Fourier Analysis: String Vibrations with Fixed Ends and Varying Height

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In summary, the conversation discusses a problem involving a string with fixed ends at t=0 and finding the first four coefficients for the given wave equation. It is determined that every fourth term is zero due to the use of trigonometric identities and the physical interpretation is that the initial wave pulse does not contain any harmonic of the frequency n*pi. The conversation also touches on graphing the wave using a 3D plot with t as the z-axis.
  • #1
Fourier mn

Homework Statement



String of length L and string mass-density of[tex]\mu[/tex] is fixed at both ends. at t=0
y(x,t)=
4xh/L 0<x<L/4
2h-4xh/L L/4<x<L/2
0 L/2<x<0
Find the first four coefficients, is anyone of them is zero? If so why is it?

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the peak is L/4 and the right corner of the triangle is L/2.

Homework Equations



2/L{[tex]\int[/tex](4xh/L) sin(nx[tex]\pi[/tex]/L) over 0<x<L/4 +
[tex]\int[/tex](2h-4xh/L)sin(nx[tex]\pi[/tex]/L) over L/4<x<L/2 +
[tex]\int[/tex] 0*sin(nx[tex]\pi[/tex]/L) over L/2<x<L}

The Attempt at a Solution



so the third integral is equal to zero, we are left with only two.
After integrating the first two I've got the following result for An=
(8h)/(n[tex]\pi[/tex])^2[2sin(n[tex]\pi[/tex]/4)-sin(n[tex]\pi[/tex]/2)]
is this result correct? I've never seen Fourier series described by two sine terms. should I change the height to (h+1) so the last integral won't be zero and then integrate?
any other ideas how to go about it?
 
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  • #2
Use the following trig identity: [itex]sin(2x)=2sin(x)cos(x)[/itex] to rewrite your second sine term as [itex]2sin(\frac{n \pi}{4})cos(\frac{n \pi}{4})[/itex]...what does that make A1,A2,A3 and A4?
 
  • #3
hmmmmm...I didnt think I can use cos as part of the Fourier series.
is it ok? but I'm still going to have two terms of sin and cos--
2sin(npi/4)(1-cos(npi/4)) it's still not uniformly zeros...
 
  • #4
every fourth term is a zero
 
  • #5
Yes, every fourth term is zero...is there any physical interpretation for this result?
 
  • #6
I'm still trying to figure out how to graph the wave.
But, I think that it means that the wave is symmetric about the x-axis and that's why it's zero, correct?
 
  • #7
Not really; it means that every fourth harmonic is zero...in other words the initial wave pulse doesn't contain any harmonic of the frequency n*pi
 
  • #8
so the other half of the string never moves?
i agree that every multiple of 4 will cause the amplitude to be zero, but i thought that the main notion from Fourier is that one should always consider harmonic frequency when using his analysis (i.e the frequency is the same for all=>harmonic)
 
  • #9
how would you go about graphing it? just choose arbitrary values?
 
  • #10
A1,A2 and A3 do contain the harmonic n*pi
 
  • #11
A1,A2,A3 contain harmonics of npi/4, npi/2 and 3npi/4 not npi...this means that the Fourier series of the initial waveform does not contain any terms with frequency (0,pi,2pi,...).

To graph the function just do a 3D plot of y(x,t) with t as your z-axis.
 
  • #12
ohhh...got you. thanks
 

FAQ: Fourier Analysis: String Vibrations with Fixed Ends and Varying Height

What is Fourier analysis?

Fourier analysis is a mathematical tool used to break down complex signals or functions into simpler components. It is based on the concept that any periodic function can be represented as a sum of simple sine and cosine waves.

How is Fourier analysis used in studying string vibrations?

In string vibrations, Fourier analysis is used to decompose the complex motion of a string into simpler harmonic components. This allows us to analyze and understand the behavior of the string in terms of its fundamental frequencies and modes of vibration.

What does it mean for string ends to be fixed and height to vary?

When studying string vibrations, the concept of fixed ends and varying height refers to a string that is firmly attached at both ends and has a changing height profile along its length. This type of string is commonly seen in instruments like guitars, where the height of the strings is adjusted to produce different notes.

What are some real-world applications of Fourier analysis in studying string vibrations?

Fourier analysis is widely used in various fields such as music, physics, engineering, and signal processing to study string vibrations. It is used to analyze the sound produced by musical instruments, understand the behavior of vibrating strings in bridges and buildings, and even in medical imaging techniques such as MRI scans.

Are there any limitations to using Fourier analysis in studying string vibrations?

One of the main limitations of Fourier analysis is that it assumes the string to be linear and perfectly elastic, which may not always be the case in real-world scenarios. Additionally, it may not accurately predict the behavior of strings with extreme non-linear characteristics or when there are multiple strings interacting with each other.

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