Decomposing Wave Patterns: ψ(x)=sin4x

In summary, the conversation discusses the concept of decomposing wave patterns into simpler components using double angle trigonometry formulas. The example given shows how ψ(x) = sin4x can be decomposed into a sum of simpler wave patterns. The general formula for decomposing a wave function is also mentioned, with coefficients cn representing the amplitudes of each component. The conversation also discusses the significance of even and odd values of n in this decomposition process.
  • #1
terp.asessed
127
3

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 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...

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) 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?
 
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  • #2
cos22x = 1 - sin22x
and use the example.
 
  • #3
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
yes.
 
  • #5
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
terp.asessed said:
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.

terp.asessed said:
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.
 
  • #7
td21 said:
yes.
No. Sorry for that.
 

Related to Decomposing Wave Patterns: ψ(x)=sin4x

1. What is the equation for "Decomposing Wave Patterns"?

The equation for "Decomposing Wave Patterns" is ψ(x)=sin4x.

2. How do you interpret the meaning of ψ(x) in the equation?

ψ(x) represents the wave amplitude at a given point x on the wave.

3. What does the number 4 represent in the equation?

The number 4 is the frequency of the wave. It indicates how many complete cycles the wave will complete in one unit of distance, in this case, one unit of x.

4. Is there a specific unit for x in this equation?

No, the unit for x can be any unit of distance, such as meters or centimeters. This equation can be used to describe any type of wave, regardless of the unit of distance.

5. How can this equation be used in real-world applications?

This equation can be used in various fields such as physics, engineering, and even music. It can help in understanding and predicting wave behavior, such as sound waves, electromagnetic waves, and water waves. It can also be used in signal processing and analyzing data in various industries.

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