exercise_13 waves on a string

chapter 6
姚媛媛 2014301020047

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Abstract

This article will research waves on a string, includes its motion and power spectra. Here the particular case of waves on a string is considered. At the beginning, a solution for the wave equation in the ideal case is introduced and developed, that is, for a perfectly flexible and frictionless string. Only one initial Gaussian wave packet and two initial Gaussian wave packets are considered to show that the wave packets are unaffected by the collisions. Besides, Fourier analysis is applied in the spectral analysis to exam the waves on a string. 　 　

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backgrond

the fundamental wave equation is

$$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}$$

We discretized the expression to calculate it numerically,

$$y(i,n+1)=2(1-r^2)y(i,n)-y(i,n-1)+r^2[y(i+1,n)+y(i-1,n)]$$

where $r=c\frac{\Delta t}{\Delta x}$, and $r=1$ is the best choice.

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mainbody

Waves on a string with fixed ends in different initial packet



Besides, we can draw them together, here we have:

Vibration of a point on this string
Considering the vibration of a point located on 5% of this string, if its intial profile is Gaussian profile, its vibration with time is shown:


Obviously, if the frequency of power spectrum is odd times of 150Hz, then there have crest values on it.
When the vibration is not on the center of the string, then the power spectrum is:

It vibrated only when the wave passed by.We can see that if the frequency of power spectrum is integer times of 150Hz, then there have crest values on it.
If we consider realistic excitation profile for a guitar string that is plucked, its vibration with time is shown as following figure, which has a strange shape.


When the angulus parietalis are not on the center of the strings:

the power spectrum is

CODE

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conclutions

If the string is excited at $\frac{l}{n}$(it must be an irreducible fraction) of it, there won't exist "peak" at $knf_0$ in its power spectra, where $k$ is an arbitrary positive integer, $f_0=\frac{c}{2L}$ is fundamental frequency of this string.
If the vibration of a point is observed at $\frac{l}{m}$(it must be an irreducible fraction) of the string, there won't exist "peak" at $kmf_0$ in its power spectra, where $k$ is an arbitrary positive integer, $f_0=\frac{c}{2L}$is fundamental frequency of this string.

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acknowledgement

Mr.Cai Hao's PPT
Computational physics by Nicholas J.Giordano,Hisao Nakanishi.
Guo Xiao's excercise for reference.