Waves on a string

未分类
Author:GUO Xiao
2013301020099

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Abstract

This article will research waves on a string, includes its motion and power spectra.

Wave Equation

$$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x ^2}$$
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 $r=c\Delta t/\Delta x$, and $r=1$ is the best choice.

Waves on a string

Consider a string whose length is $L=1m$, which is divided into 100 points.The propagation speed of wave is 300m/s.
If the initial displacement of string has Gaussian profile, its evolution with time is shown as the following figure:

Curves at different heights represent the wave shapes at different time.The initial state is at the bottom of the figure.
From this figure, we can observe half-wave loss phenomenon as well.

While realistic excitation profile for a guitar string that is plucked isn't Gaussian profile, its shape evolution is shown as following figure:

There is a serratted wave on the string, not smooth waveform!

Now we focus on a point on this string, and investigate its vibration.

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:

It vibrated only when the wave passed by.
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.

To analyse its vibration, use power spectrum which can be obtained by the FFT of signal.

Power spectrum and symmetry

As we know, the fundamental frequency of this string is $c/(2L)=150Hz$. All spectra will only have components whose frequency is integer multiple of fundamental frequency.
If the string was excited at center, due to symmetry, its power spectrum doesn't has components whose frequency is even multiple of fundamental frequency.

The lowest frequency peak in the spectrum occur at 150Hz. There are substantial peaks at 450Hz, 750Hz, 1050Hz and so on. While "peak" at 300Hz, 600Hz, 900Hz, etc. are missing or extremely small.
However, if we excite the string not at center, for instance, excite it 5% from its center. Losing symmetry, even multiples of fundamental frequency components will also appear in its power spectrum as following figure:

Program code

The program to solve former problem is
wave1.py

problem 6.13

1

If the string is excited at 50% of string (Gaussian profile), the power spectra at different nodes are

They are power spectra at 5%, 10%, 40%, 50% of string, respectively.
The power spectrum at 5% of string doesn't have 3000Hz component and the power spectrum at 10% of string doesn't have 1500Hz, 3000Hz components.(Though they are obvious)
While there don't exist "peak" at 750Hz, 1500Hz, 2250Hz,3000Hz etc. in the power spectrum at 40% ($\frac{2}{5}$) of string.
However, the power spectrum at center of string has almost odd multiples of fundamental frequency components.

2

Similarily, when the string is excited at 55%, even multiple of fundamental frequency components will also appear in its power spectrum:

For different nodes, we have similar conclusions.

3

If the string is excited at 25%, there won't exist "peak" at 600Hz, 1200Hz, 1800Hz, 2400Hz, 3000Hz, etc. in its spectra.

The program to solve problem 6.13 is
problem6.13.py

Conclusions

• 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.