Representing a wave as a complex number.

[Fallen Seraph]

[SOLVED] Representing a wave as a complex number.

I'm just a bit confused as to the validity of representing the equation of a wave or oscillatory motion as a complex number. As is my understanding the argument for doing so goes thus:

Assuming our amplitude is 1, our equation is:

$$y(t) = cos ( \omega t)$$

Which we can write as

$$cos ( \omega t) = [Re] exp(i \omega t)$$

Which it certainly is. but then people seem to go on to manipulate $$[Re] exp(i \omega t)$$ as if it were simply $$exp(i \omega t)$$ and then consider the answers correct. For example my lecturers appear to be squaring $$exp(i \omega t)$$ as if it were $$= cos( \omega t)$$. Which, as far as I can tell, reduces to absurdity quite quickly:

$$cos( \omega t) = exp(i \omega t)<br /> <br /> \Rightarrow<br /> <br /> cos ( \omega t) = cos ( \omega t) + iSin( \omega t)<br /> <br /> \Rightarrow<br /> <br /> cos^2 ( \omega t) = cos^2 ( \omega t) - sin^2 ( \omega t) +2iCos( \omega t)Sin( \omega t)$$

The real part of the right side is clearly not equal to the real part of the left side. And so, I don't understand how $$exp(i \omega t)$$ can be used, usefully, to describe a wave.

 

[mjsd]

is this done in the context of electrical engineering?
clearly, by setting R cos(x) to be equivalent to R exp(ix) is just a way to simplify computation/algebraic manipulation, it is probably understood that what is important about the use of R exp(ix) is not the actual function itself, but the information carried by R and x.

 

[Bill Foster]

This is the representation of a wave function from Cartesian coordinates (x-y plane) to the Polar coordinates (absolute value and phase).

These two sites may help explain it:

http://en.wikipedia.org/wiki/Complex_number
http://en.wikipedia.org/wiki/Euler's_formula

 

[Fallen Seraph]

is this done in the context of electrical engineering?
clearly, by setting R cos(x) to be equivalent to R exp(ix) is just a way to simplify computation/algebraic manipulation, it is probably understood that what is important about the use of R exp(ix) is not the actual function itself, but the information carried by R and x.

Well it's not actually in the context of electrical engineering, but I see what you're getting at. That would make sense to me were this notation not used to perform calculations and derive formulae, but it is.


The picture I have uploaded might be a more succint summing up of my objection to such manipulation, if someone could enlighten me on it?

 

[mjsd]

you can still probably use the complex notation to derive some formulae if you treat Real and Imaginary part carefully. Basically, knowing what's your limit in the complex notation (without abusing it )
the example you showed should probably look like this
$$x = A \cos \theta = Re(Ae^{i \theta})$$
$$x^2 = (Re(Ae^{i \theta}))^2 = A^2 \cos^2 \theta$$

as you would know.

Example 2:

$$(e^{i\theta})^2 = e^{2i\theta} = \cos 2\theta + i \sin 2\theta$$
also
$$(e^{i\theta})^2 = (\cos \theta + i \sin \theta)^2<br /> = (\cos^2 \theta - \sin^2\theta) + i (2 \sin\theta\cos\theta)$$

this gives you two relationships for double angle (re and I am part)
$$\cos 2\theta = \cos^2 \theta - \sin^2\theta$$
$$\sin 2\theta = 2 \sin\theta\cos\theta$$

you probably know these too.

 

[Fallen Seraph]

I'm not convinced that one of my lecturers isn't abusing the notation in his derivations...


Nevertheless, your help is most appreciated.

 

[Galileo]

That notation can certainly lead to misconceptions.
The complex notation is used since exponentials are more easily manipulated than sines.
Writing $$cos( \omega t) = exp(i \omega t)$$
Is just horrifying (and plain wrong). The notation is supposed to make things easier, not more difficult.

Just distinguish carefully between the real signal $$y(t)=Acos(\omega t +\phi)$$
and the complex signal derived from that:
$$\tilde y(t)=Ae^{i\phi}e^{i\omega t}=\tilde A e^{i\omega t}$$,
at least in the beginning until you've become comfortable working with complex signals only.

As you can see, the phase constant is now absorbed in the complex amplitude.
Then establish the relation between the real and the complex signal:
$$y(t)=\Re{\tilde y(t)}$$
$$A=|\tilde y(t)|$$
$$\omega t+\phi =\arctan{\Im{\tilde y(t)}/\Re{\tilde y(t)}}$$

Adding two signals (waves) of the same frequency is now easy:
$$\tilde y_1(t)+\tilde y_2(t)=(\tilde A_1+\tilde A_2)e^{i\omega t}$$
So you get a wave with the same frequency and to get the complex amplitude of you just add the complex amplitudes of the other two waves. The real amplitude of the new wave is then $$|\tilde A_1+\tilde A_2|$$. That's easier than using trig identities.

However it does go wrong when you multiply two waves, for instance when you wish to compute the energy in a wave. Don't use the complex representation for that, for the same reasons you mentioned.