you may be having is with functions like

e^{ik(x-vt)}

which are mathematically simpler "building block" functions that can be
used to create the more general cases. They take a little getting used to,
but in the long run they make things simpler...

Answer 3

\[\sin x= \frac{ie^{-ix}+-ie^{ix}}{2}\]

Answer 4

the derivation of your identity comes from Euler identity \[e^{{i}{\theta}} =\cos(\theta)+isin(\theta)\]

Answer 5

\[e^{{-i}{\theta}}= \cos(\theta)-isin(\theta)\] hence

Answer 6

\[\cos(\theta)+isin(\theta)-\cos(\theta)+isin(\theta)= 2isin(\theta)\]

Answer 7

and so we have \[[{e^{{i}{\theta}}-e^{{i}{\theta}}} ]/ 2i = \sin(\theta)\]

Answer 8

so they are interchangeable.

Answer 9

ah yes, because cosine is even

Answer 10

ops i forgot to put the negative sign into the second value for e also your equation comes by simply multiply \[i/i\]

Answer 11

\[ i/i * [e^{iθ}−e^{-iθ}]/2i=\sin(θ)\] -->\[-i [e^{iθ}−e^{-iθ}]/2=\sin(θ)\]

Answer 12

ah yes, because cosine is even .. exactly :)

Answer 13

you can understand many things about waves just looking at the sea :-)