Wave Patterns

1. Oct 15, 2014

terp.asessed

1. The problem statement, all variables and given/known data
I read from a book (obtained from a library) which stated that:

"Wave patterns, no matter how complicated, can always be written as a sum of simple wave patterns.
Ex: ψ(x) = sin2x = 1/2 + cos2x/2
"

I understand that ψ(x) has been decomposed with double angle trignometry formulas.

"More generally, it is possible to decompose the wave function into components corresponding to a constant pattern plus all possible wavelengths of hte form 2pi/n with n, an integer. That is, we can find coefficients cn such that:

ψ(x) = sigma (n=0 to infinite) cn cosnx In this ex, c0 = 1/2 and c2 = -1/2. All other coefficents are zero.
"

So...since the statement said, "wave patterns, no matter how complicated," I decided to try out with ψ(x) = sin4x out of curiosity....

2. Relevant equations

3. The attempt at a solution
ψ(x) = sin4x
I used double angle formulas to get:
ψ(x) = (1-cos2x)2/4...meaning the wave pattern is decomposed into:
ψ(x) = 1/4 + cos2x/2 + cos22x/4

However, I am trying to figure out about

"ψ(x) = sigma (n=0 to infinite) cn cosnx In this ex, c0 = 1/2 and c2 = -1/2. All other coefficents are zero.
" Could someone explain how to use this method so that I can try it out on ψ(x) I just made up?

2. Oct 15, 2014

td21

cos22x = 1 - sin22x
and use the example.

3. Oct 15, 2014

terp.asessed

ψ(x) = 1/4 + cos2x/2 + cos22x/4
= 1/4 + cos2x/2 + 1/4 - sin22x/4
= 1/2 - cos2x/x -1/4(1/2)(1-cos2x)
= 3/8 - 3cos2x/8...since it is now in two terms...I guess c0 = 3/8 and c2 = -3/8?

4. Oct 15, 2014

td21

yes.

5. Oct 15, 2014

terp.asessed

Hmm..but, I don't understand why only even values of n show up? I mean, why is c1 or c3 = 0, not not for n = 0 and 2? It seems it happens to both ψ(x) = sin2x and sin4x? Or, am I thinking too much?

6. Oct 16, 2014

td21

On line 3, cos4x will appear.

Odd values of n will show up for some other functions.

7. Oct 16, 2014

td21

No. Sorry for that.