Complex representation of a wave

[e101101]

Homework Statement

Hi there,
I'm currently taking an Optics course and the teacher is expecting us to have an understanding of the complex representation of waves. Although, hardly any of us have even heard of this yet. I've tried to google how to convert a cos(obj) and sin(obj) to an exponential... but I just don't understand. I would really love it if someone could explain this to me, as this has been really been bringing me down.

Relevant Equations

E=Eoexp[k•r±⍵t]

Homework Statement: Hi there,
I'm currently taking an Optics course and the teacher is expecting us to have an understanding of the complex representation of waves. Although, hardly any of us have even heard of this yet. I've tried to google how to convert a cos(obj) and sin(obj) to an exponential... but I just don't understand. I would really love it if someone could explain this to me, as this has been really been bringing me down.
Homework Equations: E=Eoexp[k•r±⍵t]

This is an equation we saw in class today:

E=Eocos[k•r±⍵t] (where E, Eo, k and r are vectors). The answer was given to us (equation above), but I really want to understand why that is the result.

What would the complex representation of this function be? Could you please be thorough with your explanation... I am so lost and need a clear (and simple) answer.

Also, how would this result change if there was a constant in the argument of the cos function (ex: phase constant)? What if this was a sine function (I know you can switch any sine function to a cos, but I would like to know how to do it the 'hard' way?

 

[BvU]

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LCKurtz
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@lychette: You might find my article "There's nothing imaginary about complex numbers" interesting as a way to broach the subject to beginners. It's at https://math.la.asu.edu/~kurtz/complex.html

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[DrClaude]

Homework Equations: E=Eoexp[k•r±⍵t]

You missed an i in the exponential.

What would the complex representation of this function be? Could you please be thorough with your explanation... I am so lost and need a clear (and simple) answer.

https://en.wikipedia.org/wiki/Euler's_formula

Also, how would this result change if there was a constant in the argument of the cos function (ex: phase constant)? What if this was a sine function (I know you can switch any sine function to a cos, but I would like to know how to do it the 'hard' way?

I'm not sure that this answers your question, but as you wrote, to general solution is a sum of a sine and a cosine. There can be combined into a phase-shifted cosine or sine. By convention, most of the time the cosine is used.
https://www.myphysicslab.com/springs/trig-identity-en.html

 

[jedishrfu]

Here’s a reference to the Euler equation on Khan Academy. There are other videos there that can explain this topic better than we can from memory.

https://www.khanacademy.org/math/ap...-10-14/v/euler-s-formula-and-euler-s-identity

 

[RPinPA]

Here are some useful identities.

Euler's Identity: $e^{i\theta} = \cos(\theta) + i \sin(\theta)$

From which you get:
$\cos(\theta) = \frac {(e^{i\theta}+ e^{-i\theta})} { 2}$
$\sin(\theta) = \frac {(e^{i\theta}- e^{-i\theta})} { 2i}$

And since a phase shift in sine or cosine can be written as a combination of sines and cosines, for example:
$A \cos(\omega t + \phi) = A [ \cos(\omega t) cos(\phi) - \sin(\omega t) \sin(\phi)] = (A \cos\phi) \cos(\omega t) - (A \sin\phi) \sin(\omega t) \\ = B \cos(\omega t) + C \sin(\omega t)$
then you can write a general phase-shifted sine or cosine as a combination of exponentials using the above identities.

Note that you get both a positive and negative exponent when you make that substitution. If you have something representing a real-valued signal, then when represented as exponentials the full expression will have complex conjugate terms such that there's not actually any imaginary waves around.

But most of the time you don't worry about that kind of thing, and the exponential notation makes calculation of all kinds of things a heck of a lot easier.