computationalphysics

• 列表项

exercise_13:waves

abstract

• key words: wave motion ,spectrum, python , fft , matplotlib.animation

background

• the center equation of wave motion for ideal cases: $$\frac{\partial ^2y}{\partial t^2}=c^2\frac{\partial^2}{\partial x^2}$$
• Gaussian pluck: $$y_0(x)=exp[-k(x-x_0)^2]$$
• algorithm for ideal wave motion equation ： $$y(i,n+1)=2[1-r^2]y(i,n)-y(i,n-1)+r^2[y(i+1,n)+y(i-1,n)]$$

main body

点我查看代码
对于代码的几点说明：

• 固定边界处理办法： $$y[0]=y[-1]=0$$
• 自由边界处理办法: $$y[0]=y[1]-(y[2]-y[1])=2*(y[1]-y[2])\\y[-1]=y[-2]-(y[-3]-y[-2])=2*(y[-2]-y[-3])$$
• 每一次update都可以看作是前2个时刻的状态的函数，然而初始条件只包含一个时刻的状态，所以第一次update的计算方法与后续有所不同，这里我采用的方法是： $$y_1[i+1]=2*(1-r**2)y_0[i+1]-y_0[i+1]+r**2*(y_0[i+2]+y_0[i]\\(i=0,1,2,...,n-2)$$
• 有一个最小时间间隔，为一次"update"，实际情况下可以将其设为某一值。$r$恒为1，通过改变$\Delta x$来改变传播速度。
• 做fft时，输入周期越多，频谱越清晰,这一点从下面三幅频谱图中可以看出。（以下是update-times分别为500，5000，50000 时的频谱图）
  
  
  

1.wave propagating on a fixed ends

• a $sin$ initial pluck
  
  this is a stationary wave.
• a gaussian initial pluck
  

2.wave propagating on a free ends

• a $sin$ initial pluck
  
• a gaussian initial pluck
  
  this looks alike to that on a fixed ends,but if you pay attention to the ends,you'll find the difference.

3.spectrums

• spectrum of a sine vibration
  
• spectrum of a signal with a initial gaussian pluck

first lets research how string signal varies with time for different initial pluck point（时域）

#the next plots show how the spectrum varies with different pluck point.（频域）

conclusion

• the ends condition have a significant affect on the string motion.
• It is depict in the plots above that different pluck points results in differnt string signal and power spectrum.and it explains why it sounds different when we pluck at differnt point of the string from a stringed instrument.

acknowledgement

• 用python作科学计算
• 脚本之家——Python使用matplotlib绘制动画的方法
• Nicholas J.Giodano's computational physics.