# Complex representation of a wave

• e101101
In summary, the conversation is about understanding the complex representation of waves in an Optics course. The equation E=Eoexp[k•r±⍵t] was given in class and the person is looking for a clear and simple explanation of why this is the result. They also ask about the complex representation of the function and how it would change with a phase constant or if it was a sine function. The experts provide resources and explanations on Euler's formula and useful identities to help understand the topic better.
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?

e101101 said:
Homework Equations: E=Eoexp[k•r±⍵t]
You missed an i in the exponential.

e101101 said:
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
e101101 said:
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

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.

DrClaude and Delta2

## 1. What is a complex representation of a wave?

A complex representation of a wave is a mathematical representation that describes the amplitude and phase of a wave at a specific point in space and time. It uses complex numbers to represent both the real and imaginary components of a wave, allowing for a more comprehensive understanding of how a wave behaves.

## 2. How is a complex representation of a wave different from a simple sine wave?

A simple sine wave only describes the amplitude and frequency of a wave, while a complex representation also takes into account the phase of the wave. This means that a complex representation can provide more detailed information about the behavior of a wave, such as interference and diffraction patterns.

## 3. What is the purpose of using complex representations in wave analysis?

Complex representations are useful in wave analysis because they allow for a more accurate and comprehensive understanding of how waves behave. They can be used to analyze complex wave phenomena such as interference, diffraction, and superposition, which cannot be fully described using simple sine wave representations.

## 4. How are complex representations of waves used in practical applications?

Complex representations are used in a variety of practical applications, such as in telecommunications, signal processing, and quantum mechanics. They are also used in the study of electromagnetic waves, acoustic waves, and other types of waves in physics and engineering.

## 5. What are some common examples of complex representations of waves?

Some common examples of complex representations of waves include the Fourier transform, which decomposes a wave into its frequency components, and the complex exponential representation, which describes a wave as a combination of real and imaginary exponential functions. The complex phasor representation is also commonly used in electrical engineering to analyze the behavior of AC circuits.

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