Final

homework

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Abstract

Plucked string instruments are a subcategory of string instruments that are played by plucking the strings. Plucking is a way of pulling and releasing the string in such a way as to give it an impulse that causes the string to vibrate. Plucking can be done with either a finger or a plectrum.
This article will simulating a guitar, including three parts:
(1) the motion of strings
(2) the vibrations of a two dimensional surface
(3) the production of sound waves in a room

Wave equation

As we have discussed, the wave equation for such a string is

$$\frac{\partial^{2}y}{\partial t^{2}} = c^2 \frac{\partial^{2}y}{\partial x^{2}}$$
To treat the wave rquation numerically we proceed as we did in Chapter-6, and disvretize time and space, with a time step $\Delta t$ and a spatial step $\Delta x$. We then write the string displacement as a function of these discrete variables with
$$y(x,t) \to y(i\Delta x,n\Delta t)\to y(i,n)$$
Since we have discussed in the previous chapters how to deal with this problem,
and the final results are as follows
$$y(i,n+1) = 2[1-r^2]y(i,n)-y(i,n-1)+r^2[y(i+1,n)+y(i-1,n)]$$
Where we define $$r\equiv c\Delta t/\Delta x$$
For all of the calculations we will assume that the ends of the string are rigidly held,so that $$y(i=0,n)=y(i=M,n)=0$$ where the ends of the string are at $i=0$ and $i=M$
The following picture shows how a kink splits into two separate kinks are reflected at the ends


Acknowledgement for Junyi Shangguan,because I have no idea about how to create a gif by python.(but when i convert markdown to pdf the gif do not move again...)
Here is the code

Power spectrum of waves

First we introduce a definition about $F_{bridge}$. It is the force at the end of the string at $x=i=0$

$$F_{bridge}=T\frac{\partial y}{\partial x}$$
In terms of our diesrete approximation this is
$$F_{bridge}=T\frac{\partial y}{\partial x} \vert _{x=0} =T\frac{[y(1,n)-y(0,n)]}{\Delta x}$$
Then we can plot the guitar spectrum just like text book P361 figure11.3


Here is the code2
We can calculate thie spectrum by performing a Fast Fourier Transformation analysis of the vridge force signal. but today I just directly give the consequence:
There are just the components of the sound at the fundamental frequency $f_1$, and its harmonics $nf_1$ and for our calculation with $\beta =1/5$ , the harmonics at n=5,10,etc … ,all vanish

Relationship with music theory

The vibration of a string is the superposition of a series of standing waves with the same eigenfrequency.The standing wave of n = 1 is called the fundamental wave, and the rest of the standing waves are called n-th harmonic wave respectively.The fundamental corresponds to the lowest frequency, called the tonic; the corresponding harmonics frequency $n\omega$ is called overtones, and overtones determine the timbre (Timbre distinguishes different types of sound production).
Since the amplitude is inversely proportional to $n^2$ , so in general, the amplitude of each vibration wave decreases. The sound of a stringed instrument depends on the spectral structure of the music being pronounced, ie the vibrations at various frequencies and their proportions, which are related to several factors.This is why we can distinguish between different instruments even when playing the same music.

Acknowledgement and reference

Acknowledgement for Junyi Shangguan
[1]张茹,张莉,吴高建.驻波和弦乐器[J].时代教育(教育教学),2012(07):177-178.
[2]Nicholas J.Giordano, Hisao Nakanishi. 2007. Computational Physics[M]. Beijing: Tsinghua University Press