Decomposing Wave Patterns: ψ(x)=sin4x

[terp.asessed]

Homework Statement

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 c_n such that:

ψ(x) = sigma (n=0 to infinite) c_n cos_nx 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...

Homework Equations

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) c_n cos_nx 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?

 

[td21]

cos²2x = 1 - sin²2x
and use the example.

 

[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?

 

[td21]

yes.

 

[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?

 

[td21]

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?

On line 3, cos4x will appear.

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?

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

 

[td21]

yes.

No. Sorry for that.